THE MASTER HANDBOOK
OF ACOUSTICS
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THE MASTER HANDBOOK
OF ACOUSTICS
F. Alton Everest
FOURTH EDITION
McGraw-Hill
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DOI: 10.1036/0071399747
To Bonnie Gail, whose love of art, nature, and the author
now embraces acoustics.
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Epigraph xxi
Introduction xxiii
Chapter 1 Fundamentals of Sound 1
The Simple Sinusoid 2
Sine-Wave Language 3
Propagation of sound 5
The dance of the particles 5
How a sound wave is propagated 7
Sound in free space 9
Wavelength and Frequency 10
Complex Waves 12
Harmonics 12
Phase 12
Partials 15
Octaves 15
The concept of spectrum 17
Electrical, Mechanical, and Acoustical Analogs 20
Chapter 2 Sound Levels and the Decibel 23
Ratios vs. Differences 23
Handling numbers 25
Logarithms 26
Decibels 26
Reference Levels 28
Logarithmic and Exponential Forms Compared 30
Acoustic Power 31
C O N T E N T S
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v i i i CONTENTS
Using Decibels 33
Example: Sound-pressure level 34
Example: Loudspeaker SPL 34
Example: Microphone specifications 35
Example: Line amplifier 35
Example: General-purpose amplifier 35
Example: Concert hall 35
Example: Combining decibels 36
Ratios and Octaves 37
Measuring Sound-Pressure Level 39
Chapter 3 The Ear and the Perception of Sound 41
Sensitivity of the Ear 41
A Primer of Ear Anatomy 42
The pinna: Directional encoder of sound 43
Directional cues: An experiment 44
The ear canal 44
The middle ear 45
The inner ear 48
Stereocilia 49
Loudness vs. Frequency 50
Loudness Control 51
Area of Audibility 53
Loudness vs. Sound-Pressure Level 54
Loudness and Bandwidth 56
Loudness of Impulses 59
Audibility of Loudness Changes 61
Pitch vs. Frequency 61
An experiment 63
Timbre vs. Spectrum 63
Localization of Sound Sources 64
Binaural Localization 67
Aural harmonics: Experiment #1 68
Aural harmonics: Experiment #2 69
The missing fundamental 69
The Ear as an Analyzer 70
The Ear as a Measuring Instrument 70
An auditory analyzer: An experiment 71
CONTENTS ix
Meters vs. the Ear 72
The Precedence Effect 73
Perception of Reflected Sound 75
Occupational and Recreational Deafness 76
Summary 79
Chapter 4 Sound Waves in the Free Field 83
Free Sound Field: Definition 83
Sound Divergence 84
Examples: Free-field sound divergence 84
Inverse square in enclosed spaces 87
Hemispherical propagation 88
Chapter 5 Speech, Music, and Noise 89
The Voice System 89
Artificial larynx 89
Sound spectrograph 90
Sound sources for speech 92
Vocal tract molding of speech 92
Formation of voiced sounds 94
Formation of unvoiced sounds 95
Putting it all together 95
Synthesized speech 96
Digital speech synthesis 97
Directionality of speech 98
Music 99
Wind instruments 101
Nonharmonic overtones 101
Dynamic range of speech and music 101
Power in Speech and Music 103
Frequency Range of Speech and Music 104
Future Dynamic-Range Requirements 104
Auditory Area 104
Noise 107
Noise—The good kind 108
Random noise 109
White and pink noise 111
Signal Distortion 112
Harmonic Distortion 114
x CONTENTS
Chapter 6 Analog and Digital Signal Processing 119
Resonance 120
Filters 122
Active filters 123
Analog vs. digital filters 124
Digitization 125
Quantization 126
Digital filters 126
Application of Digital Signal Processing (DSP) 105
Application of DSP to Room Equalization 106
Chapter 7 Reverberation 129
Reverberation and Normal Modes 130
Growth of Sound in a Room 132
Decay of Sound in a Room 134
Idealized Growth and Decay of Sound 134
Reverberation Time 135
Measuring Reverberation Time 137
Impulse Sound Sources 137
Steady-State Sources 138
Equipment 138
Measurement Procedure 140
Analysis of decay traces 140
Mode Decay Variations 142
Writing speed 143
Frequency effect 144
Reverberation Time Variation with Position 145
Acoustically Coupled Spaces 146
Electroacoustically Coupled Spaces 146
Decay rate 147
Eliminating decay fluctuations 147
Influence of Reverberation on Speech 148
Influence of Reverberation on Music 149
Optimum Reverberation Time 150
Bass rise of reverberation time 152
Living room reverberation time 154
CONTENTS xi
Artificial Reverberation: The Past 155
Artificial Reverberation: The Future 156
Arrival Time Gap 157
The Sabine Equation 159
Reverberation calculation: Example 1 160
Reverberation calculation: Example 2 162
Reverberant Field 162
Chapter 8 Control of Interfering Noise 165
Noise Sources and Some Solutions 166
Airborne noise 167
Noise carried by structure 167
Noise transmitted by diaphragm action 168
Sound-insulating walls 168
Porous materials 169
Sound Transmission Classification (STC) 170
Comparison of Wall Structures 171
Double Windows 173
Sound-Insulating Doors 175
Noise and room resonances 176
Active noise control 177
Chapter 9 Absorption of Sound 179
Dissipation of Sound Energy 179
Evaluation of Sound Absorption 181
Reverberation Chamber Method 182
Impedance Tube Method 182
Tone-Burst Method 185
Mounting of Absorbents 186
Mid/High Frequency Absorption by Porosity 187
Glass fiber: Building insulation 189
Glass fiber: Boards 190
Acoustical tile 190
Effect of Thickness of Absorbent 190
Effect of Airspace behind Absorbent 191
Effect of Density of Absorbent 192
x i i CONTENTS
Open-Cell Foams 192
Drapes as Sound Absorbers 193
Carpet as Sound Absorber 196
Effect of carpet type on absorbance 199
Effect of carpet underlay on absorbance 200
Carpet absorption coefficients 200
Sound Absorption by People 200
Absorption of Sound in Air 203
Low-Frequency Absorption by Resonance 203
Diaphragmatic Absorbers 205
Polycylindrical Absorbers 209
Poly Construction 212
Membrane Absorbers 213
Helmholtz Resonators 215
Perforated Panel Absorbers 218
Slat Absorbers 224
Placement of Materials 225
Reverberation Time of Helmholtz Resonators 225
Taming room modes 226
Increasing Reverberation Time 229
Modules 229
Chapter 10 Reflection of Sound 235
Reflections from Flat Surfaces 235
Doubling of Pressure at Reflection 237
Reflections from Convex Surfaces 237
Reflections from Concave Surfaces 237
Reflections from Parabolic Surfaces 238
Reflections inside a Cylinder 240
Standing Waves 240
Reflection of Sound from Impedance
Irregularities 240
The Corner Reflector 243
Echo-Sounding 243
Perceptive Effects of Reflections 244
CONTENTS x i i i
Chapter 11 Diffraction of Sound 245
Rectilinear Propagation 245
Diffraction and Wavelength 246
Diffraction of Sound by Large and Small Apertures 247
Diffraction of Sound by Obstacles 248
Diffraction of Sound by a Slit 249
Diffraction by the Zone Plate 250
Diffraction around the Human Head 251
Diffraction by Loudspeaker Cabinet Edges 253
Diffraction by Various Objects 254
Chapter 12 Refraction of Sound 257
Refraction of Sound 258
Refraction of sound in solids 258
Refraction of sound in the atmosphere 260
Refraction of sound in the ocean 263
Refraction of sound in enclosed spaces 265
Chapter 13 Diffusion of Sound 267
The Perfectly Diffuse Sound Field 267
Evaluating Diffusion in a Room 268
Steady-state measurements 268
Decay Beats 269
Exponential Decay 270
Spatial Uniformity of Reverberation Time 271
Decay Shapes 275
Microphone Directivity 275
Room Shape 275
Splaying Room Surfaces 281
Nonrectangular rooms 281
Geometrical Irregularities 282
Absorbent in Patches 282
Concave Surfaces 286
Convex Surfaces: The Poly 286
Plane Surfaces 287
xiv CONTENTS
Chapter 14 The Schroeder Diffusor 289
Schroeder’s First Acoustic Diffusor 290
Maximum-Length Sequences 292
Reflection Phase-Grating Diffusors 292
Quadratic-Residue Diffusors 293
Primitive-Root Diffusors 296
Quadratic-Residue Applications 298
Performance of Diffraction-Grating Diffusors 298
Expansion of the QRD line 304
Solving flutter problems 304
Application of fractals 306
Diffusion in three dimensions 308
Acoustic concrete blocks 309
Measuring diffusion efficiency 311
Comparison of Gratings with Conventional
Approaches 312
Chapter 15 Modal Resonances in Enclosed Spaces 317
Resonance in a Pipe 318
Bathroom Acoustics 319
Reflections Indoors 320
Two-Wall Resonance 322
Waves vs. Rays 322
Frequency Regions 323
Dividing the Audio Spectrum 325
Wave Acoustics 326
Mode calculations—An example 328
Experimental Verification 331
Mode Identification 331
Mode Decay 333
Mode Bandwidth 334
Mode Pressure Plots 339
Modal Density 341
Mode Spacing and Coloration 342
Experiments with Colorations 344
Simplified Axial Mode Analysis 346
CONTENTS xv
The Bonello Criterion 348
Controlling Problem Modes 348
Mode Summary 350
Chapter 16 Reflections in Enclosed Spaces 353
Law of the First Wavefront 353
Mean Free Path 354
The effect of single reflections 355
Perception of sound reflections 355
Perception of spaciousness 357
Image changes 357
Discrete echoes 357
Effect of angle of incidence on audibility
of reflection 357
Effect of signal type of audibility of reflection 358
Effect of spectrum on audibility of reflection 358
Using reflection data 359
Large Spaces 359
Echoes 359
Spaciousness 360
Chapter 17 Comb-Filter Effects 363
What Is a Comb Filter? 363
Superposition of Sound 364
Tonal Signals and Comb Filters 365
Combing of music and speech signals 367
Combing of direct and reflected sound 368
Comb Filters and Critical Bands 371
Comb Filters in Stereo Listening 374
Coloration and Spaciousness 374
Combing in Stereo Microphone Pickups 375
Audibility of Comb-Filter Effects 375
Comb filters in practice 376
Estimating comb-filter response 380
Chapter 18 Quiet Air for the Studio 385
Selection of Noise Criterion 386
Fan Noise 388
xvi CONTENTS
ASHRAE 389
Machinery Noise 390
Air Velocity 390
Effect of Terminal Fittings 391
“Natural” Attenuation 391
Duct Lining 392
Plenum Silencers 393
Packaged Attenuators 394
Reactive Silencers 394
Resonator Silencer 395
Duct Location 395
Some Practical Suggestions 395
Chapter 19 Acoustics of the Listening Room 399
The Acoustical Link 399
Peculiarities of Small-Room Acoustics 400
Room size 401
Room proportions 401
Reverberation time 403
The Listening Room: Low Frequencies 403
Control of modal resonances 406
Bass traps for the listening room 406
Modal colorations 408
The Listening Room: The Mid-High Frequencies 409
Identification and treatment of
reflection points 411
Lateral reflections: Control of spaciousness 413
Chapter 20 Acoustics of the Small Recording Studio 415
Acoustical Characteristics of a Studio 416
Reverberation 418
Studio Design 419
Studio Volume 419
Room Proportions 421
Reverberation Time 422
CONTENTS xvii
Diffusion 423
Noise 424
Studio Design Procedure 424
Some Studio Features 424
Elements Common to all Studios 427
Chapter 21 Acoustics of the Control Room 429
The Initial Time-Delay Gap 429
The Live End 431
Specular Reflections vs. Diffusion 432
Low-Frequency Resonances in the Control Room 434
Initial Time-Delay Gaps in Practice 436
Managing Reflections 438
The Reflection-Free-Zone Control Room 439
Control-Room Frequency Range 441
Outer Shell of the Control Room 442
Inner Shell of the Control Room 442
Representative Control Rooms 442
Some European Designs 444
Consultants 450
Chapter 22 Acoustics for Multitrack Recording 453
Flexibility 545
Advantages of Multitrack 455
Disadvantages of Multitrack 456
Achieving Track Separation 457
Studio Acoustics 458
Distance between artists 458
Microphone management 458
Barriers for separation 459
Electronic separation 459
Electronic instruments and separation 459
The Future of Multichannel 460
Automation 460
xviii CONTENTS
Chapter 23 Audio/Video Tech Room and Voice-Over
Recording 461
Selection of Space: External Factors 462
Selection of Space: Internal Factors 462
Work Space Treatment 462
Audio/Video Work Place Example 463
Appraisal of Room Resonances 463
Control of room resonances 464
Treatment of work place 465
Calculations 465
The Voice-Over Booth 468
Dead-End Live-End Voice Studio 468
Voice-Over Booths 468
The Quick Sound Field™ 469
Chapter 24 Adjustable Acoustics 473
Draperies 473
Adjustable Panels: Absorption 474
Adjustable Panels: The Abffusor™ 476
Hinged Panels 478
Louvered Panels 479
Variable Resonant Devices 480
Rotating Elements 483
Portable Units: The Tube Trap™ 484
Portable Units: The Korner Killer™ 485
Chapter 25 Acoustical Distortion 489
Acoustic Distortion and the Perception of Sound 489
Sources of Acoustic Distortion 490
Coupling of room modes 490
Speaker-boundary interference response 491
Comb filtering 493
Poor diffusion 498
Conclusion 500
Chapter 26 Room Acoustics Measurement Software 501
The Evolution of Measurement Technologies 502
CONTENTS xix
Building a Better Analyzer 504
Time-delay spectrometry (TDS) measurement
techniques 504
Maximum-length sequence (MLS) techniques 508
AcoustiSoft’s ETF Program 509
Frequency-response measurements 513
Resonance measurements 517
Fractional-octave measurements 520
Energy-time curve measurements 521
Reverberation time 524
Conclusion 526
Chapter 27 Room Optimizer 529
Introduction 529
Modal Response 530
Speaker-Boundary Interference Response 531
Optimization 533
Theory 536
Prediction of room response 536
Optimizing procedure 541
Cost parameter 543
Optimization Procedure 545
Results 549
Stereo pair 549
Stereo pair with two woofers per loudspeaker 550
THX home theater 551
Multichannel music 554
Subwoofer 556
Conclusion 558
Chapter 28 Desktop Auralization 565
Introduction 565
The Auralization Process 569
Summary 581
Appendix 585
Glossary 589
Index 599
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Directly or indirectly, all questions connected with this subject must
come for decision to the ear, as the organ of hearing; and from it there
can be no appeal. But we are not therefore to infer that all acoustical
investigations are conducted with the unassisted ear. When once we
have discovered the physical phenomena which constitute the foundation
of sound, our explorations are in great measure transferred to
another field lying within the dominion of the principles of Mechanics.
Important laws are in this way arrived at, to which the sensations of
the ear cannot but conform.
Lord Raleigh in The Theory of Sound,
First Edition 1877.
(Also in first American edition, 1945,
courtesy of Dover Publications Inc.)
E P I G R A P H
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Excerpts from the introduction to the third edition.
In 1981, the copyright year of the first edition of this book, Manfred
Schroeder was publishing his early ideas on applying number theory
to the diffusion of sound. In the third edition a new chapter has been
added to cover numerous applications of diffraction-grating diffusors
to auditoriums, control rooms, studios and home listening rooms.
Introduction to the fourth edition.
The science of acoustics made great strides in the 20th century, during
which the first three editions of this book appeared. This fourth edition,
however, points the reader to new horizons of the 21st century. A
newly appreciated concept of distortion of sound in the medium itself
(Chap. 25), a program for acoustic measurements (Chap. 26), and the
optimization of placement of loudspeakers and listener (Chap. 27), all
based on the home computer, point forward to amazing developments
in acoustics yet to come.
As in the previous three editions, this fourth edition balances treatment
of the fundamentals of acoustics with the general application of
fundamentals to practical problems.
F. Alton Everest
Santa Barbara
I N T R O D U C T I O N
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Sound can be defined as a wave motion in air or other elastic
media (stimulus) or as that excitation of the hearing mechanism
that results in the perception of sound (sensation). Which definition
applies depends on whether the approach is physical or psychophysical.
The type of problem dictates the approach to sound. If
the interest is in the disturbance in air created by a loudspeaker, it is
a problem in physics. If the interest is how it sounds to a person near
the loudspeaker, psychophysical methods must be used. Because
this book addresses acoustics in relation to people, both aspects of
sound will be treated.
These two views of sound are presented in terms familiar to those
interested in audio and music. Frequency is a characteristic of periodic
waves measured in hertz (cycles per second), readily observable
on a cathode-ray oscilloscope or countable by a frequency counter.
The ear perceives a different pitch for a soft 100 Hz tone than a loud
one. The pitch of a low-frequency tone goes down, while the pitch of
a high-frequency tone goes up as intensity increases. A famous
acoustician, Harvey Fletcher, found that playing pure tones of 168 and
318 Hz at a modest level produces a very discordant sound. At a high
intensity, however, the ear hears the pure tones in the 150-300 Hz
octave relationship as a pleasant sound. We cannot equate frequency
and pitch, but they are analogous.
Fundamentals of Sound
1
1
C H A P T E R
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2 CHAPTER ONE
The same situation exists between intensity and loudness. The relationship
between the two is not linear. This is considered later in more
detail because it is of great importance in high fidelity work.
Similarly, the relationship between waveform (or spectrum) and
perceived quality (or timbre) is complicated by the functioning of the
hearing mechanism. As a complex waveform can be described in
terms of a fundamental and a train of harmonics (or partials) of various
amplitudes and phases (more on this later), the frequency-pitch interaction
is involved as well as other factors.
The Simple Sinusoid
The sine wave is a basic waveform closely
related to simple harmonic motion. The
weight (mass) on the spring shown in Fig.
1-1 is a vibrating system. If the weight is
pulled down to the 5 mark and released,
the spring pulls the weight back toward 0.
The weight will not, however, stop at zero; its
inertia will carry it beyond 0 almost to 5.
The weight will continue to vibrate, or oscillate,
at an amplitude that will slowly
decrease due to frictional losses in the spring,
the air, etc.
The weight in Fig. 1-1 moves in what is
called simple harmonic motion. The piston
in an automobile engine is connected to the crankshaft by a connecting
rod. The rotation of the crankshaft and the up-and-down
motion of the pistons beautifully illustrate the relationship between
rotary motion and linear simple harmonic motion. The piston position
plotted against time produces a sine wave. It is a very basic type of
mechanical motion, and it yields an equally basic waveshape in sound
and electronics.
If a ballpoint pen is fastened to the pointer of Fig. 1-2, and a strip of
paper is moved past it at a uniform speed, the resulting trace is a sine
wave.
In the arrangement of Fig. 1-1, vibration or oscillation is possible
because of the elasticity of the spring and the inertia of the weight.
A weight on a spring vibrates at its natural frequency
because of the elasticity of the spring and the inertia
of the weight.
W
5
5
0
F I G U R E 1 - 1
FUNDAMENTALS OF SOUND 3
Elasticity and inertia are two things all media must possess to be capable
of conducting sound.
Sine-Wave Language
The sine wave is a specific kind of alternating signal and is described
by its own set of specific terms. Viewed on an oscilloscope, the easiest
value to read is the peak-to-peak value (of voltage, current, sound
pressure, or whatever the sine wave represents), the meaning of which
is obvious in Fig. 1-3. If the wave is symmetrical, the peak-to-peak
value is twice the peak value.
The common ac voltmeter is, in reality, a dc instrument fitted with
a rectifier that changes the alternating sine current to pulsating unidirectional
current. The dc meter then responds to the average value as
indicated in Fig. 1-3. Such meters are, however, almost universely calibrated
in terms of rms (described in the next paragraph). For pure sine
waves, this is quite an acceptable fiction, but for nonsinusoidal waveshapes
the reading will be in error.
An alternating current of one ampere rms (or effective) is exactly
equivalent in heating power to 1 ampere of direct current as it flows
through a resistance of known value. After all, alternating current can
heat up a resistor or do work no matter which direction it flows, it is
just a matter of evaluating it. In the right-hand positive loop of Fig. 1-3
the ordinates (height of lines to the curve) are read off for each marked
W
Paper motion
Time
F I G U R E 1 - 2
A ballpoint pen fastened to the vibrating weight traces a sine wave on a paper strip
moving at uniform speed. This shows the basic relationship between simple harmonic
motion and the sine wave.
4 CHAPTER ONE
increment of time. Then (a) each of these ordinate values is squared, (b)
the squared values are added together, (c) the average is found, and (d)
the square root is taken of the average (or mean). Taking the square root
of this average gives the root-mean-square or rms value of the positive
loop of Fig. 1-3. The same can be done for the negative loop (squaring a
negative ordinate gives a positive value), but simply doubling the positive
loop of a symmetrical wave is easier. In this way the root-meansquare
or “heating power” value of any alternating or periodic waves
can be determined whether the wave is for voltage, current, or sound
pressure. Such computations will help you understand the meaning of
rms, but fortunately reading meters is far easier. Figure 1-3 is a useful
summary of relationships pertaining only to the sine wave.
Amplitude

 Time
RMS Average
Amplitude relationships
for
sinusoids
Peak
RMS 
Peak
√2  (0.707) (Peak)

 (Average)
2√2  (1.1) (Average)
Peak  (√2 ) (RMS)  ( 1.414) (RMS)
 (  ) (Average)  (1.57) (Average) 2
Peak
to
peak
F I G U R E 1 - 3
Amplitude relationships for sinusoids, which apply to sine waves of electrical voltage or
current, as well as to acoustical parameters such as sound pressure. Another term which
is widely used in the audio field is crest factor, or peak divided by rms.
FUNDAMENTALS OF SOUND 5
Propagation of Sound
If an air particle is displaced from its original position, elastic forces of
the air tend to restore it to its original position. Because of the inertia
of the particle, it overshoots the resting position, bringing into play
elastic forces in the opposite direction, and so on.
Sound is readily conducted in gases, liquids, and solids such as air,
water, steel, concrete, etc., which are all elastic media. As a child, perhaps
you heard two sounds of a rock striking a railroad rail in the distance,
one sound coming through the air and one through the rail. The
sound through the rail arrives first because the speed of sound in the
dense steel is greater than that of tenuous air. Sound has been detected
after it has traveled thousands of miles through the ocean.
Without a medium, sound cannot be propagated. In the laboratory,
an electric buzzer is suspended in a heavy glass bell jar. As the button is
pushed, the sound of the buzzer is readily heard through the glass. As
the air is pumped out of the bell jar, the sound becomes fainter and
fainter until it is no longer audible. The sound-conducting medium, air,
has been removed between the source and the ear. Because air is such a
common agent for the conduction of sound, it is easy to forget that other
gases as well as solids and liquids are also conductors of sound. Outer
space is an almost perfect vacuum; no sound can be conducted except
in the tiny island of air (oxygen) within a spaceship or a spacesuit.
The Dance of the Particles
Waves created by the wind travel across a field of grain, yet the individual
stalks remain firmly rooted as the wave travels on. In a similar
manner, particles of air propagating a sound wave do not move far from
their undisplaced positions as shown in Fig. 1-4. The disturbance travels
on, but the propagating particles do their little dance close to home.
There are three distinct forms of particle motion. If a stone is
dropped on a calm water surface, concentric waves travel out from the
point of impact, and the water particles trace circular orbits (for deep
water, at least) as in Fig. 1-5(A). Another type of wave motion is illustrated
by a violin string, Fig. 1-5(B). The tiny elements of the string
move transversely, or at right angles to the direction of travel of the
waves along the string. For sound traveling in a gaseous medium such
as air, the particles move in the direction the sound is traveling. These
are called longitudinal waves, Fig. 1-5C.
6 CHAPTER ONE
Vibration of air particle
Equilibrium
position
Maximum
displacement
Maximum
displacement
Displacement
Time
F I G U R E 1 - 4
An air particle is made to vibrate about its equilibrium position by the energy of a passing
sound wave because of the interaction of the elastic forces of the air and the inertia
of the air particle.
A
Particle
motion
Direction of
wave travel
Water
surface
B Violin
string
C Sound
in air
F I G U R E 1 - 5
Particles involved in the propagation of sound waves can dance in circular, transverse,
or longitudinal motions.
FUNDAMENTALS OF SOUND 7
How a Sound Wave Is Propagated
How are air particles jiggling back and forth able to carry beautiful
music from the loudspeaker to our ears at the speed of a rifle bullet?
The little dots of Fig. 1-6 represent air molecules. There are more than
a million molecules in a cubic inch of air; hence this sketch is greatly
exaggerated. The molecules crowded together represent areas of compression
in which the air pressure is slightly greater than the prevailing
atmospheric pressure. The sparse areas represent rarefactions in
which the pressure is slightly less than atmospheric. The small arrows
indicate that, on the average, the molecules are moving to the right of
the compression crests and to the left in the rarefaction troughs
between the crests. Any given molecule will move a certain distance to
the right and then the same distance to the left of its undisplaced position
as the sound wave progresses uniformly to the right.
C
A
R C R C R C
B
C  Compression (region of high pressure)
R  Rarefaction (region of low pressure)
Direction of sound wave
F I G U R E 1 - 6
In (A) the wave causes the air particles to be pressed together in some regions and
spread out in others. An instant later (B) the wave has moved slightly to the right.
8 CHAPTER ONE
Why does the sound wave move to the right? The answer is
revealed by a closer look at the arrows of Fig. 1-6. The molecules
tend to bunch up where two arrows are pointing toward each other,
and this occurs a bit to the right of each compression. When the
arrows point away from each other the density of molecules will
decrease. Thus, the movement of the higher pressure crest and the
lower pressure trough accounts for the small progression of the
wave to the right.
As mentioned previously, the pressure at the crests is higher than
the prevailing atmospheric barometric pressure and the troughs
lower than the atmospheric pressure, as shown in the sine wave of
Fig. 1-7. These fluctuations of pressure are very small indeed. The
faintest sound the ear can hear (20 Pascal) is some 5,000 million
times smaller than atmospheric pressure. Normal speech and music
signals are represented by correspondingly small ripples superimposed
on the atmospheric pressure.
A
Compression
B
Rarefaction Atmospheric
pressure
Time
Pressure
F I G U R E 1 - 7
(A) An instantaneous view of the compressed and rarefied regions of a sound wave in
air. (B) The compressed regions are very slightly above and the rarefied regions very
slightly below atmospheric pressure. Pressure variations of sound waves are thus superimposed
on prevailing barometric pressure.
FUNDAMENTALS OF SOUND 9
Sound in Free Space
The intensity of sound decreases as the distance to the source is
increased. In free space, far from the influence of surrounding objects,
sound from a point source is propagated uniformly in all directions.
The intensity of sound decreases as shown in Fig. 1-8. The same sound
power flows out through A1, A2, A3, and A4, but the areas increase as
the square of the radius, r. This means that the sound power per unit
area (intensity) decreases as the square of the radius. Doubling the distance
reduces the intensity to one-fourth the initial value, tripling the
distance yields 19, and increasing the distance four times yields 116 of
A1
A2
A3
A4
r
2r
3r
4r
F I G U R E 1 - 8
In the solid angle shown, the same sound energy is distributed over spherical surfaces
of increasing area as r is increased. The intensity of the sound is inversely proportional
to the square of the distance from the point source.
10 CHAPTER ONE
the initial intensity. The inverse square law states that the intensity of
sound in a free field is inversely proportional to the square of the distance
from the source. This law provides the basis of estimating the
sound level in many practical circumstances and is discussed in a
later chapter.
Wavelength and Frequency
A simple sine wave is illustrated in Fig. 1-9. The wavelength is the distance
a wave travels in the time it takes to complete one cycle. A wavelength
can be measured between successive peaks or between any two
corresponding points on the cycle. This holds for periodic waves other
than the sine wave as well. The frequency is the number of cycles per
second (or hertz). Frequency and wavelength are related as follows:
Wavelength (ft)  (1-1)
which can be written as:
Frequency  (1-2)
The speed of sound in air is about 1,130 feet per second (770 miles per
hour) at normal temperature and pressure. For sound traveling in air,
Equation 1-1 becomes:
Speed of sound

Wavelength
Speed of sound (ft/sec)

Frequency (hertz)
Wavelength is the distance a wave travels in the time it takes to complete one cycle. It
can also be expressed as the distance from one point on a periodic wave to the corresponding
point on the next cycle of the wave.
Wavelength
Time
Peak
Amplitude
F I G U R E 1 - 9
FUNDAMENTALS OF SOUND 11
Wavelength  (1-3)
This relationship is used frequently in following sections. Figure
1-10 gives two graphical approaches for an easy solution to Equation 1-3.
1,130

Frequency
(A) Convenient scales for rough determination of wavelength of sound in air from
known frequency, or vice versa. (B) A chart for easy determination of the wavelength in
air of sound waves of different frequencies. (Both based on speed of sound of 1,139 ft
per second.)
50 40 20 10 5 2 1
12 8 6 4 3 2 1
20 50 100 200 500 1 k 2 kHz 5 kHz 10 kHz 20 kHz
Inches
Wavelength
Feet
A
Frequency - Hz
100
50
30
10
5
3
1
0.5
0.3
0.1
10 30 50 100 300 500 1 2 3 4 5 10
Wavelength, ft.
Hz kHz
B
F I G U R E 1 - 1 0
12 CHAPTER ONE
Complex Waves
Speech and music waveshapes depart radically from the simple sine
form. A very interesting fact, however, is that no matter how complex the
wave, as long as it is periodic, it can be reduced to sine components. The
obverse of this is that, theoretically, any complex periodic wave can be
synthesized from sine waves of different frequencies, different amplitudes,
and different time relationships (phase). A friend of Napoleon,
named Joseph Fourier, was the first to develop this surprising idea. This
idea can be viewed as either a simplification or complication of the situation.
Certainly it is a great simplification in regard to concept, but sometimes
complex in its application to specific speech or musical sounds.
As we are interested primarily in the basic concept, let us see how even
a very complex wave can be reduced to simple sinusoidal components.
Harmonics
A simple sine wave of a given amplitude and frequency, f1, is shown in
Fig. 1-11A. Figure 1-11B shows another sine wave half the amplitude and
twice the frequency (f2). Combining A and B at each point in time the
waveshape of Fig. 1-11C is obtained. In Fig. 1-11D, another sine wave half
the amplitude of A and three times its frequency (f3) is shown. Adding
this to the f1  f2 wave of C, Fig. 1-11E is obtained. The simple sine wave
of Fig. 1-11A has been progressively distorted as other sine waves have
been added to it. Whether these are acoustic waves or electronic signals,
the process can be reversed. The distorted wave of Fig. 1-11E can be disassembled,
as it were, to the simple f1, f2, and f3 sine components by either
acoustical or electronic filters. For example, passing the wave of Fig.
1-11E through a filter permitting only f1 and rejecting f2 and f3, the original
f1 sine wave of Fig. 1-11A emerges in pristine condition.
Applying names, the sine wave with the lowest frequency (f1) of
Fig. 1-11A is called the fundamental, the one with twice the frequency
(f2) of Fig. 1-11B is called the second harmonic, and the one three
times the frequency (f3) of Fig. 1-11D is the third harmonic. The fourth
harmonic, the fifth harmonic, etc., are four and five times the frequency
of the fundamental, and so on.
Phase
In Fig. 1-11, all three components, f1, f2, and f3, start from zero together.
This is called an in-phase condition. In some cases, the time relationFUNDAMENTALS
OF SOUND 13
A study in the combination of sine waves. (A) The fundamental of frequency f1. (B) A
second harmonic of frequency f2 = 2 f1 and half the amplitude of f1. (C) The sum of f1
and f2 obtained by adding ordinates point by point. (D) A third harmonic of frequency
f3 = 3 f1 and half the amplitude of f1. (E) The waveshape resulting from the addition
of f1, f2, and f3. All three components are “in phase,” that is, they all start from zero
at the same instant.
A


0
Amplitude
f1
Time


0
f2
B


0
f1  f2
C


0
f3
D


0
f1  f2  f3
E
F I G U R E 1 - 1 1
14 CHAPTER ONE
ships between harmonics or between harmonics and the fundamental
are quite different from this. Remember how one revolution of the
crankshaft of the automobile engine (360°) was equated with one cycle
of simple harmonic motion of the piston? The up-and-down travel of
the piston spread out in time traces a sine wave such as that in Fig. 1-12.
One complete sine-wave cycle represents 360° of rotation. If another
sine wave of identical frequency is delayed 90°, its time relationship to
the first one is a quarter wave late (time increasing to the right). A halfwave
delay would be 180°, etc. For the 360° delay, the wave at the bot-
90¡
180¡
270¡
360¡
Time
F I G U R E 1 - 1 2
Illustration of the phase relationships between waves with the same amplitude and frequency.
A rotation of 360 degrees is analogous to one complete sine cycle.
FUNDAMENTALS OF SOUND 15
tom of Fig. 1-12 falls in step with the top one, reaching positive peaks and
negative peaks simultaneously and producing the in-phase condition.
In Fig. 1-11, all three components of the complex wave of Fig. 1-11E
are in phase. That is, the f1 fundamental, the f2 second harmonic, and
the f3 third harmonic all start at zero at the same time. What happens if
the harmonics are out of phase with the fundamental? Figure 1-13 illustrates
this case. The second harmonic f2 is now advanced 90°, and the
third harmonic f3 is retarded 90°. By combining f1, f2, and f3 for each
instant of time, with due regard to positive and negative signs, the contorted
wave of Fig. 1-13E is obtained.
The only difference between Figs. 1-11E and 1-13E is that a phase
shift has been introduced between harmonics f2 and f3, and the fundamental
f1. That is all that is needed to produce drastic changes in the
resulting waveshape. Curiously, even though the shape of the wave is
dramatically changed by shifting the time relationships of the components,
the ear is relatively insensitive to such changes. In other words,
waves E of Figs. 1-11 and 1-13 would sound very much alike to us.
A common error is confusing polarity with phase. Phase is the time
relationship between two signals while polarity is the / or the /
relationship of a given pair of signal leads.
Partials
A musician is inclined to use the term partial instead of harmonic, but
it is best that a distinction be made between the two terms because the
partials of many musical instruments are not harmonically related to
the fundamental. That is, partials might not be exact multiples of the
fundamental frequency, yet richness of tone can still be imparted by
such deviations from the true harmonic relationship. For example, the
partials of bells, chimes, and piano tones are often in a nonharmonic
relationship to the fundamental.
Octaves
Audio and electronics engineers and acousticians frequently use the
integral multiple concept of harmonics, closely allied as it is to the
physical aspect of sound. The musician often refers to the octave, a
logarithmic concept that is firmly embedded in musical scales and terminology
because of its relationship to the ear’s characteristics. Audio
people are also involved with the human ear, hence their common use
of logarithmic scales for frequency, logarithmic measuring units, and
16 CHAPTER ONE
A study of the combination of sine waves that are not in phase. (A) The fundamental
of frequency f1. (B) The second harmonic f2 with twice the frequency and half the
amplitude of f1 advanced 90 degrees with respect to f1. (C) The combination of f1 and
f2 obtained by adding ordinates point by point. (D) The third harmonic f3 with phase
90 degrees behind f1, and with half the amplitude of f1. (E) The sum of f1, f2, and f3.
Compare this waveshape with that of Fig. 1-11(E). The difference in waveshapes
is due entirely to the shifting of the phase of the harmonics with respect to the
fundamental.
Time
f1
A


0
Amplitude
B
C
D
E
f2 

0


0


0


0
f1  f2
f3
f1  f2  f3
F I G U R E 1 - 1 3
FUNDAMENTALS OF SOUND 17
various devices based on octaves, which are more fully discussed
later. Harmonics and octaves are compared in Fig. 1-14.
The Concept of Spectrum
Chapter 3 relates the commonly accepted scope of the audible spectrum,
20 Hz to 20 kHz, to specific characteristics of the human ear.
Here, in the context of sine waves, harmonics, etc., we need to establish
the spectrum concept. The visible spectrum of light has its counterpart
in sound in the audible spectrum, the range of frequencies that
fall within the perceptual limits of the human ear. We cannot see farultraviolet
light because the frequency of its electromagnetic energy is
too high for the eye to perceive. Nor can we see the far-infrared light
because its frequency is too low. There are likewise sounds of too low
(infrasound) and too high frequency (ultrasound) for the ear to hear.
Figure 1-15 shows several waveforms that typify the infinite number
of different waveforms commonly encountered in audio. These
waveforms have been photographed directly from the screen of a cathode-
ray oscilloscope. To the right of each photograph is the spectrum
of that particular signal. The spectrum tells how the energy of the signal
is distributed in frequency. In all but the bottom signal of Fig. 1-15,
the audible range of the spectrum was searched with a wave analyzer
having a very sharp filter with a passband only 5 Hz wide. In this way,
Comparison of harmonics and octaves. Harmonics are linearly related; octaves are logarithmically
related.
Harmonics
Fundamental
100 Hz
2nd 3rd 4th 5th 6th 7th 8th
200 300 400 500 600 700 800. . .
Octaves
Fundamental
100 Hz
Octave
200
Octave
400
Octave
800
F I G U R E 1 - 1 4
18 CHAPTER ONE
6
4
2
0
10
f1 f2 f3 f4 f5 f6 f7
6
4
2
0
f1 f2 f3 f4 f5 f6 f7
10
6
4
2
0
10
f1 f2 f3 f4 f5 f6 f7
Frequency
Amplitude-arbitrary units
F I G U R E 1 - 1 5
The spectral energy of a pure sinusoid is contained entirely at a single frequency. The triangular and square
waves each have a prominent fundamental and numerous harmonics at integral multiples of the fundamental
frequency. Random noise (white noise) has energy distributed uniformly throughout the spectrum up to some
point at which energy begins to fall off due to generator limitations. Random noise may be considered a mixture
of sine waves with continuously shifting frequencies, amplitudes, and phases.
FUNDAMENTALS OF SOUND 19
concentrations of energy were located and measured with an electronic
voltmeter.
For an ideal sine wave, all the energy is concentrated at one frequency.
The sine wave produced by this particular signal generator is not
really a pure sine wave. No oscillator is perfect and all have some harmonic
content, but in scanning the spectrum of this sine wave, the harmonics
measured were too low to show on the graph scale of Fig. 1-15.
The triangular wave of this signal generator has a major fundamental
component of 10 units magnitude. The wave analyzer detected a significant
second harmonic component at f2, twice the frequency of the fundamental
with a magnitude of 0.21 units. The third harmonic showed
an amplitude of 1.13 units, the fourth of 0.13 units, etc. The seventh harmonic
still had an amplitude of 0.19 units and the fourteenth harmonic
(about 15 kHz in this case) an amplitude of only 0.03 units, but still easily
detectable. So we see that this triangular wave has both odd and even
components of modest amplitude down through the audible spectrum.
If you know the amplitude and phases of each of these, the original triangular
wave shape can be synthesized by combining them.
A comparable analysis reveals the spectrum of the square wave
shown in Fig. 1-15. It has harmonics of far greater amplitude than the
triangular wave with a distinct tendency toward more prominent odd
than even harmonics. The third harmonic shows an amplitude 34 percent
of the fundamental! The fifteenth harmonic of the square wave is
still 0.52 units! If the synthesis of a square wave stops with the fifteenth
harmonic, the wave of Fig. 1-16C results.
A glance at the spectra of sine, triangular, and square waves reveals
energy concentrated at harmonic frequencies, but nothing between.
These are all so-called periodic waves, which repeat themselves cycle
after cycle. The fourth example in Fig. 1-15 is a random noise. The
spectrum of this signal cannot be measured satisfactorily by a wave
analyzer with a 5-Hz passband because the fluctuations are so great
that it is impossible to get a decent reading on the electronic voltmeter.
Analyzed by a wider passband of fixed bandwidth and with the help
of various integrating devices to get a steady indication, the spectral
shape shown is obtained. This spectrum tells us that the energy of the
random-noise signal is equally distributed throughout the spectrum
until the drooping at high frequencies indicates that the upper frequency
limit of the random noise generator has been reached.
20 CHAPTER ONE
There is little visual similarity between
the sine and the random-noise signals as
revealed by the cathode-ray oscilloscope,
yet there is a hidden relationship. Even random
noise can be considered as being made
up of sine-wave components constantly
shifting in frequency, amplitude, and
phase. If you pass random noise through a
narrow filter and observe the filter output
on a cathode-ray oscilloscope, you will see
a restless, sinelike wave that constantly
shifts in amplitude. Theoretically, an infinitely
narrow filter would sift out a pure,
but nervous, sine wave. (See chapter 5.)
Electrical, Mechanical, and
Acoustical Analogs
An acoustical system such as a loudspeaker
can be represented in terms of an
equivalent electrical or mechanical system.
The physicist freely uses these equivalents
to set up his mathematical approach
for analyzing a given system. Although such approaches are far outside
the scope of this book, it is useful to develop some appreciation
for these methods. For example, the effect of a cabinet on the functioning
of a loudspeaker is clarified by thinking of the air in the
enclosed space as acting like a capacitor in an electrical circuit,
absorbing and giving up the energy imparted by the cone movement.
Figure 1-17 shows the graphical representation of the three basic
elements in electrical, mechanical, and acoustical systems. Inductance
in an electrical circuit is equivalent to mass in a mechanical system
and inertance in an acoustical system. Capacitance in an electrical circuit
is analogous to compliance in a mechanical system and capacitance
in an acoustical system. Resistance is resistance in all three
systems, whether it be the frictional losses offered to air-particle movement
in glass fiber, frictional losses in a wheel bearing, or resistance to
the flow of current in an electrical circuit.
In synthesizing the square wave of (A), including
only the fundamental and two harmonics yields (B).
Including fifteen components yields (C). It would
take many more than fifteen harmonics to smooth
the ripples and produce the square corners of (A).
A
B
C
F I G U R E 1 - 1 6
FUNDAMENTALS OF SOUND 21
The three basic elements of electrical systems and their analogs in mechanical and
acoustical systems.
Electrical Mechanical Acoustical
Inductance Mass Inertance
Resistance Resistance Resistance
Capacitance Compliance Capacitance
F I G U R E 1 - 1 7
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The decibel is as commonly used in audio circles as the minute or the
mile is in general usage. A superficial understanding of the decibel
can hinder the study of the science of sound and be a barrier in the
proper use and development of its many applications. The goal of this
chapter is to show the need for the decibel concept and how decibels
can be applied in many different ways.
Levels in decibels make it easy to handle the extremely wide range
of sensitivity in human hearing. The threshold of hearing matches the
ultimate lower limit of perceptible sound in air, the noise of air molecules
beating a tattoo on the eardrum. The sensitivity of normal human
eyes also matches the ultimate limit by responding to one or a very few
photons of light. From these threshold responses to the most feeble
stimuli, the ear and eye are also capable of handling high intensities of
sound and light. A level in decibels is a convenient way of handling
the billion-fold range of sound pressures to which the ear is sensitive
without getting bogged down in long strings of zeros.
Ratios vs. Differences
Imagine a sound source set up in a room completely protected from
interfering noise. (The term sound-proof is avoided because there will
be much sound in it.) The sound source is adjusted for a weak sound
with a sound pressure of 1 unit, and its loudness is carefully noted.
Sound Levels and the Decibel
2
23
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
24 CHAPTER TWO
When the sound pressure is increased until it sounds twice as loud,
the level dial reads 10 units. This completes observation A. For observation
B, the source pressure is increased to 10,000 units. To double
the loudness, you find that the sound pressure must be increased from
10,000 to 100,000 units. The results of this experiment can now be
summarized as follows:
Ratio of Two
Observations Two Pressures Pressures
A 10 – 1 10: 1
B 100,000 – 10,000 10: 1
Observations A and B accomplish the same doubling of perceived
loudness. In observation A, this was accomplished by an
increase in sound pressure of only 9 units, where in observation B it
took 90,000 units. Ratios of pressures seem to describe loudness
changes better than differences in pressure. Ernst Weber (1834),
Gustaf Fechner (1860), Hermann von Helmholtz (1873), and other
early researchers pointed out the importance of ratios, which we
know apply equally well to sensations of vision, hearing, vibration,
or even electric shock.
Many years ago, a friend working in a university research laboratory
demonstrated his experiment on the hearing of cats, which in
many ways is similar to that of humans. A tone of 250 Hz, radiated
from a nearby loudspeaker, was picked up by the ears of an anesthetized
cat, a portion of whose brain was temporarily exposed. A
delicate probe picked up the 250-Hz signal at a highly localized spot
on the auditory cortex, displaying it on a cathode-ray oscilloscope.
When the tone was shifted to 500 Hz, the signal was picked up at
another spot on the cortex. Tones of 1,000 and 2,000 Hz were detected
at other specific spots. The fascinating point here is that changing the
tone an octave resulted in the signal appearing on the auditory cortex
at discrete, equally spaced points. Frequencies in the ratio of 2 to 1
(an octave) seem to have a linear positional relationship in the cat’s
brain. This indicates a logarithmic response to frequency. Ratios of
stimuli come closer to matching up with human perception than do
differences of stimuli. This matching is not perfect, but close enough
to make a strong case for the use of levels in decibels.
SOUND LEVELS AND THE DECIBEL 25
Ratios of powers or ratios of intensities, or ratios of sound pressure,
voltage, current, or anything else are dimensionless. For instance, the
ratio of 1 watt to 100 watts is 1 watt/100 watts, and the watt unit
upstairs and the watt unit downstairs cancel, leaving 1⁄100 = 0.01, a pure
number without dimension. This is important because logarithms can
be taken only of nondimensional numbers.
Handling Numbers
Table 2-1 illustrates three different ways numbers can be expressed.
The decimal and arithmetic forms are familiar in everyday activity.
The exponential form, while not as commonly used, has the charm of
simplifying things once the fear of the unknown or little understood is
conquered. In writing one hundred thousand, there is a choice
between 100,000 watts and 105 watts, but how about a millionth of a
millionth of a watt? All those zeros behind the decimal point make it
impractical even to reproduce here, but 10–12 is easy. And the prefix
that means 10–12 is pico; so the power is 1 picowatt (shown later in
Table 2-4). Engineering-type calculators take care of the exponential
form in what is called scientific notation, by which very large or very
small numbers can be entered.
Table 2-1. Ways of expressing numbers.
Decimal Arithmetic Exponential
form form form
100,000 10 × 10 × 10 × 10 × 10 105
10,000 10 × 10 × 10 × 10 104
1,000 10 × 10 × 10 103
100 10 × 10 102
10 10 × 1 101
1 10/10 100
0.1 1/10 10–1
0.01 1/(10 × 10) 10–2
0.001 1/(10 × 10 × 10) 10–3
0.0001 1/(10 × 10 × 10 × 10) 10–4
26 CHAPTER TWO
Table 2-1. Ways of expressing numbers (Continued).
Decimal Arithmetic Exponential
form form form
100,000 (100)(1,000) 102 + 103 = 102+3 = 105
100 10,000/100 104/102 = 104–2 = 102
10 100,000/10,000 105/104 = 105–4 = 10–1 = 10
10 100 = 2 100 1001/2 = 1000.5
4.6416 3 100 1001/2 = 1000.333
31.6228 4 1003 1003/4 = 1000.75
Logarithms
Representing 100 as 102 simply means that 10 × 10 = 100 and that 103
means 10 × 10 × 10 = 1,000. But how about 267? That is where logarithms
come in. It is agreed that 100 equals 102. By definition you can
say that the logarithm of 100 to the base 10 = 2, commonly written log10
100 = 2, or simply log 100 = 2, because common logarithms are to the
base 10. Now that number 267 needn’t scare us; it is simply expressed
as 10 to some other power between 2 and 3. The old fashioned way
was to go to a book of log tables, but with a simple hand-held calculator
punch in 267, push the “log” button, and 2.4265 appears. Thus,
267 = 102.4265, and log 267 = 2.4265. Logs are so handy because, as
Table 2-1 demonstrates, they reduce multiplication to addition, and
division to subtraction. This is exactly how the now-extinct slide rule
worked, by positioning engraved logarithmic scales.
Logs should be the friend of every audio worker because they are
the solid foundation of our levels in decibels. A level is a logarithm of
a ratio. A level in decibels is ten times the logarithm to the base 10
of the ratio of two power like quantities.
Decibels
A power level of a power W1 can be expressed in terms of a reference
power W2 as follows:
SOUND LEVELS AND THE DECIBEL 27
L1 = log10 bels (2-1)
Because the decibel, from its very name, is 1⁄10 bel (from Alexander Graham
Bell), the level in decibels of a power ratio becomes:
L1 = 10 log10 decibels (2-2)
Equation 2-2 applies equally to acoustic power, electric power,
or any other kind of power. A question often arises when levels
other than power need to be expressed in decibels. For example,
acoustic intensity is acoustic power per unit area in a specified
direction, hence Equation 2-2 is appropriate. Acoustic power is
proportional to the square of the acoustic pressure, p, hence the
power level is:
Lp = 10 log
= 20 log in decibels (2-3)
The tabulation of Table 2-2 will help you decide whether the Equation
2-2 or Equation 2-3 form applies.
p1p2
p2
1p
2
2
W1
W2
W1
W2
Table 2-2. Use of 10 log and 20 log.
Eq(2-2) Eq(2-3)
Parameter 10 log10 20 log10
Acoustic
Power X
Intensity X
Air particle velocity X
Pressure X
b1b2
a1a2
28 CHAPTER TWO
Sound pressure is usually the most accessible parameter to measure
in acoustics, even as voltage is for electronic circuits. For this reason,
the Equation 2-3 form is more often encountered in day-to-day
technical work.
Reference Levels
A sound-level meter is used to read a certain sound-pressure level. If
the corresponding sound pressure is expressed in normal pressure
units, a great range of very large and very small numbers results. Ratios
are more closely related to human senses than linear numbers, and
“the level decibels approach” compresses the large and small ratios
into a more convenient and comprehensible range. Basically, our
sound-level meter reading is a certain sound-pressure level, 20 log
(p1/p2), as in Equation 2-3. Some standard reference sound pressure
for p2 is needed. The reference p2 selected must be the same as that
used by others, so that ready comparisons can be made worldwide.
Several such reference pressures have been used over the years, but
for sound in air the standard reference pressure is 20 µPa (micropascal).
This might seem quite different from the reference pressure of
0.0002 microbar or 0.0002 dyne/cm2, but it is the same standard
merely written in different units. This is a very minute sound pressure
and corresponds closely to the threshold of human hearing. The
relationship between sound pressure in Pascals, pounds per square
inch, and sound pressure level is shown in the graph of Fig. 2-1.
Table 2-2. Use of 10 log and 20 log (Continued).
Eq(2-2) Eq(2-3)
Parameter 10 log10 20 log10
Electric
Power X
Current X
Voltage X
Distance
(From source-SPL; inverse square) X
b1b2
a1a2
SOUND LEVELS AND THE DECIBEL 29
When a statement is encountered such as, “The sound pressure
level is 82 dB,” the 82-dB sound-pressure level is normally used in
direct comparison with other levels. However, if the sound pressure
were needed, it can be computed readily by working backward from
Equation 2-3 as follows:
82 = 20 log
log =
= 10
The yx button on the calculator (the Hewlett-Packard 41-C is assumed)
helps us to evaluate 104.1. Press 10, enter 4.1 press y x power button, and
the answer 12,589 appears.
82 20 p1
20 µPa
82

20
p1
20 µPa
p1
20 µPa
109 108 107 106 105 0.0001 0.001 0.01
RMS pounds per sq. in.
120
100
80
60
40
20
0
0.0001 0.00001 0.001 0.01 0.1 1 10 100 1000
RMS sound pressure - Pascals
Sound pressure level - dB
Pascals (read bottom scale)
Pounds per sq. in. (read top scale)
F I G U R E 2 - 1
The relationship between sound pressure in Pascals or pounds per square inch and
sound-pressure level (referred to 20 uPa) is shown in this graph. This is a graphical
approach to the solution of Eq. 2-2.
30 CHAPTER TWO
p1 = (20 µPa)(12,589)
p1 = 251,785 µPa
There is another lesson here. The 82 has what is called two significant
figures. The 251,785 has six significant figures and implies a
precision that is not there. Just because the calculator says so doesn’t
make it so! A better answer is 252,000 µPa, or 0.252 Pa.
Logarithmic and
Exponential Forms Compared
The logarithmic and exponential forms are equivalent as can be seen
by glancing again at Table 2-1. In working with decibels it is imperative
that a familiarity with this equivalence be firmly grasped.
Let’s say we have a power ratio of 5:
There are two tens in the exponential statement but they come from
different sources as indicated by the arrows. Now let us treat a sound
pressure ratio of 5:
Remember that sound-pressure level in air means that the reference
pressure downstairs (p2) in the pressure ratio is 20 µPa. There
are other reference quantities; some of the more commonly used
ones are listed in Table 2-3. In dealing with very large and very
small numbers, you should become familiar with the prefixes of
Table 2-4. These prefixes are nothing more than Greek names for the
powers exponents of 10.
10 log10 5 = 6.99 is exactly equivalent to
5 = 10
6.99
10
(2-4)
20 log10 5 = 13.98
5 = 10
13.98
20
(2-5)
SOUND LEVELS AND THE DECIBEL 31
Acoustic Power
It doesn’t take many watts of acoustic power to
produce some very loud sounds, as anyone who
lives downstairs from a dedicated audiophile
will testify. We are conditioned by megawatt
electrical generating plants, 350-horsepower
(261 kilowatt) automobile engines, and 1,500-
watt flatirons that eclipse the puny watt or so the
hi-fi loudspeakers might radiate as acoustic
power. Even though a hundred-watt amplifier
may be driving the loudspeakers, loudspeaker
efficiency (output for a given input) is very low,
perhaps on the order of 10 percent, and headroom
must be reserved for the occasional peaks
of music. Increasing power to achieve greater results is often frustrating.
Doubling power from 1 to 2 watts is a 3-dB increase in power level (10
log 2 = 3.01), a very small increase in loudness; however, the same 3-dB
increase in level is represented by an increase in power from 100 to 200
watts or 1,000 to 2,000 watts.
Table 2-5 lists sound pressure and sound-pressure levels of some
common sounds. In the sound-pressure column, it is a long stretch
Table 2-4. The Greeks had a word for it.
Prefix Symbol Multiple
tera T 1012
giga G 109
mega M 106
kilo k 103
milli m 10-3
micro  10-6
nano n 10-9
pico p 10-12
Table 2-3. Reference quantities in common use.
Level in decibels Reference quantity
Acoustic
Sound pressure level in 20 micropascal
air (SPL, dB)
Power level (Lp,dB) 1 picowatt(10–12 watt)
Electric
Power level re 1 mW 10–3 watt (1 milliwatt)
Voltage level re 1 V 1 volt
Volume level, VU 10–3 watt
32 CHAPTER TWO
from 100,000 Pa (100 kPa), which is atmospheric pressure to, 0.00002
Pa (20 µPa), but this range is reduced to quite a convenient form in the
level column. The same information is present in graphical form in
Fig. 2-2.
Another way to generate a 194-dB sound-pressure level, besides
launching a Saturn rocket, is to detonate 50 pounds of TNT 10 feet
away. Common sound waves are but tiny ripples on the steady-state
atmospheric pressure. A 194-dB sound-pressure level approaches the
atmospheric and, hence, is a ripple of the same order of magnitude as
Table 2-5. Some common sound-pressure levels and sound pressures.
Sound
Sound level*
pressure (decibels,
Sound Source (Pa) A-weighted)
Saturn rocket 100,000. 194
(one atmosphere)
Ram jet 2,000. 160
Propeller aircraft 200. 140
Threshold of pain 135
Riveter 20. 120
Heavy truck 2. 100
Noisy office, } 0.2 80
Heavy traffic
Conversational speech 0.02 60
Private office 50
Quiet residence 0.0002 40
Recording studio 30
Leaves rustling 0.0002 20
Hearing threshold, good ears at
frequency of maximum sensitivity 10
Hearing threshold, excellent
ears at frequency maximum
response 0.00002 0
* Reference pressure (take your pick, these are identical):
20 micropascal (µPa)
0.00002 pascal
2×10–5 newton/meter2
0.0002 dyne/cm2 or microbar
SOUND LEVELS AND THE DECIBEL 33
atmospheric pressure. The 194-dB sound pressure is an rms (root
mean square) value. A peak sound pressure 1.4 times as great would
modulate the atmospheric pressure completely.
Using Decibels
A level is a logarithm of a ratio of two powerlike quantities. When levels
are computed from other than power ratios, certain conventions are
Sound-pressure
level-dB
Riveter 110
100
90
80
70
60
50
40
30
20
10
0
Noise near
freeway
Background noise
in studio
Sound pressure
Pascal
1 Pa
0.1 Pa
0.01 Pa
0.001 Pa
0.0001 Pa
Standard reference
pressure: 20 106 Pa
20 Pa
(Approximate
threshold of hearing) {
An appreciation of the relative magnitude of a sound pressure of 1 Pascal can be gained
by comparison to known sounds. The standard reference pressure for sound in air is
20Pa, which corresponds closely to the minimum audible pressure.
F I G U R E 2 - 1
34 CHAPTER TWO
observed. The convention for Equation 2-3 is that sound power is proportional
to (sound pressure)2. The voltage-level gain of an amplifier
in decibels is 20 log (output voltage/input voltage), which holds true
regardless of the input and output impedances. However, for powerlevel
gain, the impedances must be considered if they are different. If
it is a line amplifier with 600-ohm input and output impedances, well
and good. Otherwise, a correction is required. The important lesson is
to clearly indicate what type of level is intended, or else label the gain
in level as “relative gain, dB.” The following examples illustrate the
use of the decibel.
Example: Sound-Pressure Level
A sound-pressure level (SPL) is 78 dB. What is the sound pressure, p?
78 dB = 20 log p/(20 × 10–6)
log p/(20 × 10– 6) = 78/20
p/(20 × 10– 6) = 103.9
p = (20 × 10– 6) (7,943.3)
p = 0.159 pascals
Remember that the reference level in SPL measurements is 20 µPa.
Example: Loudspeaker SPL
An input of 1 watt produces a SPL of 115 dB at 1 meter. What is the
SPL at 20 ft (6.1 meters)?
SPL = 115 – 20 log (6.1/1)
= 115 – 15.7
= 99.3 dB
The assumption made in the 20 log 6.1 factor is that the loudspeaker is
operating in a free field and that the inverse square law is operating.
This is a reasonable assumption for a 20-foot distance if the loudspeaker
is remote from reflecting surfaces.
An Electro-Voice “constant directivity” horn Model HP9040 is rated at
a sound pressure level of 115 dB on axis at 1 meter with 1 watt into 8 ohms.
If the input were decreased from 1 watt to 0.22 watts, what would be the
sound-pressure level at 1 meter distance?
SPL = 115 – 10 log (0.22/1)
= 115 – 6.6
= 108.4 dB
SOUND LEVELS AND THE DECIBEL 35
Note that 10 log is used because two powers are being compared.
Example: Microphone Specifications
A Shure Model 578 omnidirectional dynamic microphone open-circuit
voltage is specified as –80 dB for the 150-ohm case. They also
specify that 0 dB = 1 volt per µbar. What would be the open-circuit
voltage,
, be in volts?
–80 dB = 20 log
/1
log
/1 = –80/20

= 0.0001 volt
= 0.1 millivolt
Example: Line Amplifier
A line amplifier (600 ohms in, 600 ohms out) has a gain of 37 dB. With
an input of 0.2 volts, what is the output voltage?
37 dB = 20 log (
/0.2)
log (
/0.2) = 37/20
= 1.85

/0.2 = 101.85

= (0.2)(70.79)

= 14.16 volts
Example: General-Purpose Amplifier
This amplifier has a bridging input of 10,000 Ω impedance and an output
impedance of 600 Ω. With a 50 m
input, an output of 1.5 V is
observed. What is the gain of the amplifier? The so-called voltage gain
is:
voltage gain = 20 log (1.5/0.05)
= 29.5 dB
It must be emphasized that this is not a power level gain because of the
differences in impedance. However, voltage gain may serve a practical
purpose in certain cases.
Example: Concert Hall
Seat x in a concert hall is 84 feet from the tympani drums. The tympanist
strikes a single, mighty note. The sound-pressure level of the
direct sound of the note at seat x is measured to be 55 dB. The first
36 CHAPTER TWO
reflection from the nearest sidewall arrives at seat x 105 milliseconds
after the arrival of the direct sound. (A) How far does the reflection
travel to reach seat x? (B) What is the SPL of the reflection at seat x,
assuming perfect reflection at the wall? (C) How long will the reflection
be delayed after arrival of the direct sound at seat x?
(A) Distance = (1,130 ft/sec) (0.105 sec)
= 118.7 ft
(B) First, the level, L, 1 foot from the tympani drum must be estimated:
55 = L – 20 log (84/1)
L = 55 + 38.5
L = 93.5 dB
The SPL of the reflection at seat x is:
dB = 93.5 – 20 log (118.7/1)
= 93.5 – 41.5
= 52 dB
(C) The reflection will arrive after the direct sound at seat x after:
Delay = (118.7 – 84)/1,130 ft/sec
= 30.7 milliseconds
A free field is also assumed here. In chapter 3, the 30.7 ms reflection
might be called an incipient echo.
Example: Combining Decibels
Let’s say it is warm in our studio and a fan is brought in to augment the
air conditioning (A/C) system. If both fan and the A/C are turned off, a
very low noise level prevails, low enough to be neglected in the calculation.
If the A/C alone is running, the sound-pressure level at a given
position is 55 dB. If the fan alone is running, the sound-pressure level
is 60 dB. What will be the sound-pressure level if both are running at
the same time?
Combined dB = 10 log (10 + 10 )
= 61.19 dB
If the combined level of two noise sources is 80 dB and the level with
one of the sources turned off is 75 dB, what is the level of the remaining
source?
60 10
55 10
SOUND LEVELS AND THE DECIBEL 37
Difference dB = 10 log (10 – 10 )
= 78.3 dB
In other words, combining the 78.3 dB level with the 75 dB level gives
the combined level of 80 dB.
Ratios and Octaves
An octave is defined as a 2:1 ratio of two frequencies. For example,
middle C (C4) on the piano has a frequency close to 261 Hz. The
next highest C (C5) has a frequency of about 522 Hz. Ratios of frequencies
are very much a part of the musical scale. The frequency
ratio 2:1 is the octave; the ratio 3:2 is the fifth; 4:3 is the fourth, etc.
Because the octave is very important in acoustical work, it is well to
consider the mathematics of the octave.
As the ratio of 2:1 is defined as the octave, its mathematical
expression is:
= 2n
(2-6)
in which:
f2 = the frequency of the upper edge of the octave interval.
f1 = the frequency of the lower edge of the octave interval.
n = the number of octaves.
For 1 octave, n = 1 and Equation (2-6) becomes f2/f1 = 2, which is the
definition of the octave. Other applications of Equation (2-6) are now
explored:
Example
The low-frequency edge of a band is 20 Hz, what is the high-frequency
edge of a band 10 octaves wide?
= 210
f2 = (20) (210)
f2 = (20) (1,024)
f2 = 20,480 Hz
f220 Hz
f2
f1
75 10
80 10
38 CHAPTER TWO
Example
If 446 Hz is the lower edge of a 1⁄3 octave band, what is the frequency of
the upper edge?
= 21/3
f2 = (446) (21/3)
f2 = (446) (1.2599)
f2 = 561.9 Hz
Example
What is the lower edge of a 1⁄3 octave band centered on 1,000 Hz? The
f1 is 1,000 Hz but the lower edge would be 1⁄6 octave lower than the 1⁄3
octave, so n = 1⁄6:
= =21/6
f1 =
f1 =
f1 = 890.9 Hz
Example
At what frequency is the lower edge of an octave band centered on
2,500 Hz?
= 21/2
f1 =
f1 =
f1 = 1,767.8 Hz
What is the upper edge?
= 21/2 f22,500
2,500
1.4142
2,500
2
1/2
2,500
f
1
1,000
1.12246
1,000
2
1/6
1,000
f
1
f2
f1
f2446
SOUND LEVELS AND THE DECIBEL 39
f2 = (2,500) (21/2)
f2 = (2,500) (1.4142)
f2 = 3,535.5 Hz
Measuring Sound-Pressure Level
A sound level meter is designed to give readings of sound-pressure
level; sound pressure in decibels referred to the standard reference
level, 20 µPa. Sound level meters usually offer a selection of weighting
networks designated A, B, and C having frequency responses shown in
Fig. 2-3. Network selection is based on the general level of sounds to
be measured (background noise? jet engines?), such as:
• For sound-pressure levels of 20–55 dB...use network A.
• For sound-pressure levels of 55–85 dB...use network B.
• For sound-pressure levels of 85–140 dB...use network C.
These network response shapes are designed to bring the sound level
meter readings into closer conformance to the relative loudness of
sounds.
10
0
10
20
30
40
20 100 1 kHz 10 kHz
Frequency - Hz
Relative response - dB
C
B
A
F I G U R E 2 - 3
A, B, and C weighting response characteristics for sound level meters. (ANSI S1 .4-1971.)
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The study of the structure of the ear is a study in physiology. The
study of human perception of sound comes under the general heading
of psychology. Psychoacoustics is an inclusive term embracing the
physical structure of the ear, the sound pathways, the perception of
sound, and their interrelationships. Psychoacoustics, quite a recent
term, is especially pertinent to this study because it emphasizes both
structure and function of the human ear.
The stimulus sound wave striking the ear sets in motion mechanical
movements that result in neuron discharges that find their way to
the brain and create a sensation. Then comes the question, “How are
these sounds recognized and interpreted?” In spite of vigorous
research activities on all aspects of human hearing, our knowledge is
still woefully incomplete.
Sensitivity of the Ear
The delicate and sensitive nature of our hearing can be underscored dramatically
by a little experiment. A bulky door of an anechoic chamber is
slowly opened, revealing extremely thick walls, and three-foot wedges of
glass fiber, points inward, lining all walls, ceiling, and what could be
called the floor, except that you walk on an open steel grillwork.
A chair is brought in, and you sit down. This experiment takes time,
and as a result of prior briefing, you lean back, patiently counting
The Ear and the
Perception of Sound
3
41
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
42 CHAPTER THREE
the glass fiber wedges to pass the time. It is very eerie in here. The sea
of sound and noises of life and activity in which we are normally
immersed and of which we are ordinarily scarcely conscious is now
conspicuous by its absence.
The silence presses down on you in the tomblike silence, 10 minutes,
then a half hour pass. New sounds are discovered, sounds that
come from within your own body. First, the loud pounding of your
heart, still recovering from the novelty of the situation. An hour goes
by. The blood coursing through the vessels becomes audible. At last, if
your ears are keen, your patience is rewarded by a strange hissing
sound between the “ker-bumps” of the heart and the slushing of blood.
What is it? It is the sound of air particles pounding against your
eardrums. The eardrum motion resulting from this hissing sound is
unbelievably small—only 1⁄100 of a millionth of a centimeter—or 1⁄10
the diameter of a hydrogen molecule!
The human ear cannot detect sounds softer than the rain of air particles
on the eardrum. This is the threshold of hearing. There would be
no reason to have ears more sensitive, because any lower-level sound
would be drowned by the air-particle noise. This means that the ultimate
sensitivity of our hearing just matches the softest sounds possible
in an air medium. Accident? Adaptation? Design?
At the other extreme, our ears can respond to the roar of a cannon, the
noise of a rocket blastoff, or a jet aircraft under full power. Special protective
features of the ear protect the sensitive mechanism from damage
from all but the most intense noises.
A Primer of Ear Anatomy
The three principal parts of the human auditory system, shown in Fig.
3-1, are the outer ear, the middle ear, and the inner ear. The outer ear
is composed of the pinna and the auditory canal or auditory meatus.
The auditory canal is terminated by the tympanic membrane or the
eardrum. The middle ear is an air-filled cavity spanned by the three
tiny bones, the ossicles, called the malleus, the incus, and the stapes.
The malleus is attached to the eardrum and the stapes is attached to
the oval window of the inner ear. Together these three bones form a
mechanical, lever-action connection between the air-actuated eardrum
and the fluid-filled cochlea of the inner ear. The inner ear is terminated
in the auditory nerve, which sends impulses to the brain.
THE EAR AND THE PERCEPTION OF SOUND 43
The Pinna: Directional Encoder of Sound
In ancient times, the pinna was regarded as either a vestigial organ or
a simple sound-gathering device. True, it is a sound-gathering device.
The pinna offers a certain differentiation of sounds from the front as
compared to sound from the rear. Cupping your hand behind the ear
increases the effective size of the pinna and thus the apparent loudness
by an amount varying with frequency. For the important speech
frequencies (2,000 to 3,000 Hz), sound pressure at the eardrum is
increased about 5 dB. This front-back differentiation is the more modest
contribution of the pinna.
Recent research has revealed that the pinna performs a very crucial
function in imprinting directional information on all sounds picked
up by the ear. This means that information concerning the direction to
the source is superimposed on the sound content itself so that the
resultant sound pressure on the eardrum enables the brain to interpret
both the content of the sound and the direction from which it comes.
Pinna
Semicircular
canals
Cochlea Middle
ear
Round
window
Eardrum
Auditory
canal
F I G U R E 3 - 1
The four principal parts of the human ear: the pinna, the auditory canal, the middle ear,
and the inner ear.
44 CHAPTER THREE
Directional Cues: An Experiment
If the equipment is available, a simple psychoacoustical experiment can
illustrate how subjective directional impressions result from simple
changes in sounds falling on the ear. Listen with a headphone on one ear
to an octave bandwidth of random noise centered on 8 kHz arranged
with an adjustable notch filter. Adjusting the filter to 7.2 kHz will cause
the noise to seem to come from a source on the level of the observer. With
the notch adjusted to 8 kHz the sound seems to come from above. With
the notch at 6.3 kHz the sound seems to come from below. This experiment1
demonstrates that the human hearing system extracts directional
information from the shape of the sound spectra at the eardrum.
The Ear Canal
The ear canal also increases the loudness of the sounds traversing it. In
Fig. 3-2 the ear canal, with an average diameter of about 0.7 cm and
length of about 3 cm, is idealized by straightening and giving it a uniform
diameter throughout its length.
Acoustically, this is a reasonable approximation.
It is a pipe-like duct, closed at the
inner end by the eardrum.
Organ pipes were studied intensely by
early investigators when the science of
acoustics was in its infancy. The acoustical
similarity of this ear canal to an organ pipe
was not lost on early workers in the field.
The resonance effect of the ear canal
increases sound pressure at the eardrum at
certain frequencies. The maximum is near
the frequency at which the 3-cm pipe is
one-quarter wavelength—about 3,000 Hz.
Figure 3-3 shows the increase in sound
pressure at the eardrum over that at the
opening of the ear canal. A primary peak is
noted around 3,000 Hz caused by the quarter-
wave pipe resonance effect. The primary
pipe resonance amplifies the sound
pressure at the eardrum approximately
12 dB at the major resonance at about
Pressure
Distance
4
Eardrum
F I G U R E 3 - 2
The auditory canal, closed at one end by the eardrum,
acts as a quarter-wavelength “organ pipe.” Resonance
provides acoustic amplification for the important
voice frequencies.
THE EAR AND THE PERCEPTION OF SOUND 45
4,000 Hz. There is a secondary resonance nearer 9,000 Hz of lower
peak pressure.2
The Middle Ear
Transmitting sound energy from a tenuous medium such as air into a
dense medium like water is a serious problem. Without some very special
equipment, sound originating in air bounces off water like light off
a mirror. It boils down to a matter of matching impedances, and in this
case the impedance ratio is something like 4,000:1. Consider how satisfactory
it would be to drive the 1-ohm voice coil of a loudspeaker
with an amplifier having an output impedance of 4,000 ohms! Clearly
not much power would be transferred.
The object is to get the feeble energy represented by the vibratory
motion of a rather flimsy diaphragm, transferred with maximum efficiency
to the fluid of the inner ear. The two-fold solution is suggested in
Fig. 3-4. The three ossicles (hammer, anvil, and stirrup) form a mechanical
linkage between the eardrum and the oval window, which is in intimate
contact with the fluid of the inner ear. The first of the three bones,
the malleus, is fastened to the eardrum. The third, the stapes, is actually
a part of the oval window. There is a lever action in this linkage with a
15
10
5
0
5
10
0.1 0.2 0.5 1 2 5 10 20
Relative sound-pressure level - dB
Frequency - kHz
F I G U R E 3 - 3
The transfer function (frequency response) of the ear canal. This is a fixed component that is combined with
every directionally-encoded sound reaching the eardrum. See also Figs. 3-15 and 3-16. (After Mehrgardt and
Mellart.2)
46 CHAPTER THREE
ratio leverage ranging from 1.3:1 to 3.1:1. That is, the eardrum motion is
reduced by this amount at the oval window of the inner ear.
This is only part of this fascinating mechanical-impedance-matching
device. The area of the eardrum is about 80 sq mm, and the area of
the oval window is only 3 sq mm. Hence, a given force on the eardrum
is reduced in the ratio of 80/3, or about 27-fold.
In Fig. 3-4B, the action of the middle ear is likened to two pistons
with area ratios of 27:1 connected by an articulated connecting rod
having a lever arm ranging from 1.3:1 to 3.1:1, making a total mechanical
force increase of between 35 and 80 times. The acoustical impedance
ratio between air and water being on the order of 4,000:1, the
pressure ratio required to match two media would be 4,000 , or about
63.2, and we note that this falls within the 35 to 80 range obtained
from the mechanics of the middle ear illustrated in Fig. 3-4B.
Ossicles
Eardrum Oval window
Eardrum
80 Sq mm
1.3–3.1 to 1
1 3 Sq mm
oval window
B
A
F I G U R E 3 - 4
(A) The ossicles (hammer, anvil, and stirrup) of the middle ear, which transmit mechanical
vibrations of the eardrum to the oval window of the cochlea. (B) A mechanical analog
of the impedance-matching function of the middle ear. The difference in area
between the eardrum and the oval window, coupled with the step-down mechanical linkage,
match the motion of the air-actuated eardrum to the fluid-loaded oval window.
THE EAR AND THE PERCEPTION OF SOUND 47
The problem of matching sound in air to sound in the fluid of
the inner ear is beautifully solved by the mechanics of the middle
ear. The evidence that the impedance matching plus the resonance
amplification of Fig. 3-3 really work is that a diaphragm motion comparable
to molecular dimensions gives a threshold perception.
A schematic of the ear is given in Fig. 3-5. The conical eardrum at
the inner end of the auditory canal forms one side of the air-filled middle
ear. The middle ear is vented to the upper throat behind the nasal
cavity by the Eustachian tube. The eardrum operates as an “acoustic
suspension” system, acting against the compliance of the trapped air
in the middle ear. The Eustachian tube is suitably small and constricted
so as not to destroy this compliance. The round window separates
the air-filled middle ear from the practically incompressible
fluid of the inner ear.
The Eustachian tube fulfills a second function by equalizing the
static air pressure of the middle ear with the outside atmospheric pressure
so that the eardrum and the delicate membranes of the inner ear
can function properly. Whenever we swallow, the Eustachian tubes
Oval window
Middle
ear
Inner ear
Fluid
Basilar membrane
Round window
Eustachian
tube
Ear canal
Eardrum
F I G U R E 3 - 5
Highly idealized sketch of the human ear showing the unrolled fluid-filled cochlea.
Sound entering the ear canal causes the eardrum to vibrate. This vibration is transmitted
to the cochlea through the mechanical linkage of the middle ear. The sound is analyzed
through standing waves set up on the basilar membrane.
48 CHAPTER THREE
open, equalizing the middle ear pressure. When an aircraft (at least
those without pressurized cabins) undergoes rapid changes in altitude,
the occupants might experience momentary deafness or pain
until the middle ear pressure is equalized by swallowing. Actually, the
Eustachian tube has a third emergency function of drainage if the middle
ear becomes infected.
The Inner Ear
Only the acoustical amplifiers and the mechanical impedance matching
features of the middle ear have been discussed so far. These are relatively
well understood. The intricate operation of the cochlea is still clouded in
mystery, but extensive research is steadily adding to our knowledge.
Figure 3-1 shows the close proximity of the three mutually-perpendicular,
semicircular canals of the vestibular mechanism, the balancing
organ, and the cochlea, the sound-analyzing organ. The same fluid
permeates all, but their functions are independent.
The cochlea, about the size of a pea, is encased in solid bone. It is
coiled up like a cockleshell from which it gets its name. For the purposes
of illustration, this 23⁄4-turn coil has been stretched out its full
length, about one inch, as shown in Fig. 3-5. The fluid-filled inner ear
is divided lengthwise by two membranes, Reissner’s membrane and
the basilar membrane. Of immediate interest is the basilar membrane
and its response to sound vibrations in the fluid.
Vibration of the eardrum activates the ossicles. The motion of the
stapes, attached to the oval window, causes the fluid of the inner ear to
vibrate. An inward movement of the oval window results in a flow of
fluid around the distant end of the basilar membrane, causing an outward
movement of the membrane of the round window. Sound actuating
the oval window results in standing waves being set up on the
basilar membrane. The position of the amplitude peak of the standing
wave on the basilar membrane changes as the frequency of the exciting
sound is changed.
Low-frequency sound results in maximum amplitude near the distant
end of the basilar membrane; high-frequency sound produces
peaks near the oval window. For a complex signal such as music or
speech, many momentary peaks are produced, constantly shifting in
amplitude and position along the basilar membrane. These resonant
peaks on the basilar membrane were originally thought to be so broad
THE EAR AND THE PERCEPTION OF SOUND 49
as to be unable to explain the sharpness of frequency discrimination
displayed by the human ear. Recent research is showing that at low
sound intensities, the basilar membrane tuning curves are very sharp,
broadening only for intense sound. It now appears that the sharpness
of the basilar membrane’s mechanical tuning curves is comparable to
the sharpness of single auditory nerve fibers, which innervate it.
Stereocilia
Waves set up on the basilar membrane in the fluid-filled duct of the
inner ear stimulate hairlike nerve terminals that convey signals to the
brain in the form of neuron discharges, about 15,000 outer hair cells
with about 140 tiny hairs called stereocilia jutting from each one. In
addition, there are about 3,500 inner hair cells, each having about 40
stereocilia attached. These stereocilia are the true transducers of
sound energy to electrical discharges. There are two types of hair cells,
inner and outer, so-called by their placement and arrangement. As
sound causes the cochlear fluid and the basilar membrane to move, the
stereocilia on the hair cells are bent, initiating neural discharges to the
auditory cortex.
When sound excites the fluid of the inner ear, membrane and hair
cells are stimulated, sending an electrical wave through the surrounding
tissue. These so-called microphonic potentials (analog) can be
picked up and amplified, reproducing the sound falling on the ear,
which acts as a biological microphone. These potentials are proportional
to the sound pressure and linear in their response over an 80-dB
range. While interesting, this microphonic potential must not be confused
with the action potentials of the auditory nerve, which convey
information to the brain.
Bending the stereocilia triggers the nerve impulses that are carried
by the auditory nerve to the brain. While the microphonic signals are
analog, the impulses sent to the acoustic cortex are impulses generated
by neuron discharges. A single nerve fiber is either firing or not
firing (binary!). When it fires, it causes an adjoining one to fire, and so
on. Physiologists liken the process to a burning gunpowder fuse. The
rate of travel bears no relationship to how the fuse was lighted. Presumably
the loudness of the sound is related to the number of nerve
fibers excited and the repetition rates of such excitation. When all the
nerve fibers (some 15,000 of them) are excited, this is the maximum
50 CHAPTER THREE
loudness that can be perceived. The threshold sensitivity would be
represented by a single fiber firing. An overall, well-accepted theory of
how the inner ear and the brain really function has not yet been formulated.
3–6
This has been a highly simplified presentation of a very complex
mechanism to which much current research is being devoted. Some of
the numbers used and theories discussed are not universally accepted.
Popularization of a subject such as the ear is an occupation that might
be hazardous to my health, but audio workers must surely be amazed
at the delicate and effective workings of the human ear. It is hoped that
a new awareness of, and respect for, this delicate organism will be
engendered and that damaging high levels of sound be avoided.
Loudness vs. Frequency
The seminal work on loudness was done at Bell Laboratories by
Fletcher and Munson and reported in 1933,7 and refinements have
been added by others since that time. The family of equal-loudness
contours of Fig. 3-6, the work of Robinson and Dadson,8 has been
adopted as an international standard (I.S.O. 226).
Each equal-loudness contour is identified by its value at 1,000 Hz,
and the term loudness level in phons is thus defined. For example, the
equal-loudness contour passing through 40-dB sound-pressure level at
1,000 Hz is called the 40-phon contour. Loudness is a subjective term;
sound-pressure level is strictly a physical term. Loudness level is also
a physical term that is useful in estimating the loudness of a sound (in
units of sones) from sound-level measurements. The shapes of the
equal-loudness contours contain subjective information because they
were obtained by a subjective comparison of the loudness of a tone to
its loudness at 1,000 Hz.
The surprising thing about the curves of Fig. 3-6 is that they reveal
that perceived loudness varies greatly with frequency and sound-pressure
level. For example, a sound-pressure level of 30 dB yields a loudness
level of 30 phons at 1,000 Hz, but it requires a sound-pressure
level of 58 dB more to sound equally loud at 20 Hz as shown in Fig.
3-7. The curves tend to flatten at the higher sound levels. The 90-phon
curve rises only 32 dB between 1,000 Hz and 20 Hz. Note that inverting
the curves of Fig. 3-7 gives the frequency response of the ear in
THE EAR AND THE PERCEPTION OF SOUND 51
terms of loudness level. The ear is less sensitive to bass notes than
midband notes at low levels. There are wiggles in the ear’s highfrequency
response that are relatively less noticeable. This bass problem
of the ear means that the quality of reproduced music depends on
the volume-control setting. Listening to background music at low levels
requires a different frequency response than listening at higher levels.
Loudness Control
Let us assume that the high fidelity enthusiast adjusts the volume control
on his or her amplifier so that the level of recorded symphony
music is pleasing as a background to conversation (assumed to be
Loudness level-phons
120
110
100
90
80
70
60
50
40
30
20
10
120
100
80
60
40
20
0
Sound-pressure level - dB
20 50 100 300 1 kHz 3 kHz 10 kHz 20 kHz
Frequency - Hz
Minimum
audible
F I G U R E 3 - 6
Equal-loudness contours of the human ear. These contours reveal the relative lack of sensitivity of the ear to
bass tones, especially at lower sound levels. Inverting these curves give the frequency response of the ear in
terms of loudness level. (After Robinson and Dadson.8)
52 CHAPTER THREE
about 60 phons). As the passage was played at something like an 80-
phon loudness level in the concert hall, something needs to be done to
give the bass and treble of the music the proper balance at the lowerthan-
concert-hall level. Our enthusiast would find it necessary to
increase both bass and treble for good balance.
The loudness control found on many amplifiers adjusts electrical networks
to compensate for the change in frequency response of the ear for
different loudness levels. But the curve corresponding to a given setting
of the loudness control applies only to a specific loudness level of reproduced
sound. The loudness control is far from a complete solution to the
problem. Think of all the things that affect the volume-control setting in
a particular situation. The loudspeakers vary in acoustic output for a
given input power. The gain of preamplifiers, power amplifiers, tuners,
120
100
80
60
40
20
0
32 dB
58 dB
90 Phons
30 Phons
20 50 100 300 1 kHz 3 kHz 10 kHz 20 kHz
Frequency - Hz
Sound-pressure level - dB
F I G U R E 3 - 7
A comparison of the ear’s response at 20 Hz compared to that at 1,000 Hz. At a loudness
level of 30 phons, the sound-pressure level of a 20-Hz tone must be 58 dB higher
than that at 1,000 Hz to have the same loudness. At 90 phons loudness level, an
increase of only 32 dB is required. The ear’s response is somewhat flatter at high loudness
levels. Loudness level is only an intermediate step to true subjective loudness as
explained in the text.
THE EAR AND THE PERCEPTION OF SOUND 53
and phono pickups differs from brand to brand and circuit to circuit. Listening-
room conditions vary from dead to highly reverberant. With all of
these variables, how can a manufacturer design a loudness control truly
geared to the sound-pressure level at the ear of listener x with the particular
variables of x’s equipment and x’s listening environment? For a
loudness control to function properly, x’s system must be calibrated and
the loudness control fitted to it.9
Area of Audibility
Curves A and B of Fig. 3-8 were obtained from groups of trained listeners.
In this case, the listeners face the sound source and judge
whether a tone of a given frequency is barely audible (curve A) or
120
100
80
60
40
20
0
Threshold of feeling
Auditory area
Threshold of hearing
Sound-pressure level - dB
20 50 100 300 1 kHz 3 kHz 10 kHz 20 kHz
Frequency - Hz
A
B
F I G U R E 3 - 8
The auditory area of the human ear is bounded by two threshold curves, (A) the threshold
of hearing delineating the lowest level sounds the ear can detect, and (B) the
threshold of feeling at the upper extreme. All of our auditory experiences occur within
this area.
54 CHAPTER THREE
beginning to be painful (curve B). These two curves represent the
extremes of our perception of loudness.
Curve A of Fig. 3-8, the threshold of hearing, tells us that human
ears are most sensitive around 3 kHz. Another way to state this is that
around 3 kHz a lower-level sound elicits a greater threshold response
than higher or lower frequencies. At this most sensitive region, a
sound-pressure level of 0 dB can just barely be heard by a person of
average hearing acuity. Is it fortuitous that this threshold is at a nice,
round, 0-dB level? No, the reference level of pressure of 20 mPa (20
micropascals) was selected for this reason. It is both instructive and
comforting to know that a sound-pressure level of 60 dB turns out to
be approximately 60 dB above our threshold of hearing.
Curve B of Fig. 3-8 represents the level at each frequency at which
a tickling sensation is felt in the ears. This occurs at a sound-pressure
level of about 120 or 130 dB. Further increase in level results in an
increase in feeling until a sensation of pain is produced. The threshold
tickling is a warning that the sound is becoming dangerously loud and
that ear damage is either imminent or has already taken place.
In between the threshold of hearing (curve A of Fig. 3-8) and the
threshold of feeling (curve B) is the area of audibility. This is an area
with two dimensions: the vertical dimension of sound-pressure level
and the horizontal range of frequencies that the ear can perceive. All
the sounds that humans experience must be of such a frequency and
level as to fall within this auditory area. Chapter 5 details more specifically
how much of this area is used for common music and speech
sounds.
The area of audibility for humans is quite different from that of
many animals. The bat specializes in sonar cries that are far above the
upper frequency limit of our ears. The hearing of dogs extends higher
than ours, hence the usefulness of ultrasonic dog whistles. Sound in
the infrasonic and ultrasonic regions, as related to the hearing of
humans, is no less true sound in the physical sense, but it does not
result in human perception.
Loudness vs. Sound-Pressure Level
The phon is the unit of loudness level that is tied to sound-pressure
level at 1,000 Hz as we have seen in Figs. 3-6, 3-7, and 3-8. This is
THE EAR AND THE PERCEPTION OF SOUND 55
useful up to a point, but it tells us little about human reaction to loudness
of sound. We need some sort of subjective unit of loudness. Many
experiments conducted with hundreds of subjects and many types of
sound have yielded a consensus that for a 10-dB increase in soundpressure
level, the average person reports that loudness is doubled.
For a 10-dB decrease in sound level, subjective loudness is cut in half.
One researcher says this should be 6 dB, others say 10 dB, so work on
the problem continues. However, a unit of subjective loudness has
been adopted called the sone. One sone is defined as the loudness
experienced by a person listening to a tone of 40-phon loudness level.
A sound of 2 sones is twice as loud, and 0.5 sone half as loud.
Figure 3-9 shows a graph for translating sound-pressure levels to
loudness in sones. One point on the graph is the very definition of the
10
5
2
1
0.4
0.3
0.5
0.2
0.1
0 10 20 30 40 50 60 70
Loudness level - phons
Loudness - sones
F I G U R E 3 - 9
The graphical relationship between the physical loudness level in phons and subjective loudness in sones.
56 CHAPTER THREE
sone, the loudness experienced by a person hearing a 1,000-Hz tone at
40-dB sound-pressure level, or 40 phons. A loudness of 2 sones is then
10 dB higher; a loudness of 0.5 sones is 10 dB lower. A straight line can
be drawn through these three points, which can then be extrapolated
for sounds of higher and lower loudness.
As crude as this graph may be, it is a way of getting at the subjective
factor of loudness. The value of this line of reasoning is that if a
consultant is required by a court to give his or her opinion on the loudness
of an industrial noise that bothers neighbors, he or she can make
a one-third octave analysis of the noise, translate the sound-pressure
levels of each band to sones by the help of a series of graphs such as
Fig. 3-9, and by adding together the sones of each band, arrive at an
estimate of the loudness of the noise. This idea of being able to add
component sones is very nice; adding decibels of sound-pressure levels
is a path that leads only to confusion.
Table 3-1 shows the relationship between loudness level in phons
to the subjective loudness in sones. Although most audio workers will
have little occasion to become involved in phons or sones, it is good to
realize that a true subjective unit of loudness (sone) is related to loudness
level (phon), which is in turn related by definition to what we can
measure with a sound-level meter. There are highly developed empirical
methods of calculating the loudness of sound as they would be
perceived by humans from purely physical measurements of sound
spectra, such as those measured with a sound-level meter and an
octave or one-third octave filter.10
Loudness and Bandwidth
In the discussion of loudness we have
talked tones up to this point, but singlefrequency
tones do not give all the information
we need to relate subjective loudness
to meter readings. The noise of a jet aircraft
taking off sounds much louder than a tone
of the same sound-pressure level. The
bandwidth of the noise affects the loudness
of the sound, at least within certain limits.
Table 3-1. Loudness level in phons vs.
loudness in sones.
Loudness Subjective
level loudness
(phons) (sones) Typical examples
100 64 Heavy truck passing
80 16 Talking loudly
60 4 Talking softly
40 1 Quiet room
20 0.25 Very quiet studio
THE EAR AND THE PERCEPTION OF SOUND 57
Figure 3-10A represents three sounds having the same soundpressure
level of 60 dB. Their bandwidths are 100, 160, and 200 Hz,
but heights (representing sound intensity per Hz) vary so that areas
are equal. In other words, the three sounds have equal intensities.
(Sound intensity has a specific meaning in acoustics and is not to be
equated to sound pressure. Sound intensity is proportional to the
square of sound pressure for a plane progressive wave). The catch is
that all three sounds of Fig.3-10A do not have the same loudness.
The graph in Fig. 3-10B shows how a bandwidth of noise having a
constant 60-dB sound-pressure level and centered on 1,000 Hz is
related to loudness as experimentally determined. The 100-Hz noise
A
100
Hz
1 kHz 1 kHz 1 kHz
160
Hz 200
Hz
80
75
70
65
60
55
80 160 330 670 1,400
16
8
4
Loudness level - phons
Loudness - sones
Bandwidth - Hz
B
F I G U R E 3 - 1 0
(A) Three noises of different bandwidths, but all having the same sound-pressure level
of 60 dB. (B) The subjective loudness of the 100- and 160-Hz noise is the same, but the
200-Hz band sounds louder because it exceeds the 160-Hz critical band width of the ear
at 1,000 Hz.
58 CHAPTER THREE
has a loudness level of 60 phons and a loudness of 4 sones. The 160-
Hz bandwidth has the same loudness, but something mysterious
happens as the bandwidth is increased beyond 160 Hz. The loudness
of the noise of 200-Hz bandwidth is louder, and from 160 Hz or
up, increasing bandwidth increases loudness. Why the sharp change
at 160 Hz?
It turns out that 160 Hz is the width of the ear’s critical band at
1,000 Hz. If a 1,000-Hz tone is presented to a listener along with random
noise, only the noise in a band 160 Hz wide is effective in
masking the tone. In other words, the ear acts like an analyzer composed
of a set of bandpass filters stretching throughout the audible
spectrum. This filter set is nothing like that found in the electronics
laboratory. The common 1⁄3-octave filter set may have 28 adjacent filters
overlapping at the –3 dB points. The set of critical band filters is
continuous; that is, no matter where you might choose to set the signal
generator dial, there is a critical band centered on that frequency.
Many years of research on this problem has yielded a modicum of
agreement on how the width of the critical-band filters varies with frequency.
This classical bandwidth function is shown in the graph of
Fig. 3-11. There has been some question as to the accuracy of this
graph below about 500 Hz that has led to other methods of measuring
the bandwidth. Out of this has come the concept of the equivalent rectangular
bandwidth (ERB) that applies to young listeners at moderate
sound levels.11 This approach is based on mathematical methods and
offers the convenience of being able to calculate the ERB from the
equation given in Fig. 3-11.
One-third-octave filter sets have been justified in certain measurements
because the filter bandwidths approach those of the critical
bands of the ear. For comparison, a plot of one-third-octave
bandwidths is included in Fig. 3-11. One-third-octave bands are 23.2
percent of the center frequency. The classical critical-band function
is about 17 percent of the center frequency. It is interesting to note
that the ERB function (12 percent) is very close to that of one-sixthoctave
bands (11.6 percent). This suggests the possibility of onesixth-
octave filter sets playing a larger role in sound measurements
of the future.
THE EAR AND THE PERCEPTION OF SOUND 59
Loudness of Impulses
The examples discussed up to this point have been concerned with
steady-state tones and noise. How does the ear respond to transients of
short duration? This is important because music and speech are essentially
made up of transients. To focus attention on this aspect of speech
and music, play some tapes backward. The initial transients now
appear at the ends of syllables and musical notes and stand out prominently.
These transients justify a few words on the ear’s response to
short-lived sounds.
A 1,000-Hz tone sounds like 1,000 Hz in a 1-second tone burst, but
an extremely short burst sounds like a click. The duration of such a
2,000
1,000
500
200
100
50
30
Bandwidth - Hz
100 200 500 1k 2k 5k 10k
Center frequency - Hz
1/3 Octave
1/6 Octave
Classical
Equivalent
rectangular
band (ERB)
ERB  6.23f2  93.3f  28.52 Hz
f  freq. in kHz
F I G U R E 3 - 1 1
A comparison of bandwidths of 1⁄3- and 1⁄6-octave bands, critical bands of the ear, and
equivalent rectangular critical bands (ERB) calculated from the above equation.11
60 CHAPTER THREE
burst also influences the perceived loudness. Short bursts do not
sound as loud as longer ones. Figure 3-12 shows how much the level
of the shorter pulses has to be increased to have the same loudness as
a long pulse or steady tone. A pulse 3 milliseconds long must have a
level about 15 dB higher to sound as loud as a 0.5-second (500 millisecond)
pulse. Tones and random noise follow roughly the same relationship
in loudness vs. pulse length.
The 100-msec region is significant in Fig. 3-12. Only when the
tones or noise bursts are shorter than this amount must the soundpressure
level be increased to produce a loudness equal to that of long
pulses or steady tones or noise. This 100 msec appears to be the integrating
time or the time constant of the human ear.
In reality, Fig. 3-12 tells us that our ears are less sensitive to short
transients. This has a direct bearing on understanding speech. The consonants
of speech determine the meaning of many words. For instance,
the only difference between bat, bad, back, bass, ban, and bath are the
consonants at the end. The words led, red, shed, bed, fed, and wed have
the all-important consonants at the beginning. No matter where they
20
15
10
5
0
Random
noise
Tones
1 10 100 1000
Pulse duration - milliseconds
Sound-pressure level increase tomaintain same loudness, dB
F I G U R E 3 - 1 2
Short pulses of tones or noise are less audible than longer pulses as these graphs indicate.
The discontinuity of the 100- to 200-msec region is related to the integrating time
of the ear.
THE EAR AND THE PERCEPTION OF SOUND 61
occur, these consonants are genuine transients having durations on the
order of 5 to 15 msec. A glance at Fig. 3-12 tells you that transients this
short must be louder to be comparable to longer sounds. In the above
words, each consonant is not only much shorter than the rest of the
word, it is also at a lower level. Thus you need good listening conditions
to distinguish between such sets of words. Too much background noise
or too much reverberation can seriously impair the understandability of
speech because of the consonant problem.12
Audibility of Loudness Changes
Modern faders are of the composition type giving gradations in level
so small as to be inaudible. Wire-wound faders of early mixing consoles
produced discrete steps in level that could be audible. Steps of
5 dB were definitely audible, steps of 0.5 dB were inaudible, but these
steps cost too much to produce and 0.5 dB steps were not necessary.
Steps of 2 dB, an economic compromise, produced changes in signal
level that were barely detectable by an expert ear. Detecting differences
in intensity varies somewhat with frequency and also with
sound level.
At 1 kHz, for very low levels, a 3-dB change is the least detectable
by the ear, but at high levels the ear can detect a 0.25-dB change. A
very low level 35-Hz tone requires a 9-dB level change to be
detectable. For the important midfrequency range and for commonly
used levels, the minimum detectable change in level that the ear can
detect is about 2 or 3 dB. Making level changes in increments less than
these is usually unnecessary.
Pitch vs. Frequency
Pitch, a subjective term, is chiefly a function of frequency, but it is not
linearly related to it. Because pitch is somewhat different from frequency,
it requires another subjective unit—the mel. Frequency is a
physical term measured in cycles per second, now called Hertz.
Although a weak 1,000-Hz signal is still 1,000 Hz if you increase its
level, the pitch of a sound may depend on sound-pressure level. A reference
pitch of 1,000 mels has been defined as the pitch of a 1,000-Hz
tone with a sound-pressure level of 60 dB. The relationship between
62 CHAPTER THREE
pitch and frequency, determined by experiments with juries of listeners,
is shown in Fig. 3-13. Notice that on the experimental curve 1,000
mels coincides with 1,000 Hz, which tells us that the sound-pressure
level for this curve is 60 dB. It is interesting to note that the shape of
the curve of Fig. 3-13 is quite similar to a plot of position along the
basilar membrane of the inner ear as a function of frequency. This suggests
that pitch is related to action on this membrane, but much work
remains to be done to be certain of this.
Researchers tell us that there are about 280 discernible steps in
intensity and some 1,400 discernible steps in pitch that can be
3,500
3,000
2,500
2,000
1,500
1,000
500
0
20 100 500 1 kHz 5 kHz 10 kHz
Frequency - Hz
Pitch - mels
SPL = 60 dB
F I G U R E 3 - 1 3
Pitch (in mels, a subjective unit) is related to frequency (in Hz, a physical unit) according
to this curve obtained by juries of listeners. (After Stevens and Volkman.13)
THE EAR AND THE PERCEPTION OF SOUND 63
detected by the human ear. As changes in intensity and pitch are the
very stuff of communication, it would be interesting to know how
many combinations are possible. Offhand, it might seem that there
would be 280 × 1,400 = 392,000 combinations detectable by the ear.
This is overly optimistic because the tests were conducted by comparing
two simple, single-frequency sounds in rapid succession and bears
little resemblance to the complexities of commonly heard sounds.
More realistic experiments show that the ear can detect only about 7
degrees of loudness and 7 degrees of pitch or only 49 pitch-loudness
combinations. This is not too far from the number of phonemes (the
smallest unit in a language that distinguishes one utterance from
another) which can be detected in a language.
An Experiment
The level of sound affects the perception of pitch. For low frequencies,
the pitch goes down as the level of sound is increased. At high frequencies,
the reverse takes place—the pitch increases with sound level.
The following is an experiment within the reach of many readers that
was suggested by Harvey Fletcher. Two audio oscillators are required, as
well as a frequency counter. One oscillator is fed to the input of one
channel of a high-fidelity system, the other oscillator to the other channel.
After the oscillators have warmed up and stabilized, adjust the frequency
of the left channel oscillator to 168 Hz and that of the right
channel to 318 Hz. At low level these two tones are quite discordant.
Increase the level until the pitches of the 168-Hz and 318-Hz tones
decrease to the 150-Hz–300-Hz octave relationship, which gives a pleasant
sound. This illustrates the decrease of pitch at the lower frequencies.
An interesting follow-up would be to devise a similar test to show that
the pitch of higher frequency tones increases with sound level.
Timbre vs. Spectrum
Timbre has to do with our perception of complex sounds. The word is
applied chiefly to the sound of various musical instruments. A flute
and oboe sound different even though they are both playing A. The
tone of each instrument has its own timbre. Timbre is determined by
the number and relative strengths of the instrument’s partials. Tonal
quality comes close to being a synonym for timbre.
64 CHAPTER THREE
Timbre is another subjective term. The analogous physical term is
spectrum. A musical instrument produces a fundamental and a set of
partials (or harmonics) that can be analyzed with a wave analyzer and
plotted as in Fig. 1-15. Suppose the fundamental is 200 Hz, the second
harmonic 400 Hz, the third harmonic 600 Hz, etc. The subjective pitch
that the ear associates with our measured 200 Hz, for example, varies
slightly with the level of the sound. The ear also has its own subjective
interpretation of the harmonics. Thus, the ear’s perception of the overall
timbre of the instrument’s note might be considerably different from the
measured spectrum in a very complex way.
In listening to an orchestra in a music hall, the timbre you hear is
different for different locations in the seating area.14 The music is composed
of a wide range of frequencies, and the amplitude and phase of
the various components are affected by reflections from the various
surfaces of the room. The only way to get one’s analytical hands on
studying such differences is to study the sound spectra at different
locations. However, these are physical measurements, and the subjective
timbre still tends to slip away from us. The important point of this
section is to realize that a difference exists between timbre and
spectrum.
Localization of Sound Sources
The perception of a direction to a source of a sound is, at least partially,
the result of the amazing encoding function of the external ear,
the pinna. Sound reflected from the various ridges, convolutions, and
surfaces of the pinna combines with the unreflected (direct) sound at
the entrance to the auditory canal. This combination, now encoded
with directional information, passes down the auditory canal to the
eardrum and thence to the middle and inner ear and on to the brain for
interpretation.
This directional encoding process of the sound signal is indicated
in Fig. 3-14. The sound wavefront can be considered as a multiplicity
of sound rays coming from a specific source at a specific horizontal
and vertical angle. As these rays strike the pinna they are reflected
from the various surfaces, some of the reflections going toward the
entrance to the auditory canal. At that point these reflected components
combine with the unreflected (direct) component.
THE EAR AND THE PERCEPTION OF SOUND 65
For a sound coming
directly from the front of
the observer (azimuth
and vertical angle = 0°),
the “frequency response”
of the combination at the
opening of the ear canal
will be that shown in
Fig. 3-15. Instead of frequency
response, a curve
of this type is called a
transfer function because
it represents a vector
combination involving
phase angles.
For the sound at the
entrance of the ear canal
(Fig. 3-15) to reach the
eardrum, the auditory
canal must be traversed.
As the transfer function
at the entrance to the ear
canal (Fig. 3-15) and that
of the ear canal (Fig. 3-3) are combined, the shape of the resulting
transfer function impinging on the eardrum is radically changed. Figure
3-3 showed a typical transfer function of the ear canal alone. It is a
static, fixed function that does not change with direction of arrival of
the sound. The ear canal acts like a quarter-wave pipe closed at one
end by the eardrum exhibiting two prominent resonances.
The transfer function representing the specific direction to the
source of Fig. 3-15 combining with the fixed transfer function of the
ear canal (Fig. 3-3) gives the combined transfer function at the eardrum
of Fig. 3-16. The brain translates this to a perception of sound coming
from directly in front of the observer.2
The transfer function at the entrance to the ear canal (such as Fig.
3-15) is shaped differently for each horizontal and vertical direction.
This is how the pinna encodes all arriving sound enabling the brain to
yield different perceptions of direction. The sound arriving at the
Wavefront
F I G U R E 3 - 1 4
A wavefront of a sound can be considered as numerous
rays perpendicular to that wavefront. Such rays,
striking a pinna, are reflected from the various ridges
and convolutions. Those reflections directed to the
opening of the ear canal combine vectorially (according
to relative amplitudes and phases). In this way
the pinna encodes all sound falling on the ear with
directional information, which the brain decodes as a
directional perception.
66 CHAPTER THREE
20
15
10
5
0
5
10
15
0.1 0.2 0.5 1 2 5 10 20
Frequency - kHz
Related-sound pressure level - dB
F I G U R E 3 - 1 5
A measured example of the sound pressure (transfer function) at the opening of the ear
canal corresponding to sound arriving from a point immediately in front of the subject.
The shapes of such transfer functions vary with the horizontal and vertical angles at
which the sound arrives at the pinna. (After Mehrgardt and Mellert.2)
20
15
10
5
0
5
10
0.1 0.2 0.5 1 2 5 10 20
Frequency - kHz
Related-sound pressure level - dB
F I G U R E 3 - 1 6
The transfer function of Fig. 3-15 at the opening of the ear canal is altered to this shape
at the eardrum after being combined with the transfer function of the ear canal. In
other words, a sound arriving at the opening of the ear canal from a source directly in
front of the observer (Fig. 3-15) looks like Fig. 3-16 at the eardrum because it has been
combined with the characteristics of the ear canal itself (Fig. 3-3). The brain has no
trouble subtracting the fixed influence of the ear canal from every changing arriving
sound.
THE EAR AND THE PERCEPTION OF SOUND 67
eardrum is the raw material for all directional perceptions. The brain
neglects (sees through?) the fixed component of the ear canal and translates
the differently shaped transfer functions to directional perceptions.
Another more obvious directional function of the pinna is that of
forward-backward discrimination, which does not depend on encoding
and decoding. At the higher frequencies (shorter wavelengths), the
pinna is an effective barrier. The brain uses this front-back differentiation
to convey a general perception of direction.
A crucial question at this juncture is, “How about sounds arriving
in the median plane?” The median plane is a vertical plane passing
symmetrically through the center of the head and nose. Sources of
sound in this plane present identical transfer functions to the two ears.
The auditory mechanism uses another system for such localization,
that of giving a certain place identity to different frequencies. For
example, signal components near 500 and 8,000 Hz are perceived as
coming from directly overhead, components near 1,000 and 10,000 Hz
as coming from the rear.15 This is an active area of research that is
being continually refined.
The pinna, originally suspected of being only a useless vestigial
organ, turns out to be a surprisingly sophisticated sound directional
encoding mechanism.
Sound arriving from directly in front of an observer results in a
peak in the transfer function at the eardrum in the 2- to 3-kHz region.
This is the basis of the successful technique of old-time sound mixers
adding “presence” to a recorded voice by adding an equalization boost
in this frequency region. A voice can also be made to stand out from a
musical background by adding such a peak to the voice response.
Binaural Localization
Stereophonic records and sound systems are a relatively new development.
Stereo hearing has been around at least as long as man. Both are
concerned with the localization of the source of sound. In early times
some people thought that having two ears was like having two lungs or
two kidneys, if something went wrong with one the other could still
function. Lord Rayleigh laid that idea to rest by a simple experiment
on the lawn of Cambridge University. A circle of assistants spoke or
struck tuning forks and Lord Rayleigh in the center with his eyes
68 CHAPTER THREE
closed pointed to the
source of sound with
great accuracy, confirming
the fact that two ears
function together in binaural
localization.
Two factors are
involved, the difference
in intensity and the difference
in time of arrival
(phase) of the sound
falling on the two ears.
In Fig. 3-17 the ear nearest
the source receives a
greater intensity than the
far ear because the hard
skull casts a “sound shadow.” Because of the difference of distance to
the source, the far ear receives sound somewhat later than the near ear.
Below 1 kHz the phase (time) effect dominates while above 1 kHz the
intensity effect dominates. There is one localization blind spot. A listener
cannot tell whether sounds are coming from directly in front or
from directly behind because the intensity of sound arriving at each
ear is the same and in the same phase.
Another method of perception of direction comes into play in a relatively
small room. The sound reaches the person over a direct path
followed by many reflections from many different directions. The
sound that arrives first creates in the hearer the main perception of
direction. This has been called the law of the first wavefront.
Aural Harmonics: Experiment #1
This experiment, suggested by Craig Stark,16 can be performed easily
with your home high-fidelity system and two audio oscillators. Plug one
oscillator into the left channel and the other into the right channel, and
adjust both channels for an equal and comfortable level at some midband
frequency. Set one oscillator to 24 kHz and the other to 23 kHz without
changing the level settings. With either oscillator alone, nothing is heard
because the signal is outside the range of the ear. (He notes here, however,
that the dog might leave the room in disgust!) When both oscillators
Sound source
F I G U R E 3 - 1 7
Our binaural directional sense is dependent in part
on the difference in intensity and phase of the sound
falling on two ears.
THE EAR AND THE PERCEPTION OF SOUND 69
are feeding their respective channels, one at 24 kHz and the other at 23
kHz, a distinct 1,000-Hz tone is heard if the tweeters are good enough
and you are standing in the right place.
The 1,000-Hz tone is the difference between 24,000 and 23,000 Hz.
The sum, or 47,000 Hz, which even the dog may not hear even if it
were radiated, is another sideband. Such sum and difference sidebands
are generated whenever two pure tones are mixed in a nonlinear
element. The nonlinear element in the above experiment is the
middle and inner ear. In addition to the intermodulation products discussed
earlier, the nonlinearity of the ear generates new harmonics
that are not present in the sound falling on the eardrum.
Aural Harmonics: Experiment #2
The distortion introduced by the auditory system cannot be measured
by ordinary instruments. It is a subjective effect requiring a different
approach. Another demonstration of distortion in the ear can be
accomplished by the following method with the same equipment used
above, with the addition of a pair of headphones.
First, a 150-Hz tone is applied to the left earphone channel. If the
hearing mechanism were perfectly linear, no aural harmonics would
be heard as the exploratory tone is swept near the frequencies of the
second, third, and other harmonics. If it is nonlinear, the presence of
aural harmonics is indicated by the generation of beats. When 150 Hz
is applied to the left ear, and the exploratory tone of the right ear is
slowly varied about 300 Hz, the second harmonic is indicated by the
presence of beats between the two. If you change the exploratory oscillator
to a frequency around 450 Hz, the presence of a third harmonic
will be revealed by beats.
Experts have even estimated the magnitude of the harmonics by the
strength of such beats. The amount of distortion produced in the ear is
modest at lower levels but becomes appreciable at high levels. Running
the above experiment with tones of a higher level will make the
presence of aural harmonics even more obvious.
The Missing Fundamental
If tones such as 1,000, 1,200, and 1,400 Hz are reproduced together, a
pitch of 200 Hz is heard. This can be interpreted as the fundamental
with 1,000 Hz as the 5th harmonic, 1,200 Hz as the 6th harmonic, etc.
70 CHAPTER THREE
At one time this 200 Hz was called “periodicity pitch” but so-called
“pattern” theories dominate today. The auditory system is supposed to
recognize that the upper tones are harmonics of the 200 Hz and supplies
the missing fundamental that would have generated them. This is
a very interesting effect but explanations of it are highly controversial.
The Ear as an Analyzer
Listening to a good symphony orchestra in your favorite concert hall,
concentrate first on the violins. Now focus your attention on the clarinets,
then the percussion section. Next listen to a male quartet and
single out the first tenor, the baritone, the bass. This is a very remarkable
power of the human ear/brain combination. In the ear canal, all
these sounds are mixed together; how does the ear succeed in separating
them? The sea surface might be disturbed by many wave systems,
one due to local wind, one from a distant storm, and several wakes
from passing vessels. The eye cannot separate these, but this is essentially
what the ear is constantly doing with complex sound waves. By
rigorous training, a keen observer can listen to the sound of a violin
and pick out the various overtones apart from the fundamental!
The Ear as a Measuring Instrument
The emphasis on the distinction between physical measurements and
subjective sensation would seem to rule out the possibility of using the
ear for physical measurements. True, we cannot obtain digital readouts
by looking in someone’s eyes (or ears), but the ears are very keen
at making comparisons. People are able to detect sound-level differences
of about 1 dB throughout most of the audible band if the level is
reasonable. Under ideal conditions, a change of a third this amount is
perceptible. At ordinary levels, and for frequencies less than 1,000 Hz,
the ear can tell the difference between tones separated by as little as
0.3%. This would be 0.3 Hz at 100 Hz and 3 Hz at 1,000 Hz.
The eminent Harvey Fletcher17 has pointed out how the remarkable
keenness of the human ear saved the day in many of his researches in
synthesizing musical sounds. For example, in his study of piano sounds,
he initially postulated that all that is necessary is to measure the frequency
and magnitude of fundamental and harmonics and then combine
THE EAR AND THE PERCEPTION OF SOUND 71
them with the measured values of attack and decay. When this was done,
the listening jury unanimously voted that the synthetic sounds did not
sound like piano sounds but more like organ tones. Further study
revealed the long-known fact that piano strings are very stiff and have
properties of both solid rods and stretched strings. The effect of this is
that piano partials are nonharmonic! By correcting the frequencies of
what were assumed to be harmonics in integral multiples, the jury could
not distinguish between the synthetic piano sounds and the real thing.
The critical faculty of the ears of the jury in comparing sound qualities
provided the key.
An Auditory Analyzer: An Experiment
Knowledge of the ear’s filterlike critical bands leads to the tantalizing
idea of analyzing continuous noises such as traffic noises, underwater
background noises, etc., by using the ear instead of heavy and expensive
sound-analyzing gear. This must have occurred to Harvey
Fletcher, who first proposed the idea of critical bands, and to many
investigators in this field who have dealt with critical bands through
the years.
The general approach is illustrated in Fig. 3-18.18 A tape recording
of the noise to be analyzed is played back and mixed with a tone
from a variable-frequency oscillator. The combination is amplified
and listened to with a pair of headphones having a flat frequency
response. The oscillator is set, say, at 1,000 Hz and its output
V
Recording
of
sound
Osc.
Ampl.
F I G U R E 3 - 1 8
Equipment arrangement for using the critical bands of the human ear for sound
analysis.
72 CHAPTER THREE
adjusted until the tone is just hidden or masked by the noise. Only
the noise in the critical band centered on 1,000 Hz is effective in
masking the tone. If the noise is expressed in sound-pressure level
of a band 1 Hz wide, the voltage of the tone then corresponds to the
1-Hz sound-pressure level of the noise at the masked point. Adjusting
the voltage until the tone is just masked should yield one of the
points on our noise spectrum graph. For convenience, let us assume
that this voltmeter is calibrated in dB referred to some arbitrary base
such as 1 volt (dBv). Referring to Fig. 3-11, note that the critical
band centered on 1,000 Hz is 160 Hz wide. This can also be
expressed in decibels by taking log10160 = 22 dB; this 22 dB, representing
the width of the critical band as it does, must be subtracted
from the voltmeter reading in dB. This gives one point on the noise
spectrum graph. By repeating the process for other frequencies, a
series of points is obtained that reveal the shape of the noise spectrum.
If the recording and the entire measuring system (including
the observer’s ears) were calibrated, the absolute levels for the noise
spectrum could be obtained.
The important point here is that there is such a set of filters in our
head that could be put to such a task, not that this method will ever
replace a good sound level meter equipped with octave or one-third
octave filters. Surely human variables would far exceed sound-level
meter fluctuations from day to day, and what the observer eats for
breakfast has no effect on the sound-level meter, although it might
affect the dependability of the readings made with physiological
equipment.
Meters vs. the Ear
There still remains a great chasm between subjective judgments of
sound quality, room acoustics, etc., and objective measurements. Considerable
attention is being focused on the problem. Consider the following
descriptive words, which are often applied to concert-hall
acoustics19,20:
warmth clarity
bassiness brilliance
definition resonance
reverberance balance
THE EAR AND THE PERCEPTION OF SOUND 73
fullness of tone blend
liveness intimacy
sonority shimmering
What kind of an instrument measures warmth or brilliance? How
would you devise a test for definition? Progress, however, is being
made. Take definition for instance. German researchers have adopted
the term deutlichkeit, which literally means clearness or distinctness,
quite close to definition. It can be measured by taking the energy in an
echogram during the first 50 to 80 milliseconds and comparing it to
the energy of the entire echogram. This compares the direct sound and
early reflections, which are integrated by the ear, to the entire reverberant
sound. This relatively straightforward measurement of an
impulsive sound from a pistol or pricked balloon holds considerable
promise for relating the descriptive term definition to an objective
measurement. It will be a long time before all of these and a host of
other subjective terms can be reduced to objective measurements, but
this is a basic problem in acoustics and psychoacoustics.
There comes a time at which meter readings must give way to observations
by human subjects. Experiments then take on a new, subjective
quality. For example, in a loudness investigation, panels of listeners are
presented with various sounds, and each observer is asked to compare
the loudness of sound A with the loudness of B or to make judgments
in other ways. The data submitted by the jury of listeners are then subjected
to statistical analysis, and the dependence of a human sensory
factor, such as loudness, upon physical measurements of sound level is
assessed. If the test is conducted properly and sufficient observers are
involved, the results are trustworthy. It is in this way that we discover
that there is no linear relationship between sound level and loudness,
pitch and frequency, or between timbre and sound quality.
The Precedence Effect
Our hearing mechanism integrates sound intensities over short intervals
and acts somewhat like a ballastic measuring instrument. In simpler
terms, in an auditorium situation, the ear and brain have the
remarkable ability to gather all reflections arriving within about 50
msec after the direct sound and combine (integrate) them to give the
impression that all this sound is from the direction of the original
74 CHAPTER THREE
source, even though reflections from other directions are involved.
The sound energy integrated over this period also gives an impression
of added loudness.
It should not be too surprising that the human ear fuses all sounds
arriving during a certain time window. After all, our eyes fuse a series
of still pictures at the cinema, giving us the impression of continuous
movement. The rate of presentation of the still pictures is important;
there must be at least 16 pictures per second (62-millisecond interval)
to avoid seeing a series of still pictures or a flicker. Auditory fusion
works best during the first 20 or 30 milliseconds; beyond 50 to 80 milliseconds
discrete echoes dominate.
Haas21 set his subjects 3 meters from two loudspeakers arranged so
that they subtended an angle of 45 degrees, the observer’s line of symmetry
splitting this angle. The conditions were approximately anechoic.
The observers were called upon to adjust an attenuator until the sound
from the “direct” loudspeaker equaled that of the “delayed” loudspeaker.
He then proceeded to study the effects of varying the delay.
A number of researchers had previously found that very short
delays (less then 1 msec) were involved in our discerning the direction
of a source by slightly different times of arrival at our two ears. Delays
greater than this do not affect our directional sense.
As shown in Fig. 3-19, Haas found that in the 5 to 35 msec delay
range the sound from the delayed loudspeaker has to be increased
more than 10 dB over the direct before it sounded like an echo. This
is the precedence effect, or Haas effect. In a room, reflected energy
arriving at the ear within 35 msec is integrated with the direct sound
and is perceived as part of the direct sound as opposed to reverberant
sound. These early reflections increase the loudness of the
sound, and as Haas said, result in “...a pleasant modification of the
sound impression in the sense of broadening of the primary sound
source while the echo source is not perceived acoustically.”
The transition zone between the integrating effect for delays less
than 35 msec and the perception of delayed sound as discrete echo is
gradual, and therefore, somewhat indefinite. Some peg the dividing
line at a convenient 1⁄16 second (62 msec), some at 80 msec, and some at
100 msec beyond which there is no question about the discreteness of
the echo. In this book we will consider the first 30 msec as in Fig. 3-19,
the region of definite integration.
THE EAR AND THE PERCEPTION OF SOUND 75
Perception of Reflected Sound
In the preceding section, “reflected” sound was considered in a rather
limited way. A more general approach is taken in this section. It is interesting
that the loudspeaker arrangement Haas used was also used by
dozens of other researchers and that this is basically the familiar stereo
setup; two separated loudspeakers with the observer (listener) located
symmetrically between the two loudspeakers. The sound from one
loudspeaker is designated as the direct sound, that from the other, the
delayed sound (the reflection). The delay injected between the two signals
and their relative levels is adjustable. Speech is used as the signal.22
With the sound of the direct loudspeaker set at a comfortable level,
and with a delay of, say 10 ms, the level of the reflected, or delayed,
loudspeaker sound is slowly increased from a very low value. The
sound level of the reflection at which the observer first detects a difference
in the sound is the threshold of reflection detection. For levels
15
10
5
0
Echo level direct - dB
0 10 20 30 40 50
Echo delay - milliseconds
F I G U R E 3 - 1 9
The precedence effect (Haas effect) in the human auditory system. In the 5 to 35 msec
region, the echo level has to be about 10 dB higher than the direct sound to be discernible
as an echo. In this region, reflected components arriving from many directions
are gathered by the ear. The resulting sound seems louder, because of the reflections
and appears to come from the direct source. For delays 50 to 100 msec and longer
reflections are perceived as discrete echoes. (After Haas.21)
76 CHAPTER THREE
less than this, the reflection is inaudible; for levels greater than this,
the reflection is clearly audible.
As the reflection level is gradually increased above the threshold
value, a sense of spaciousness is imparted to the combined sound. This
sense of spaciousness prevails, even though the experiment is conducted
in an anechoic space. As the level of the reflection is increased about
10 dB above the threshold value, another change is noticed in the sound;
a broadening of the sound image and possibly a shifting of the image
toward the direct loudspeaker is now added to the increasing spaciousness.
As the reflection level is increased another 10 dB or so above the
image broadening threshold, another change is noted; discrete echoes are
heard.
This is all very interesting, but what practical value does it have?
Consider a specific example: a listening room in which recorded music
will be played. Figure 3-20 contains answers to the effect of sound
reflected from floor, ceiling, and walls being added to the direct sound
from the loudspeakers. Reflections below the threshold of perception are
unusable; reflections perceived as discrete echoes are also unusable. The
usable area is the unshaded area between those two threshold curves, A
and C. Simple calculations can give estimates of the level and delay of
any specific reflection, knowing the speed of sound, the distance traveled
and applying the inverse square law. Figure 3-20 gives the subjective
reactions the listener will probably have to the combination of any
reflection and the direct sound.
To assist in the “simple” calculations mentioned previously, the
following equations can be applied:
Reflection delay 
This assumes 100% reflection at the reflecting surface.
Reflection level at listening position  20 log
This assumes the inverse square propagation.
Occupational and Recreational Deafness
The hearing of workers in industry is now protected by law. The
higher the environmental noise, the less exposure allowed (Table 3-2).
direct distance, ft
 reflection distance, ft
(reflected path, ft) – (direct path, ft)
 1,130 ft/sec
THE EAR AND THE PERCEPTION OF SOUND 77
Researchers are trying to determine what noise exposure workers are
subjected to in different plants. This is not easy as noise levels fluctuate
and workers move about, but wearable dosimeters are often used to
integrate the exposure over the work day. Industries are hard pressed
to keep up with changes in regulations, let alone the installation of
noise shields around offending equipment and keeping ear plugs in or
ear muffs on the workers. Nerve deafness resulting from occupational
noise is recognized as a distinct health hazard.
20
10
0
10
20
30
40
0 20 40 60 80
Reflection level (redirect) - dB
Reflection delay - ms
Reflection
inaudible
Spaciousness
Image broadening
Echo
A
B
C
F I G U R E 3 - 2 0
The effects of lateral reflections on the perception of the direct sound in a simulated
stereo arrangement. These measurements were made in anechoic conditions, lateral
angles 45–90 degrees, with speech as the signal. (A) Absolute threshold of audibility
of the reflection. (B) Image shift/broadening threshold (A & B After Olive and Toole,22
and Toole.23) (C) Lateral reflection perceived as a discrete echo (After Meyer and Schodder,
24 and Lochner and Burger25).
78 CHAPTER THREE
It is especially bad when one works all
day in a high-noise environment, then
engages in motorcycle or automobile racing,
listens to a 400-watt stereo at high
level, or spends hours in a discotheque.
The professional audio engineer operating
with high monitoring levels is risking
irreparable injury to the basic tools of the
trade—his ears. As high-frequency loss
creeps in, the volume control is turned up
to compensate, and the rate of deterioration
is accelerated.
The key to conservation of hearing is
the audiogram. Comparing today’s audiogram
with earlier ones establishes the
trend; if downward, steps can be taken to
check it. The audiogram of Fig. 3-21,
which looks something like the Big Dipper
Table 3-2. OSHA permissible noise exposure
times.*
Sound pressure level, Maximum daily
dB, A-weighting, slow exposure
response hours
85 16
90 8
92 6
95 4
97 3
100 2
102 1.5
105 1
110 0.5
115 0.25 or less
*Reference: OSHA 2206 (1978)
0
20
40
60
80
Hearing loss - dB (ISO)
250 500 1,000 2,000 4,000 8,000
Frequency - Hz
Right
ear
Left
ear
F I G U R E 3 - 2 1
Audiograms showing serious loss centered on 4 kHz, presumably resulting from years of
exposure to high-level sound in the control room of a recording studio.
THE EAR AND THE PERCEPTION OF SOUND 79
constellation, is that of a 50-ish sound mixer in a recording studio. The
indications are that this loss, centered on 4 kHz, is the accumulation of
many years of listening to high-level sounds in the control room.
Summary
 The ear is sensitive enough to hear the tattoo of air particles on
the eardrums in the quietude of an anechoic chamber.
 The auditory canal, acting as a quarter-wave pipe closed at one
end by the eardrum, contributes an acoustical amplification of
about 10 dB, and the head diffraction effect produces another 10
dB near 3 kHz. These are vital speech frequencies.
 The leverage of the ossicle bones of the middle ear and the ratio
of areas of the eardrum and oval window successfully match the
impedance of air to the fluid of the inner ear.
 The Eustachian tube and round window provide pressure
release and equalization with atmospheric pressure.
 Waves set up in the inner ear by vibration of the oval window
excite the sensory hair cells, which are connected to the brain.
There is a “place effect,” the peak of hair cell agitation for higher
frequencies being nearer the oval window, and low frequencies
at the distal end.
 The area of audibility is bounded by two threshold curves, the
threshold of audibility at the lower extreme and the threshold
of feeling or pain at the loud extreme. Our entire auditory
experience occurs within these two extremes.
 The loudness of tone bursts decreases as the length of the burst
is decreased. Bursts greater than 200 msec have full loudness,
indicating a time constant of the ear at about 100 msec.
 Our ears are capable of accurately locating the direction of a
source in the horizontal plane. In a vertical median plane, however,
localization ability is less accurate.
80 CHAPTER THREE
 Pitch is a subjective term. Frequency is the associated physical
term, and the two have only a general relationship.
 Subjective timbre or quality of sound and the physical spectrum
of the sound are related, but not equal.
 The nonlinearity of the ear generates intermodulation products
and spurious harmonics.
 The Haas, or precedence, effect describes the ability of the ear to
integrate all sound arriving within the first 50 msec, making it
sound louder.
 Although the ear is not effective as a measuring instrument
yielding absolute values, it is very keen in comparing frequencies,
levels, or sound quality.
 Occupational and recreational noises are taking their toll in permanent
hearing loss. Definite precautionary steps to minimize
this type of environmentally caused deafness are recommended.
Endnotes
1Bloom, P.J., Creating source illusions by special manipulation, J. Audio Eng. So., 25 (1977)
560-565.
2Mehrgardt, S and V. Mellert, Transformation characteristics of the external human ear, J.
Acous. Soc. Am., 61, 6 (1977) 1567-1576.
3Moore, Brian C.J., An introduction to the psychology of hearing, New York, Academic Press
(1982).
4Pickles, James D., An introduction to the physiology of hearing, 2nd. ed., San Diego,
Academic Press (1988).
5Tobias, Jerry V. Ed., Foundations of modern auditory theory, Vol. 1, New York, Academic
Press (1970).
6Tobias, Jerry V., Ed., Foundations of modern auditory theory, Vol. 2, New York, Academic
Press (1972).
7Fletcher, H. and W.A. Munson, Loudness, its definition, measurement, and calculations, J.
Acous. Soc. Am., 5 (1933) 82-108.
8Robinson, D.W. and R.S. Dadson, A re-determination of the equal-loudness relations for
pure tones, British J. of Appl. Psychology., 7 (1956) 166-181. (Adopted by the International
Standards Organization as ISO-226.)
THE EAR AND THE PERCEPTION OF SOUND 81
9Toole, Floyd E., Loudness—Applications and implications for audio, dB Magazine, Part I: 7,
5 (May 1973) 7-30, Part II: 7, 6 (June 1973) 25-28.
10Zwicker, C.G., G. Flottorp, and S.S. Stevens, Critical bandwidths in loudness summation, J.
Acous. Soc. Am., 29, 5 (1957) 548-557.
11Moore, Brian C.J. and Brian Glasberg, Suggested formulae for calculating auditory filter
bandwidths and excitation patterns, J. Acous. Soc. Am., 74, 3 (1983) 750-753.
12Meyer, Erwin, Physical measurements in rooms and their meaning in terms of hearing
conditions, Proc. 2nd Int. Congr. on Acous. (1956) 59-68.
13Stevens, S.S. and J. Volkman, The relation of pitch to frequency: a revised scale, Am. J.
Psychology, 53 (1940) 329-353.
14Plomb, R. and H.J.M. Steeneken, Place dependence of timbre in reverberant sound fields,
Acustica, 28, 1 (1973) 50-59.
15Blauert, Jens, Spatial Hearing, Cambridge, MIT Press (1983).
16Stark, Craig, The sense of hearing, Stereo Review, (September 1969) 66, 71-74.
17Fletcher, Harvey, The ear as a measuring instrument, J. Audio Eng. Soc., 17, 5 (1969) 532-
534.
18Everest, F. Alton, The filters in our ears, Audio, 70, 9 (1986) 50-59.
19Schroeder, M.R., D. Gottlob, and K.F. Siebrasse, Comparative study of European concert
halls: correlation of subjective presence with geometric and acoustic parameters, J. Acous.
Soc. Am., 56, 4 (1974) 1195-1201.
20Hawkes, R.J. and H. Douglas, Subjective experience in concert auditoria, Acustica, 28, 5
(1971) 235-250.
21Haas, Helmut, The influence of a single echo on the audibility of speech, J. Audio Eng. Soc.,
20, 2 (1972) 146-159. (This is an English translation from the German by Dr. Ingr. K.P.R.
Ehrenberg of Haas’ original paper in Acustica, 1, 2 (1951).
22Olive, S.E. and F.E. Toole, The detection of reflections in typical rooms, J. Audio Engr. Soc.
37 (1989) 539-553.
23Toole, F.E., Loudspeakers and rooms for stereophonic sound reproduction, Proc. AES 8th
International Conference, Washington, D.C., 3-6 May, 1990, pp 71-91.
24Meyer, E. and G.R. Schodder, On the influence of reflected sounds on directional
localization and loudness of speech, Nachr. Akad. Wiss., Göttingen, Math., Phys., Klasse
IIa, 6 (1952) 31-42.
25Lochner, J.P.A. and J.F. Burger, The subjective masking of short time delayed echoes by their
primary sounds and their contribution to the intelligibility of speech, Acustica, 8 (1958)
1-10.
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Practical acoustic problems are invariably associated with people,
buildings, rooms, airplanes, automobiles, etc. These can generally
be classified either as problems in physics (sound as a stimulus) or
problems in psychophysics (sound as a perception), and often as both.
Acoustical problems can be very complex in a physical sense, for
example, thousands of reflected components might be involved or
obscure temperature gradients might bend the sound in such a way as
to affect the results. When acoustical problems involve human beings
and their reactions, “complexity” takes on a whole new meaning.
Don’t be discouraged if you want a practical understanding of acoustics,
but your background is in another field, or you have little technical
background at all. The inherent complexity of acoustics is pointed out
only to justify going back to the inherent simplicity of sound in a free field
as a starting point in the study of other types of practical sound fields.
Free Sound Field: Definition
Sound in a free field travels in straight lines, unimpeded and undeflected.
Unimpeded sound is sound that is unreflected, unabsorbed,
Sound Waves in
the Free Field
4
83
C H A P T E R
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84 CHAPTER FOUR
undiffracted, unrefracted, undiffused, and not subjected to resonance
effects. These are all hazards that could (and do) face a simple ray of
sound leaving a source.
Free space must not be confused with cosmological space. Sound
cannot travel in a vacuum; it requires a medium such as air. Here, free
space means any air space in which sound acts as though it is in the
theoretical free space. Limited free space can even exist in a room
under very special conditions.
Sound Divergence
The point source of Fig. 4-1 radiates sound at a fixed power. This
sound is of uniform intensity (power per unit area) in all directions.
The circles represent spheres having radii in simple multiples. All of
the sound power passing through the small square area at radius d also
passes through the areas at 2d, 3d, 4d, etc. This increment of the total
sound power traveling in this single direction is spread over increasingly
greater areas as the radius is increased. Intensity decreases with
distance. As the area of a sphere is 4πr2, the area of a small segment on
the surface of the sphere also varies as the square of the radius. Doubling
the distance from d to 2d reduces the intensity to 14, tripling the
distance reduces the intensity to 19, and quadrupling the distance
reduces intensity to 116. Intensity of sound is inversely proportional to
the square of the distance in a free field.
Intensity of sound (power per unit area) is a difficult parameter to
measure. Sound pressure is easily measured. As intensity is proportional
to the square of sound pressure, the inverse square law (for
intensity) becomes the inverse distance law (for sound pressure). In
other words, sound pressure varies inversely as the first power of the
distance. In Fig. 4-2, the sound-pressure level in decibels is plotted
against distance. This illustrates the basis for the common and very
useful expression, 6 dB per doubling of the distance that, again,
applies only for a free field.
Examples: Free-Field Sound Divergence
When the sound-pressure level L1 at distance d1 from a point source
is known, the sound-pressure level L2 at another distance d2 can be
calculated from:
SOUND WAVES IN THE FREE FIELD 85
L2  L1  20 log , decibels (4-1)
In other words, the difference in sound-pressure level between two
points that are d1 and d2 distance from the source is:
L2  L1  20 log , decibels (4-2)
For example, if a sound-pressure level of 80 dB is measured at 10 ft,
what is the level at 15 ft?
d2d1
d2d1
In the solid angle shown, the same sound energy is distributed over spherical surfaces
of increasing area as d is increased. The intensity of sound is inversely proportional to
the square of the distance from the point source.
d
2d
3d
4d
F I G U R E 4 - 1
86 CHAPTER FOUR
Solution:
20 log 10/15  3.5 dB; the level is 80  3.5  76.5 dB.
What is the sound-pressure level at 7 ft?
Solution:
20 log 10/7 3.1 dB, and level is 80  3.1  83.1 dB.
All this is for a free field in which sound diverges spherically, but
this procedure may be helpful for rough estimates even under other
conditions.
If a microphone is 5 feet from an enthusiastic soprano and the VU
meter in the control room peaks +6, moving the microphone to 10 feet
would bring the reading down approximately 6 dB. The word
“approximately” is important. The inverse square law holds true only
for free field conditions. The effect of sound energy reflected from
The inverse square law of sound intensity becomes the inverse distance law for sound
pressure. This means that sound-pressure level is reduced 6 dB for each doubling of
the distance.
0
10
20
30
Relative sound-pressure level, dB
1 2 4 8 16 32
Distance
double
6 dB
Distance units
F I G U R E 4 - 2
SOUND WAVES IN THE FREE FIELD 87
walls would be to make the change for a doubling of the distance
something less than 6 dB.
An awareness of the inverse square law is of distinct help in estimating
acoustical situations. For instance, a doubling of the distance
from 10 to 20 feet would, for free space, be accompanied by the same
sound-pressure level decrease, 6 dB, as for a doubling from 100 to 200
feet. This accounts for the great carrying power of sound outdoors.
Inverse Square in Enclosed Spaces
Free fields exist in enclosed spaces only in very special and limited
circumstances. The reflections from the enclosing surfaces affect the
way sound level decreases with distance. No longer does the inverse
square law or the inverse distance law describe the entire sound field.
For example, assume that there is an installed loudspeaker in an
enclosed space that is capable of producing a sound-pressure level of
100 dB at a distance of 4 ft. As shown in the graph of Fig. 4-3, free field
Even in an enclosed space the inverse square law is followed close to the source. By
definition, the critical distance is that distance at which the direct sound pressure is
equal to the reverberant sound pressure.
100
90
80
70
Sound-pressure level - dB
4 10 20 30 50 100
Distance - ft
Inverse square
(6 dB/dist. double)
Critical
distance
Reverberant
sound level
Direct sound
F I G U R E 4 - 3
88 CHAPTER FOUR
conditions exist close to the loudspeaker. This means that spherical
divergence prevails in this limited space, and reflections from the surfaces
are of negligible comparative level. Moving away from the loudspeaker,
the effects of sound reflected from the surfaces of the room
begin to be effective. At the critical distance the direct and the reflected
sound are equal. The critical distance may be taken as a rough singlefigure
description of the acoustics of the environment.
In the region very close to the loudspeaker, the sound field is in
considerable disarray. The loudspeaker, at such close distances, can
in no way be considered a point source. This region is called the near
field. Only after moving several loudspeaker dimensions away from it
can significant measurements be made in the far field.
Hemispherical Propagation
True spherical divergence implies no reflecting surfaces at all. Tied to
this earth’s surface as we are, how about hemispherical sound propagation
over the surface of this planet? Estimates made by the very convenient
“6 dB per distance double” rule are only rough approximations.
Reflections from the surface of the earth outdoors usually tend to
make the sound level with distance something less than that indicated
by the 6 dB per distance double. The reflective efficiency of the earth’s
surface varies from place to place. Note the sound level of a sound at
10 ft and again at 20 ft from the source. The difference between the two
will probably be closer to 4 dB than 6 dB. For such outdoor measurements
the distance law must be taken at “X dB (4?, 5?) per distance
double.” There is also the effect of general environmental noise that
can influence the measurement of specific sound sources.
Speech, music, and noise are common in that they are within the
experience of everyone. Noise is also a common thread that runs
through speech and music. Speech sounds are but modulated noise.
Noise is a close companion to every musical instrument. The highest
skill of every musician must be exerted to minimize such incidental
noises as thumps, scratches, and wheezes. The close relationship of
speech, music, and noise is made evident in this chapter.
The Voice System
One of the many amazing things about the human body is the high
degree of efficiency associated with the multiple use of organic systems.
The functions of eating, breathing, and speaking all take place in
relative simultaneous harmony. We can eat, breathe, talk practically at
the same time through the interworking of muscle action and valves,
without food going down the wrong hatch. If we sometimes try to do
too many things at once, the system is momentarily thwarted, and we
agonize as a bit of food is retrieved from the wrong pipe.
Artificial Larynx
Noise that contains energy over a wide range of constantly shifting frequencies,
phases, and amplitudes can be shaped even into speech.
Speech, Music,
and Noise
5
89
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
90 CHAPTER FIVE
Sometimes people lose their voices. Perhaps the vocal cords are paralyzed,
or the larynx was removed surgically. For these people, the
Western Electric Company offers a prosthetic device, which when
held against the throat, produces pulses of sound that simulate the
sounds produced by the natural vocal cords as they interrupt the air
stream. This battery-operated device even has a pitch control for controlling
“voice” pitch. Then the tongue, lips, teeth, nasal passages, and
throat perform their normal function of molding the pulsed noise into
words. Even if the overall effect has a somewhat duck-like quality, it
enables the user to speak by shaping the noise appropriately.
Sound Spectrograph
An understanding of speech sounds is necessary to understand how
the sounds are produced. Speech is highly variable and transient in
nature, comprising energy chasing up and down the three-dimensional
scales of frequency, sound level, and time. It takes the sound
spectrograph to show all three on the same flat surface such as the
pages of this book. Examples of several commonly experienced
sounds revealed by the spectrograph are shown in Fig. 5-1. In these
spectrographs, time progresses horizontally to the right, frequency
increases from the origin upward, and the sound level is indicated
roughly by the density of the trace—the blacker the trace, the more
intense the sound at that frequency and at that moment of time. Random
noise on such a plot shows up as a gray, slightly mottled rectangle
as all frequencies in the audible range and all intensities are
represented as time progresses. The snare drum approaches random
noise at certain points, but it is intermittent. The “wolf whistle”
opens on a rising note followed by a gap, and then a similar rising
note that then falls in frequency as time goes on. The police whistle
is a tone, slightly frequency modulated. Each common noise has its
spectrographic signature that reveals the very stuff that characterizes
it.
The human voice mechanism is capable of producing many
sounds other than speech. Figure 5-2 shows a number of these as
revealed by sound spectrograms. It is interesting to note that harmonic
trains appear on a spectrogram as more or less horizontal lines
spaced vertically in frequency. These are particularly noticeable in
the trained soprano’s voice and the baby’s cry, but traces are evident
SPEECH, MUSIC, AND NOISE 91
Miscellaneous sounds
Snare drum
Telephone bell
Wolf whistle
Police whistle
Riffling cards
Bubbles blown through water
F I G U R E 5 - 1
Sound spectrographic recordings of common sounds. Time progresses to the right, the
vertical scale is frequency, and the intensity of components is represented by the intensity
of the trace. AT&T Bell Laboratories.
92 CHAPTER FIVE
in other spectrograms. The following discussion owes much to the
clear presentation of Flanagan.1
Sound Sources for Speech
The artificial larynx is based on the fact that there are really two more
or less independent functions in the generation of speech sounds: the
sound source and the vocal system. In general, it is a series flow as pictured
in Fig. 5-3A, in which the raw sound is produced by a source
and subsequently shaped in the vocal tract. To be more exact, there are
really three different sources of sound to be shaped by the vocal tract
as indicated in Fig. 5-3B. First, there is the one we naturally think of—
the sounds emitted by the vocal cords. These are formed into the
voiced sounds. They are produced by air from the lungs flowing past
the slit between the vocal cords (the glottis), which causes the cords to
vibrate. The air stream, broken into pulses of air, produces a sound
that can almost be called periodic, that is, repetitive in the sense that
one cycle follows another.
The second source of sound is that made by forming a constriction
at some point in the vocal tract with the teeth, tongue, or lips and forcing
air through it under high enough pressure to produce significant
turbulence. Turbulent air creates noise. This noise is shaped by the
vocal tract to form the fricative sounds of speech such as the consonants
f, s, v, and z. Try making these sounds, and you will see that
high-velocity air is very much involved.
The third source of sound is produced by the complete stoppage of
the breath, usually toward the front, a building up of the pressure, and
then the sudden release of the breath. Try speaking the consonants k,
p, and t, and you will sense the force of such plosive sounds. They are
usually followed by a burst of fricative or turbulent sound. These three
types of sounds—voiced, fricative, and plosive—are the raw sources
that are shaped into the words we casually speak without giving a
thought to the wonder of their formation.
Vocal Tract Molding of Speech
The vocal tract can be considered as an acoustically resonant system.
This tract, from the lips to the vocal cords, is about 6.7 in (17 cm) long.
Its cross-sectional area is determined by the placement of the lips, jaw,
tongue, and velum (a sort of trapdoor that can open or close off the
SPEECH, MUSIC, AND NOISE 93
Voice sounds
Song—trained soprano voice
Cry—baby
Laugh
Snore
Cough
Gargling water
F I G U R E 5 - 2
Sound spectrograms of human sounds other than speech. AT&T Bell Laboratories.
94 CHAPTER FIVE
nasal cavity) and varies from zero to about 3 sq in (20 sq cm). The nasal
cavity is about 4.7 in (12 cm) long and has a volume of about 3.7 cu in
(60 cu cm). These dimensions are mentioned because they have a bearing
on the resonances of the vocal tract and their effect on speech
sounds.
Formation of Voiced Sounds
If the symbolic boxes of Fig. 5-3 are elaborated into source spectra and
modulating functions, we arrive at something everyone in audio is
interested in—the spectral distribution of energy in the voice. We also
get a better understanding of the aspects of voice sounds that contribute
to the intelligibility of speech in reverberation, noise, etc. Figure 5-4
shows the steps in producing voiced sounds. First, there is the sound
Sound
source
A
Vocal
tract
shaping
Speech
sounds
Vocal
cord
vibration
Constriction
turbulence
B
Vocal
tract
shaping
Speech
sounds
Plosive
F I G U R E 5 - 3
(A) The human voice is produced through the interaction of two essentially independent
functions, a sound source and a time-varying-filter action of the vocal tract.
(B) The sound source can be broken down into vocal-cord vibration for voiced sounds,
the fricative sounds resulting from air turbulence, and the plosive sounds.
SPEECH, MUSIC, AND NOISE 95
produced by the vibration of the vocal cords, pulses of sound having a
fine spectrum that falls off at about 10 dB per octave as frequency is
increased as shown in Fig. 5-4A. The sounds of the vocal cords pass
through the vocal tract, which acts as a filter varying with time. The
humps of Fig. 5-4B are due to the acoustical resonances, called formants
of the vocal pipe, which is open at the mouth end and essentially
closed at the vocal cord end. Such an acoustical pipe 6.7 inches
long has resonances at odd quarter wavelengths, and these peaks occur
at approximately 500, 1,500, and 2,500 Hz. The output sound, shaped
by the resonances of the vocal tract, is shown in Fig. 5-4C. This applies
to the voiced sounds of speech.
Formation of Unvoiced Sounds
Unvoiced sounds are shaped in a similar manner as indicated in Fig.
5-5. Unvoiced sounds start with the distributed, almost random-noiselike
spectrum of the turbulent air as fricative sounds are produced.
The distributed spectrum of Fig. 5-5A is generated near the mouth end
of the vocal tract, rather than the vocal cord end; hence, the resonances
of Fig. 5-5B are of a somewhat different shape. Figure 5-5C shows the
sound output shaped by the time-varying filter action of Fig. 5-5B.
Putting It All Together
The voiced sounds, originating in vocal cord vibrations, the unvoiced
sounds, originating in turbulences, and plosives, which originate near
the lips, go together to form all of our speech sounds. As we speak, the
A
Amplitude
Sound source
vocal cords
Slope
10 dB/octave
Frequency
Amplitude
Frequency
Vocal tract
time-varying filter
Amplitude
Frequency
Sound
output
B C
F I G U R E 5 - 4
Sound spectrograms of human sounds other than speech. AT&T Bell Laboratories.
96 CHAPTER FIVE
formant resonances shift about in frequency as the lips, jaw, tongue,
and velum change position to shape the desired words. The result is
the unbelievable complexity of human speech evident in the spectrogram
of Fig. 5-6. Information communicated via speech is a pattern of
frequency and intensity shifting rapidly with time. Notice that there is
little speech energy above 4 kHz in Fig. 5-6, nor (which does not show)
below 100 Hz. Now it’s understandable why the presence filter peaks
in the 2- to 3-kHz region; that is where the pipes resonate!
Synthesized Speech
Mechanical speaking machines date back to 1779, when Kratzenstein
of St. Petersburg constructed a set of acoustical resonators to emulate
the human mouth. These were activated with reeds such as those of a
mouth organ. He was able to produce reasonably recognizable vowel
sounds with the contraption. Wolfgang von Kempelen of Vienna did a
much better job in 1791, which Wheatstone later improved upon. This
machine used a bellows to supply air to a leather tube that was manipulated
by hand to simulate mouth action and included an “S” whistle,
a “SH” whistle, and a nostril cutoff valve. After experimenting with a
copy of Kempelen’s machine in boyhood, Alexander Graham Bell
patented a procedure for producing speech in 1876. One important
precursor of the modern digital devices for synthesizing speech was
A
Amplitude
Frequency
Sound source
fricative turbulence
Amplitude
Frequency
Vocal tract
time-varying filter
Amplitude
Frequency
Sound
output
B C
F I G U R E 5 - 5
A diagram of the production of unvoiced fricative sounds such as f, s, v, and z. (A) The distributed spectrum
of noise due to air turbulence resulting from constrictions in the vocal tract. (B) The time-varying filter action
of the vocal tract. (C) The output sound resulting from the filter action of the distributed sound of (A).
SPEECH, MUSIC, AND NOISE 97
the analog Voder from Bell Laboratories that was demonstrated at the
World Fairs in New York (1939) and in San Francisco (1940). It took a
year to train operators to play the machine to produce simple, but recognizable,
speech.
Digital Speech Synthesis
Techniques for storing human speech in computer memory and playing
it back under specified, fixed conditions are widely used. Electrical
machines of this type now talk to us in the form of language translators,
talking calculators, spelling machines, as well as telephone-information
services. We will be seeing (rather, hearing) a stream of other answerback
applications of this technique in the days ahead, including both
storage and recall, and true speech synthesis.
It is interesting to note that to program a computer to talk, a model
of speech production is necessary and that the models of Figs. 5-3, 5-4,
and 5-5 have been applied in just this way. Figure 5-7 shows a diagram
of a digital synthesis system. A random-number generator produces the
digital equivalent of the s-like sounds for the unvoiced components. A
counter produces pulses simulating the pulses of sound of the vocal
cords for the voiced components. These are shaped by time-varying
digital filters simulating the ever-changing resonances of the vocal
tract. Special signals control each of these to form digitized speech,
which is then changed to analog form in the digital-to-analog converter.
6000
5000
4000
3000
2000
1000
0
0 0.5 1.0 1.5 2.0 Seconds
Should we chase those young o u- -t-law c- -o w- - -boy - - - s
Frequency - Hz
F I G U R E 5 - 6
Sound spectrogram of a sentence spoken by a male voice. AT&T Bell Laboratories.
98 CHAPTER FIVE
Other applications of digital speech synthesis include voice-recognition
systems for “hands-free” typing and computer programs for
recording that automatically adjust the intonation of a vocalist who
might otherwise be singing sharp or flat.
Directionality of Speech
Speech sounds do not have the same strength in all directions. This is
due primarily to the sound shadow cast by the head. A question arises
as to just how such directionality can be measured. Should a sound
source be placed in the mouth? Well, the mouth itself is a continuous
source of speech sounds, so why not use these speech sounds for the
measurement of directionality? That is what Kuttruff has done with
the results shown in Fig. 5-8.2 Because speech sounds are highly variable
and extremely complex, careful averaging is necessary to give an
accurate measure of directional effects.
The horizontal directional effects, shown in Fig. 5-8A, show only a
modest directional effect of about 5 dB in the 125- to 250-Hz band.
Randomnumber
generator
“Unvoiced”
Noise amplitude
control signal
Digital
formant
filter
Digital
to
analog
Pitch
pulse
generator
“Voiced”
Pitch
frequency
control
signal
Voice
intensity
control
signal
Filter
control
signal
F I G U R E 5 - 7
A digital system for synthesizing speech. Note the similarity to the models of Figs. 5-3, 5-4, and 5-5.
SPEECH, MUSIC, AND NOISE 99
This is to be expected because the head is
small compared to wavelengths of 4.5 to 9
feet associated with this frequency band.
There are significant directional effects,
however, for the 1,400- to 2,000-Hz band.
For this band, which contains important
speech frequencies, the front-to-back difference
is about 12 dB.
In the vertical plane, Fig. 5-8B, the 125-
to 250-Hz band shows about 5 dB front-toback
difference again. For the 1,400- to
2,000-Hz band, the front-to-back difference
is also about the same as the horizontal
plane, except for the torso effect. The discrimination
against high speech frequencies
picked up on a lapel microphone
becomes obvious in Fig. 5-8B, although the
measurements were not carried to angles
closer to 270 degrees.
Music
Musical sounds are extremely variable in
their complexity and can range from a near
sine-wave form of a single instrument or
voice to the highly complex mixed sound
of a symphony orchestra. Each instrument
and each voice has a different tonal texture
for each note. Many musical instruments,
such as the violin, viola, cello, or bass, produce
their tones by vibration of strings. On
a stretched string, the overtones are all
exact multiples of the fundamental, the
lowest tone produced. These overtones
may thus be called harmonics. If the string
is bowed in the middle, odd harmonics are
emphasized because the fundamental and
odd harmonics have maximum amplitude
A
0
5 dB
10 dB
270 90
1400–2000 Hz
125–250 Hz
180
90
5 dB
10 dB
0 180
1400–2000 Hz
125–250 Hz
270
B
F I G U R E 5 - 8
Human voice directionality measured using the voice
as a sound source. (A) Front-to-back directional
effects of about 12 dB are found for the important
speech frequencies. (B) In the vertical plane, the
front-to-back directional effects for the 1,400- to
2,000-Hz band are about the same as for the horizontal
plane. By permission of Heinrich Kuttruff and
Applied Science Publishers Ltd., London.
100 CHAPTER FIVE
there. Because the even harmonics have nodes in the center of the
string, they will be subdued if bowed there. The usual place for bowing
is near one end of the strings, which gives a better blend of even
and odd harmonics. There is a problem with the seventh harmonic
because it belongs to a different musical family. By bowing 17 of the
distance from one end, this harmonic is decreased.
The harmonic content of the E and G notes of a violin are displayed
graphically in Fig. 5-9. Harmonic multiples of the higher E tone are
spaced wider and hence have a “thinner” timbre. The lower frequency
tone, on the other hand, has a closely spaced spectral distribution and
a richer timbre. The small size of the violin relative to the low frequency
of the G string means that the resonating body cannot produce
a fundamental at as high a level as the higher harmonics. The harmonic
content and spectral shape depend on the shape and size of the
resonating violin body, the type and condition of the wood, and even
the varnish. Why there are so few superb violins among the many good
ones is a problem that has not yet been solved completely.3
Relative amplitude, dB
Tone: E
(659 Hz)
100 300 500 1 kHz 3 kHz 5 kHz 10 kHz
100 300 500 1 kHz 3 kHz 5 kHz 10 kHz
Tone: G
(196 Hz)
Frequency - Hz
F I G U R E 5 - 9
Harmonic content of open strings of the violin. The lower tones sound richer because of
the closely packed harmonics.
SPEECH, MUSIC, AND NOISE 101
Wind Instruments
Resonances in the three-dimensional room are discussed in detail in
Chap. 15. In many musical instruments, resonance in pipes or tubes
must be considered primarily one dimensional. Standing-wave effects
are dominant in pipes. If air is enclosed in a narrow pipe closed at both
ends, the fundamental (twice the length of the pipe) and all its harmonics
will be formed. Resonances are formed in a pipe open at only
one end at the frequency at which the pipe length is four times the
wavelength, and results in odd harmonics. Wind instruments form
their sounds this way; the length of the column of air is continuously
varied, as in the slide trombone, or in jumps as in the trumpet or
French horn, or by opening or closing holes along its length as in the
saxophone, flute, clarinet, and oboe.
The harmonic content of several wind instruments is compared to
that of the violin in the spectrograms of Fig. 5-10. Each instrument has
its characteristic timbre as determined by the number and strength of
its harmonics and by the formant shaping of the train of harmonics by
the structural resonances of the instrument.
Nonharmonic Overtones
Harvey Fletcher tried to synthesize piano sounds.4 It was emphasized
that piano strings are stiff strings and vibrate like a combination of solid
rods and stretched strings. This means that the piano overtones are not
strictly harmonic. Bells produce a wild mixture of overtones, and the
fundamental is not even graced with that name among specialists in the
field. The overtones of drums are not harmonically related, although
they give a richness to the drum sound. Triangles and cymbals give
such a mixture of overtones that they blend reasonably well with other
instruments. Nonharmonic overtones produce the difference between
organ and piano sounds and give variety to musical sounds in general.
Dynamic Range of Speech and Music
In the concert hall, a full symphony orchestra is capable of producing
some very loud sounds when the score says so, but also soft, delicate
passages. Seated in the audience, one can fully appreciate this grand
sweep of sound due to the great dynamic range of the human ear. The
dynamic range between the loudest and the softest passage will be on
the order of 60 to 70 dB. To be effective, the soft passages must still be
102 CHAPTER FIVE
above the ambient background noise in the hall, hence the emphasis
on adequate structural isolation to protect against traffic and other outside
noises, and precautions to ensure that air-handling equipment
noise is low.
For those not present in the music hall, AM or FM radio, television,
magnetic recordings, or disc recordings must suffice. These conventional
media are unable to handle the full dynamic range of the orchestra.
Noise at the lower extreme and distortion at the upper extreme
introduce limitations. In the case of broadcast media, there are the
added regulatory restrictions prohibiting interference with adjacent
channels.
Digital audio has brought some major revisions of our thinking in
regard to dynamic range and signal-to-noise ratio. The dynamic range
in a digital system is directly related to the range of binary digits (bits).
Kilocycles
harmonicnumber
7
6
5
4
3
2
1
0
16
14
12
10
8
6
4
2
Flute Clarinet Oboe Trumpet Trumpet
mouthpiece
Violin
Tone: middle A (440 cycles)
F I G U R E 5 - 1 0
Spectrogram comparison of the harmonic content of woodwind instruments and the violin as middle A (440
Hz) is played. The differences displayed account for the difference in timbre of the different instruments.
AT&T Bell Laboratories.
SPEECH, MUSIC, AND NOISE 103
The theoretical 96-dB dynamic range provided by the 16-bit digital
system is staggering to someone steeped in the traditional techniques.
At last there is a recording system that handles the concert hall
dynamic range reasonably well. The commercial compact disc (CD) is
capable of carrying 74 minutes of full-fidelity music with a 96 dB signal-
to-noise ratio. The digital audio cassette is another worthy addition
to the recording/playback arsenal, as well as digital audio tape
(DAT) systems. Digital techniques have transferred dynamic range
limitations from the medium to the concert hall on the one hand and
the playback environment on the other.
However, in recent years the audio community has grown dissatisfied
with the quantization noise and “graininess” of 16-bit digital
audio. New 24-bit formats are on the rise, such as “Super Audio CD”
and “Audio DVD.” In addition, professional audio mastering engineers
are now working in 24-bit resolution to avoid the audibility of
digital artifacts resulting from lower-resolution processing.
Power in Speech and Music
In learning more about the various signals to be handled, one must
consider the peak power of various sources. For speech, the average
power is only about 10 microwatts, but peaks might reach a milliwatt.
Most of the power of speech is in the low frequencies, with 80 percent
below 500 Hz, yet there is very little power below 100 Hz. On the other
hand, the small amount of power in the high frequencies determines
the intelligibility of speech and thus is very important because that is
where the consonants are. The peak power of various musical instruments
is listed in Table 5-1.
Number of binary digits Dynamic range, dB
4 24
8 48
12 72
16 96
24 144
104 CHAPTER FIVE
Frequency Range of Speech and
Music
It is instructive to compare the frequency range
of the various musical instruments with that of
speech. This is best done graphically. Figure 5-11
includes the ranges only of the fundamental
tones, and not of the harmonic tones of the
instruments. The very low piano and organ
notes, which are below the range of audibility of
the ear, are perceived by their harmonics. Certain
high-frequency noise accompanying musical
instruments is not included, such as reed noise
in woodwinds, bowing noise of strings, and key
clicks and thumps of piano and percussion
instruments.
Future Dynamic-Range
Requirements
If the peak instantaneous sound levels and noise
thresholds are regarded as determining dynamic
range requirements, much greater ranges are required. Fiedler’s study6
has shown that a dynamic range of up to 118 dB is necessary for subjectively
noise-free reproduction of music (see Fig. 5-12). He considered the
peak instantaneous sound level of various sources, as shown at the top of
the figure, and the just-audible threshold for white noise added to the
program source when the listener is in a normal listening situation, as
shown at the bottom of the figure. He used musical performances of high
peak levels in a quiet environment and a very simple recording setup.
The results are summarized in Fig. 5-12. The signal-to-noise ratio offered
by a 16-bit PCM (pulse code modulation) system is shown to be inadequate
for all but the piano solo. Future developments will undoubtedly
require greater dynamic range than that offered by 16-bit digital systems.
Auditory Area
The frequency range and the dynamic range of speech, music, and all
others sounds places varying demands on the human ear. The auditory
area back in Fig. 3-8 describes the capability of the ear. Both speech
Table 5-1 Power of Musical Sources.5
Instrument Peak, power (watts)
Full orchestra 70
Large bass drum 25
Pipe organ 13
Snare drum 12
Cymbals 10
Trombone 6
Piano 0.4
Trumpet 0.3
Bass saxophone 0.3
Bass tuba 0.2
Double bass 0.16
Piccolo 0.08
Flute 0.06
Clarinet 0.05
French horn 0.05
Triangle 0.05
Harp
GuitarAccordion
Timpani
Marimba-xylophoneChimes
FlutePiccolo
B Soprano saxophoneE Alto saxophoneB Tenor saxophoneE Baritone saxophone
B Soprano clarinetB Bass saxophone
E Alto clarinetB Bass clarinet
Oboe
BassoonEnglish horn
E Contrabass sarrusophone
B Cornet, trumpet, fluegelhornE Alto (mellophone)F french hornB Trombone and euphonium (3v.)
B Bass tuba (3v.)E Bass tuba (3v.)
ViolaViolin
Cello
Bass
Alto voiceSoprano voice
Bass voiceBaritone voice
A0
B0
C1
D1
E1
F1
G1
A1
B1
C2
D2
E2
F2
G2
A2
B2
D3
E3
F3
G3
A3
B3
C4
D4
E4
F4
G4
A4
B4
C5
D5
E5
F5
G5
A5
B5
C6
D6
E6
F6
G6
A6B6
C7
D7
E7
F7
C3
G7
A7
B7
C7
Cyclespersecond
Pianokey board(numbersof keys)
27.500
30.868
32.703
41.203
43.654
48.999
55.000
61.735
65.406
73.416
82.407
87.307
97.999
110.00
123.47
130.81
146.83
164.81
174.61
196.00
220.00
246.94
261.63
293.66
329.63
349.23
392.00
440.00
493.88
523.25
587.33
659.26
698.46
880.00
783.99
987.77
1046.5
1174.7
1318.5
1396.9
1568.0
1760.0
1975.5
2093.0
2349.3
2637.0
2793.8
3136.0
3520.0
4186.0
3951.1
29.135
34.048
38.891
46.249
51.913
58.270
69.296
77.782
92.499
116.54
103.83
138.59
155.56
185.00
207.65
233.08
277.18
311.13
369.99
415.30
466.16
622.25
554.37
739.99
830.61
932.33
1108.7
1244.5
1480.0
1661.2
2217.5
2489.0
2960.0
3322.4
3729.3
36.708
1864.7
1
3
4
6
8
9
1113151618202123252728
303233353739
404244454749
5254
57
56
59
61
63
64
66
68
69
73
75
71
78
76
80
81
83
85
87
88
51
FIGURE 5-11
The audible frequency range of various musical instruments and voices. Only the fundamental tones are included; the partials go much
higher. The very low piano and organ notes are perceived largely through their partials. Not shown are the many high-frequency incidental
noises produced by the instruments. C. G. Conn, Ltd., Oak Brook, Illinois.
105
106 CHAPTER FIVE
and music use only a portion of this area. The portion of the auditory
area used in speech is shown by the shaded area of Fig. 5-13. This kidney-
bean shape is located centrally in the auditory area, which makes
sense because neither the extremely soft or extremely loud sounds, nor
sounds of very low or very high frequency, are used in common speech
sounds. The speech area of Fig. 5-13 is derived from long-time averages,
and its boundaries should be fuzzy to represent the transient
excursions in level and frequency. The speech area, as represented,
shows an average dynamic range of about 42 dB. The 170- to 4,000-Hz
frequency range covers about 4.5 octaves.
The music area of Fig. 5-14 is much greater than the speech area of
Fig. 5-13. Music uses a much greater proportion of the full auditory
area of the ear. Its excursions in both level and frequency are correspondingly
greater than speech, as would be expected. Here again,
long-time averages are used in establishing the boundaries of the music
area, and the boundaries really should be fuzzy to show extremes. The
music area shown has a dynamic range of about 75 dB and a frequency
range of about 50 to 8,500 Hz. This frequency span is about 7.5 octaves,
compared to the 10-octave range of the human ear. High-fidelity standards
demand a much wider frequency range than this, and rightly so.
Without the averaging process involved in establishing the speech and
Typical classical
symphony
Percussive
classical
100
50
0
Sound pressure level, dB
Piano
solo
103
With audience
113
W/o audience
113
Close
mic
113
Close
mic
122
90
100 105 109 118
Audience noise
13-dB eqv.
Mic
noise
8-dB eqv.
Hearing acuity
4-dB eqv.
F I G U R E 5 - 1 2
Fiedler’s study6 has shown that a dynamic range of up to 118 dB is necessary for subjectively
noise-free reproduction of music.
SPEECH, MUSIC, AND NOISE 107
music areas, both the dynamic range and the frequency range would be
greater to accommodate the short-term transients that contribute little
to the overall average, but are still of great importance.
Noise
The word “signal” implies that information is being conveyed. How can
noise be considered an information carrier? An enthusiastic “Bronx
cheer” raspberry conveys considerable information about the perpetrator’s
attitude toward someone or something. Noise is the basic part of
such a communication, modulated in just the right way. Interrupting
noise to form dots and dashes is another way to shape noise into communication.
We will also see how a decaying band of noise can give
information on the acoustical quality of a room. There are types of noise
that are undesirable. Sometimes it is difficult to tell whether it is the
120
100
80
60
40
20
0
Sound - pressure level, dB
20 50 100 300 500 1 k 3 k 5 k 10 k 20 k
Frequency - Hz
42 dB
170–4,000 Hz
Speech
F I G U R E 5 - 1 3
The portion of the auditory region utilized for typical speech sounds.
108 CHAPTER FIVE
unpleasant thing we call noise or a carrier of information. The noise of
an automobile conveys considerable information on how well it is running.
One person’s noise might be someone else’s communication. A
high-fidelity system can produce some beautiful sounds deemed very
desirable by the owner, but to a neighbor they might not be considered
beautiful at all. Sometimes it isn’t easy to distinguish between information
and noise. The same sound can be both. Society establishes limits
to keep objectionable noise to a minimum while ensuring that information-
carrying sounds can be heard by those who need to hear them.
Noise—The Good Kind
A good kind of noise? Defining noise as unwanted sound fits the system
noise considered previously, but noise is becoming an increasingly
important tool for measurements in acoustics as discussed in
Chaps. 5 and 7. This good noise is not necessarily different from the
120
100
80
60
40
20
0
Sound - pressure level, dB
20 50 100 300 500 1 k 3 k 5 k 10 k 20 k
50–8,500 Hz
75 dB
Music
Frequency - Hz
F I G U R E 5 - 1 4
The portion of the auditory region utilized for typical music sounds.
SPEECH, MUSIC, AND NOISE 109
bad noise interfering with our listening to a favorite recording, it is just
that the noise is put to a specific use.
In acoustical measurements, the use of pure tones is often very difficult
to handle while a narrow band of noise centered on the same frequency
would make satisfactory measurements possible. For example,
if a studio is filled with a pure tone signal of 1,000 Hz from a loudspeaker,
a microphone picking up this sound will have an output that
varies greatly from position to position due to room resonances. If,
however, a band of noise one octave wide centered at 1,000 Hz were
radiated from the same loudspeaker, the level from position to position
would tend to be more uniform, yet the measurement would contain
information on what is happening in the region of 1,000 Hz. Such
measuring techniques make sense because we are usually interested in
how a studio or listening room reacts to the very complex sounds
being recorded or reproduced, rather than to steady, pure tones.
Random Noise
Random noise is generated in any electrical
circuit and minimizing its effect often
becomes a very difficult problem. Heavy
ions falling back on the cathode of a
thermionic vacuum tube produce noise of a
relatively high amplitude and wide spectrum.
Furthermore, the introduction of some
gas molecules into the evacuated space will
produce even more noise. Today a random
noise generator is made with a silicon diode
or other solid-state device followed by an
amplifier, voltmeter, and attenuator.
In Fig. 5-15 a pure sine wave and a
random noise signal are compared as
viewed on a cathode ray oscilloscope.
The regularity of the one is in stark contrast
to the randomness of the other. If the
horizontal sweep of the oscilloscope is
expanded sufficiently and a snapshot is
taken of the random noise signal, it
would appear as in Fig. 5-16.
Amplitude




Time
Pure
sine
wave
Random
noise
F I G U R E 5 - 1 5
Cathode ray oscillograms of a pure sine wave and of
random noise. Random noise may be considered
made up of sine waves, which are continually shifting
in amplitude, phase, and frequency.
110 CHAPTER FIVE
Noise is said to be purely random in character if it has a “normal”
or “Gaussian” distribution of amplitudes.7 This simply means that if
we sampled the instantaneous voltage at a thousand equally spaced
times, some readings would be positive, some negative, some greater,
some smaller, and a plot of these samples would approach the familiar
Gaussian distribution curve of Fig. 5-17.
Time
F I G U R E 5 - 1 6
A section of the random noise signal of Fig. 5-15 spread out in time. The nonperiodic
nature of a noise signal is evident, the fluctuations are random.
0.4
0.3
0.2
0.1
0
Fraction of samples
3 2 1 0 1 2 3
Instantaneous noise voltage
F I G U R E 5 - 1 7
The proof of randomness of a noise signal lies in the sampling of instantaneous voltage,
say, at 1,000 points equally spaced in time and plotting the results. The familiar bellshaped
Gaussian distribution curve results if the noise is truly random.
SPEECH, MUSIC, AND NOISE 111
White and Pink Noise
References to white noise and pink noise are common and sometimes
confusing. What is the difference? White noise is analogous to white
light in that the energy of both is distributed uniformly throughout the
spectrum. In other words, white noise energy exhibits a flat distribution
of energy with frequency (Fig. 5-18A).
White light sent through a prism is broken down into a range of
colors. The red color is associated with the longer wavelengths of
light, that is, light in the lower frequency region. Pink noise is noise
having higher energy in the low frequencies. In fact, pink noise has
come to be identified specifically as noise exhibiting high energy in
the low-frequency region with a specific downward slope of 3 dB per
octave (Fig. 5-18C). There is a practical reason for this specific slope.
These two colorful terms arose because there are two types of
spectrum analyzers in common use. One is the constant bandwidth
B
40
20
0
20
40
3 dB per octave
3 dB per octave
White noise
Pink noise filter characteristic
A
C
20 Hz 100 Hz 1 kHz 10 kHz 20 kHz
Frequency - Hz
dB
F I G U R E 5 - 1 8
Random noise has constant energy per Hz. If the spectrum of random noise is measured
(white) with a wave analyzer of fixed bandwidth, the resulting spectrum will be flat
with frequency as in A. If measured with an analyzer whose passband width is a given
percentage of the frequency to which it is tuned, the spectrum will slope upward at 3
dB per octave, as in B. By processing the white noise spectrum of A with a filter that
slopes downward at 3 dB per octave, such as in C, a flat response results when constant
percentage bandwidth filters are used such as octave or one-third octave filters. In measuring
a system, pink noise is applied to the input and, if the system is flat, the read
out response will be flat if one-third octave filters, for example, are used.
112 CHAPTER FIVE
analyzer, which has a passband of fixed width as it is tuned throughout
the spectrum. One well-known analyzer of this type has a bandwidth
of 5 Hz. If white noise with its flat spectrum were measured
with a constant-bandwidth analyzer, another flat spectrum would
result because the fixed bandwidth would measure a constant energy
throughout the band shown in Fig. 5-18A.
Another very popular and convenient spectrum analyzer is the
constant percentage bandwidth analyzer. In this instrument the bandwidth
changes with frequency. An example of this is the one-thirdoctave
analyzer, commonly used because its bandwidth follows
reasonably well with the critical bandwidth of the human ear throughout
the audible frequency range. At 100 Hz the bandwidth of the onethird-
octave analyzer is only 23 Hz but at 10 kHz the bandwidth is
2,300 Hz. Obviously, it intercepts much greater noise energy in a onethird
octave band centered at 10 kHz than one centered at 100 Hz.
Measuring white noise with a constant-percentage analyzer would
give an upward-sloping result with a slope of 3 dB/octave, as shown in
Fig. 5-18B.
In audio-frequency measurements, the desired characteristic of many
instruments, rooms, etc. is a flat response throughout the frequency
range. Assume that the system to be measured has a characteristic almost
flat with frequency. If this system is excited with white noise and measured
with the very convenient constant-percentage analyzer, the result
would have an upward slope of 3 dB/octave. It would be far more desirable
if the measured result would be close to flat so that deviations from
flatness would be very apparent. This can be accomplished by using a
noise with a downward slope of 3 dB/octave. By passing white noise
through a filter, such as that of Fig. 5-19, such a downward sloping excitation
noise can be obtained. Such a noise, sloping downward at 3
dB/octave, is called pink noise. A close-to-flat system (amplifier, room)
excited with this pink noise would yield a close-to-flat response, which
would make deviations from flatness very obvious. For such reasons
pink noise is here to stay.
Signal Distortion
Our discussion of the various signals encountered in audio is
incomplete without at least an acknowledgment of what can happen
SPEECH, MUSIC, AND NOISE 113
to the precious signal in passing through transducers, amplifiers,
and various forms of signal processing gear. Here is an incomplete
list:8,9,10
Bandwidth limitation If the passband of an amplifier cuts lows or
highs, the signal output is different from the input. If the scratch filter
reduces record surface noise, the overall effect can be improved, but
basically the signal itself is the poorer for it.
Nonuniform response Peaks and valleys within the passband also
alter the signal waveshape.
Distortions in time If tape travels across the head at any other
than the recording speed, the frequency components are shifted up or
down in frequency. If there are slow or fast fluctuations in that speed,
wow and flutter are introduced and the signal is degraded.
Phase distortion Any phase shifts introduced upset the time relationship
between signal components.
Dynamic distortion A compressor or expander changing the
original dynamic range of a signal is a form of distortion.
Crossover distortion In class-B amplifiers, in which the output
devices conduct for only half of the cycle, any discontinuities near
zero output result in what is called crossover distortion.
Nonlinear distortion If an amplifier is truly linear, there is a
one-to-one relationship between input and output. Feedback helps
to control nonlinear tendencies. The human ear is not linear. When
5600
White
noise
in
Pink
noise
out
2700
820
270
1.0
0.316
0.1
0.05
F I G U R E 5 - 1 9
A simple filter for changing white noise to pink noise. It changes random noise of constant
energy per Hz to pink noise of constant energy per octave. Pink noise is useful in
acoustical measurements utilizing analyzers having passbands with bandwidth of a constant
percentage of the center frequency. General Radio Company1
114 CHAPTER FIVE
a pure tone is impressed on the ear, harmonics can be heard. If two
loud tones are presented simultaneously, sum and difference tones
are generated in the ear itself; and these tones can be heard as can
their harmonics. A cross-modulation test on an amplifier does
essentially the same thing. If the amplifier (or the ear) were perfectly
linear, no sum or difference tones or harmonics would be generated.
The production within the component of frequency elements that
were not present in the input signal is the result of nonlinear distortion.
Transient distortion Strike a bell and it rings. Apply a steep wavefront
signal to an amplifier and it might ring a bit too. For this reason,
signals such as piano notes are difficult to reproduce. Tone burst test
signals are an attempt to explore the transient response characteristics
of equipment, as are square waves. Transient intermodulation (TIM)
distortion, slew induced distortion, and other sophisticated measuring
techniques have been devised to evaluate transient forms of distortion
in systems.
Harmonic Distortion
The harmonic distortion method of evaluating the effects of circuit
nonlinearities is probably the oldest and the most universally
accepted method. It certainly is the easiest to understand. In this
method the device under test is driven with a sine wave of high purity.
If the signal encounters any nonlinearity, the output waveshape is
changed, i.e., harmonic components appear that were not in the pure
sine wave. A spectral analysis of the output signal is made to measure
these harmonic distortion products. The most revealing method is to
use a wave analyzer having a constant passband width of, say, 5 Hz,
which can be swept through the audio spectrum. Figure 5-20 shows
illustrated results of such a measurement. The wave analyzer is first
tuned to the fundamental, fo  1 kHz, and the level is set for a convenient
1.00 volt. The wave analyzer is then tuned to the 2 kHz region
until the 2fo second harmonic is found. The voltmeter, which is a part
of the analyzer, reads 0.10 volt. The third harmonic at 3 kHz gives a
reading of 0.30 volt, the fourth a reading of 0.05 volt and so on up the
frequency scale. Beyond 6fo  6 kHz no measurable components were
found after diligent search. The data are then assembled in Table 5-2.
SPEECH, MUSIC, AND NOISE 115
fo
1.0
0.8
0.6
0.4
0.2
0
Voltage
2fo 3fo 4fo 5fo 6fo
1.00
0.10
0.30
0.05
0.20
0.025
Frequency
F I G U R E 5 - 2 0
A distorted periodic wave is measured with a constant bandwidth wave analyzer. The
fundamental, fo, is set for some reference voltage, taken here as 1.00 volt. Tuning the
wave analyzer to 2fo the second harmonic amplitude is measured as 0.10 volt. The wave
analyzer is tuned successively to 3fo, 4fo and other harmonics yielding amplitudes of
each harmonic as shown. The root-mean-square of the harmonic voltages is then compared
to the 1.00 volt fundamental to find the total harmonic distortion expressed in
percentage.
Table 5-2 Harmonic Distortion Products.
Harmonic Volts (Volts)2
2nd Harmonic 2fo 0.10 0.01
3rd Harmonic 3fo 0.30 0.09
4th Harmonic 4fo 0.05 0.0025
5th Harmonic 5fo 0.20 0.04
6th Harmonic 6fo 0.025 0.000625
7th and higher (negligible)
Sum 0.143125
Fundamental f
o
1 kHz, 1.00 volt amplitude
116 CHAPTER FIVE
The total harmonic distortion (THD) may then be found from the
expression:
THD  100 (5-1)
where e2, e3, e4…en  voltages of 2nd, 3rd, 4th, etc. harmonics
eo  voltage of fundamental
In Table 5-2 the harmonic voltages have been squared and added
together reducing Eq. 5-1 to:
THD  100
 37.8%
A total harmonic distortion of 37.8% is a
very high distortion that would make any
amplifier sound horrible on any type of
signal, but the example has served our
purpose in illustrating just what THD is
and one method for obtaining it.
Wave analyzers are expensive, highprecision
instruments that are rarely
found in equipment service shops. A
very simple adaptation of the THD
method is, however, widely used. Consider
Fig. 5-20 again. If the fo fundamental
were adjusted to some known value
and then a notch filter were adjusted to fo
essentially eliminating it, only the harmonics
would be left. Measuring these
harmonics all lumped together with an
RMS (root mean square) meter, accomplishes
what was done in the square root
portion of Eq. 5-1. Comparison of this
RMS measured value of the harmonic
components with that of the fundamental
and expressing it as a percentage
gives the total harmonic distortion.
0.143 125 

1.00
(e2)2  (e3)2  (e4)2… (en)2 

eo
Undistorted sine wave
5% THD 10% THD
Positive peaks clipped
This is what the harmonics look like on the cathode
ray oscilloscope to the same scales as above.
F I G U R E 5 - 2 1
Cathode ray oscillograms show pure, undistorted
sine wave, which is applied to the input of an
amplifer that clips the positive peaks of the signal.
The appearance of the clipped sine wave for 5% and
10% total harmonic distortion is shown. If the fundamental
is rejected by a notch filter, the summed
harmonics appear as shown.
SPEECH, MUSIC, AND NOISE 117
In Fig. 5-21 an undistorted sine wave is
sent through an amplifier, which clips
positive peaks. On the left, the flattening
of the positive peaks with 5% THD is evident,
and shown below is what the combined
total of all the harmonic products
look like with the fundamental rejected.
On the right is shown the effect of greater
clipping to yield 10% THD. Figure 5-22
shows what happens when the sine wave
passing through the amplifier is symmetrically
clipped on both positive and negative
peaks. The combined distortion
products for symmetrical clipping have a
somewhat different appearance, but they
measure the same 5% and 10% THD.
In all of this exercise keep in mind that
consumer-type power amplifiers commonly
have specifications listing total harmonic
distortion nearer 0.05% rather than 5% or
10%. In a series of double-blind subjective
tests Clark found that 3% distortion was
audible on different types of sounds.11 With
carefully selected material (such as a flute
solo) detecting distortions down to 2% or
1% might be possible. A distortion of 1%
with sine waves is readily audible.
Endnotes
1Flanagan, J. L, Voices of Men and Machines, J. Acous. Soc. Am., 51, 5, Part 1, May 1972,
1375–1387.
2Kuttruff, Heinrich, Room Acoustics, London, Applied Science Publishers, Ltd., 1979, 16, 17.
3Hutchins, Carleen M., and Francis L. Fielding, Acoustical Measurement of Violins, Physics
Today, July 1968, 35–41. Contains extensive bibliography.
4Fletcher, Harvey, The Ear As a Measuring Instrument, J. Audio Eng. Soc., 17, 5, October
1969, 532–534.
5Sivian, L.J., H.K. Dunn, and S.D. White, Absolute Amplitudes and Spectra of Certain Musical
Instruments and Orchestras, J. Acous. Soc. Am., 2, 3, January 1931, 330–371.
Undistorted sine wave
Symmetrically clipped
5% THD 10% THD
This is what the harmonics look like on the cathode
ray oscilloscope to the same scale as above.
F I G U R E 5 - 2 2
Cathode ray oscillograms show a pure, undistorted
sine wave, which is applied to the input of an
amplifer which clips both positive and negative peaks
in a symmetrical fashion. The appearance of the
clipped sine wave is shown for 5% and 10% total harmonic
distortion. The appearance of the harmonics
alone, with the fundamental filtered out, is also
shown.
118 CHAPTER FIVE
6Fiedler, Louis D., Dynamic-Range Requirements for Subjectively Noise-Free Reproduction
of Music, J. Aud. Eng. Soc., 30, 7/8, 1982, 504–511.
7Peterson, Arnold P.G. and Ervin E. Gross, Jr., Handbook of Noise Measurements, Concord,
Mass., GenRad. 7th Ed., 1974.
8Cabot, Richard C., A Comparison of Nonlinear Distortion Measurement Methods, Presented
at the 66th convention of the Audio Eng. Soc., Los Angeles, 1980, Preprint 1638.
9Buff, Paul C., Perceiving Audio Noise and Distortion, Recording Eng./Prod., 10, 3, June
1979, 84.
10Jung, Walter G., Mark L. Stephens, and Craig C. Todd, An Overview of SID and TIM, Audio,
Part I—63, 6 (June 1979), 59–72
Part II—63, 7 (July 1979), 38–47
11Clark, David, High-Resolution Subjective Testing Using a Double-Blind Comparator, J.
Audio Eng. Soc., 30, 5 (May 1982), 330–338.
In the early days of sound recording, signal storage was a major problem.
The final recording was laid down directly, without benefit of
stop-and-go recording of portions that could be patched together later.
This had the advantage of minimizing the number of recording generations,
but the format was a stringent one with little latitude for artistic
enhancement.
With the introduction of high-quality signal storage (magnetic tape
or digital memory) many creative decisions, normally reserved for the
recording session, were moved to the mix-down session. This opens
up the opportunity of making quality enhancements in the mix-down
session, long after the recording session is finished and forgotten.
These quality enhancements are often made with filters of one kind or
another. Is there a short traffic rumble on take 6 of the recording? A
high-pass filter might cure it. Did that narrator’s dentures contribute
an occasional high-frequency hiss? Run it through the “de-esser.”
How about that congenital 3 dB sag in the response at 4 kHz? Easy,
add a 3 dB peak of appropriate width with the parametric equalizer,
and so on.
The introduction of integrated circuits in the 1960s made signal
processing equipment lighter, more compact, and less expensive. The
coming of the digital revolution made it possible to routinely accomplish
sophisticated signal processing tasks, heretofore impractical or
Analog and Digital
Signal Processing
6
119
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
120 CHAPTER SIX
impossible. This chapter is a very brief overview of both analog and
digital sound processing principles and practice, especially the principles.
Resonance
The man on the stage is doing the old trick of breaking the wine glass
with sound. Instead of doing it with the diva’s voice, he holds the goblet
in front of the loudspeaker, which shatters it as a high-intensity
tone is emitted. The secret lies in the brief preparation made before the
audience was assembled. At that time he placed a small coin in the
goblet and held it in front of the loudspeaker as the frequency of the
sine generator was varied at a low level. He carefully adjusted the generator
until the frequency was found at which the coin danced wildly
in the glass. During the demonstration no tuning was necessary, a blast
of sound at this predetermined frequency easily shattered the glass.
The wild dancing of the coin in the glass in the preliminary adjustment
indicated that the excitation frequency from the loudspeaker was
adjusted to the natural frequency or resonance of the goblet. At that
frequency of resonance a modest excitation resulted in very great
vibration of the glass, exceeding its breaking
point. As shown in Fig. 6-1, the amplitude
of the vibration of the glass changes
as the frequency of excitation is varied,
going through a peak response at the frequency
of resonance, fo.
Such resonance effects appear in a
wide variety of systems: the interaction of
mass and stiffness of a mechanical system,
such as a tuning fork, or the acoustical resonance
of the air in a bottle, as the mass of
the air in the neck of the bottle reacts with
the springiness of the air entrapped in the
body of the bottle. See Helmholtz resonators,
Chap. 9.
Resonance effects are also dominant in
electronic circuits as the inertia effect of
an inductance reacts with the storage
Amplitude
Frequency
fo
F I G U R E 6 - 1
The amplitude of vibration of any resonant system is
maximum at the natural frequency or resonant frequency
(f) and is less at frequencies below and above
that frequency.
ANALOG AND DIGITAL SIGNAL PROCESSING 121
effect of a capacitance. Figure 6-2 shows
the symbols for inductance (L), commonly
a coil of wire, and capacitance (C), commonly
made of sheets of conducting material
separated by nonconducting sheets.
Energy can be stored in the magnetic field
of an inductance as well as in the electrical
charges on the plates of a capacitance. The
interchange of energy between two such
storage systems can result in a resonance
effect. Perhaps, the simplest example of
this is a weight on a spring.
Figure 6-3 shows two forms in which
an inductance and a capacitance can
exhibit resonance. Let us assume that an
alternating current of constant amplitude,
but varying frequency is flowing in the
parallel resonant circuit of Fig. 6-3A. As
V
A
Voltage
L
C
Freq.
V
Voltage
L C
Freq.
Generator
B
F I G U R E 6 - 3
A comparison of (A) parallel resonance and (B) series resonance. For a constant alternating
current flowing, the voltage across the parallel resonant circuit peaks at the resonance
frequency while that of the series resonant circuit is a minimum.
L
C
F I G U R E 6 - 2
Symbols for inductance (L) and capitance (C).
122 CHAPTER SIX
the frequency is varied, the voltage at the terminals reaches a maximum
at the natural frequency of the LC system, falling off at lower and
higher frequencies. In this way the typical resonance curve shape is
developed. Another way of saying this is that the parallel resonant circuit
exhibits maximum impedance (opposition to the flow of current)
at resonance.
Figure 6-3B illustrates the series resonant arrangement of an
inductance L and a capacitance C. As the alternating current of constant
magnitude and varying frequency flows in the circuit, the voltage
at the terminals describes an inverted resonance curve in which
the voltage is minimum at the natural frequency and rising at both
lower and higher frequencies. It can also be said that the series resonant
circuit presents minimum impedance at the frequency of resonance.
Filters
The common forms of filters are the low-pass filter, the high-pass filter,
the band-pass filter, and the band-reject filter as illustrated in Fig.
6-4. Figure 6-5 shows how inductors and capacitors may be arranged
in numerous ways to form very simple high- and low-pass filters. Filters
of Figure 6-5C will have much sharper cut-offs than the simpler
ones in (A) and (B).
There are many other highly specialized filters with specific and
unusual features. With such filters, a wideband signal such as speech
or music can be altered at will.
Response
Low-pass filter High-pass filter Band-pass filter Band-reject filter
Frequency
F I G U R E 6 - 4
Basic response shapes for the low-pass, high-pass, band-pass, and band-reject filters.
ANALOG AND DIGITAL SIGNAL PROCESSING 123
Adjustable filters can be readily shifted to any frequency within their
design band. One type is the constant bandwidth filter which offers the
same bandwidth at any frequency. For example, a spectrum analyzer may
have a 5-Hz bandwidth whether it is tuned to 100 Hz or 10,000 Hz, or any
other frequency within its operating band. An even more widely used
adjustable filter offers a pass band-width that is a constant percentage of
the frequency to which it is tuned. The 1⁄3-octave filter is such a device. If
it is tuned to 125 Hz the 1⁄3-octave bandwidth is 112 to 141 Hz. If it is tuned
to 8,000 Hz the 1⁄3-octave bandwidth is 7,079 Hz to 8,913 Hz. The bandwidth
is about 23% of the frequency to which it is tuned in either case.
Active Filters
Active filters depend on integrated circuits for their operation. An
integrated circuit can have many hundreds of components in a small
enclosure. Their fabrication depends on growing transistors and
A
High-pass filter Low-pass filter
B
C
F I G U R E 6 - 5
High-pass and low-pass filters of the simplest form. The filters in (C) will have sharper
cut-off than the others.
124 CHAPTER SIX
resistors on a semiconductor wafer and interconnecting the components
by an evaporated metal pattern. Great circuit complexity can
be compressed into unbelievably small space in this way.
A low-pass filter assembled from inductors and capacitors in the
old-fashioned way is shown in Fig. 6-6A. Another low-pass filter
based in an integrated circuit is shown in Fig. 6-6B. The four small
resistors and two small capacitors plus the integrated circuit illustrate
the space-saving advantage of the active filter.
It is interesting that the feedback capacitor C1 in Fig. 6-6B has the
electrical effect of an inductance.
Analog vs. Digital Filters
Filters can be constructed in analog or digital form. All the filters discussed
to this point have been of the analog type and applied widely in
equalizers. By adjusting the values of the resistors, inductors, and capacitors
any type of analog filter can be constructed to achieve almost any
frequency and impedance matching characteristic desired.
A
C1


B
F I G U R E 6 - 6
Two low-pass filters, (A) traditional analog type, (B) an active filter utilizing an integrated
circuit.
ANALOG AND DIGITAL SIGNAL PROCESSING 125
Digitization
A digital filter contains no such physical components as inductors and
capacitors. It is basically a computer program that operates on a sample
of the signal. This process is described in Fig. 6-7. The incoming analog
signal is represented in Fig. 6-7A. Through a multiplication or modulation
process, the analog signal of Fig. 6-7A is combined with the sam-
A
B
C
D
F I G U R E 6 - 7
(A) the analog signal, (B) digitizing pulses, (C) the digitized analog signal resulting
from the modulation of (A) with (B). Application of a sample-and-hold circuit to (D)
which completes the quantization process reducing the analog samples to discrete values
suitable for storage in memory.
126 CHAPTER SIX
pling pulses of Fig. 6-7B. These pulses, in effect, break down the analog
signal into a series of very brief samples having amplitudes equal to the
instantaneous value of the signal amplitudes, as shown in Fig. 6-7C.
This process is called digitization. With no loss of information, energy
between the sampling points is discarded. The sampling rate must be at
least twice the highest frequency of interest. If a sampling rate less than
this amount is used, spurious signals are generated.
Quantization
It is now necessary to convert the samples of Fig. 6-7C to discrete values
that can be stored in a computer. This is done by a “sample-andhold”
circuit so that the amplitude of each digitized pulse is converted
into discrete values suitable for computer storage. The sample-andhold
circuit forces the amplitude of the sample to have a constant
value throughout the sample period. This process is essentially a stepping
backward through a digitized signal, sample by sample, subtracting
from each sample some large proportion of the sample before it.
The resulting samples are thus mainly changes in the signal sample.
The closer the spacing of the digitized samples, the more accurately
the analog signal is represented. However, a restraining influence on
increasing the number of samples is that more computer memory is
required to store the data. The calculations required are intensive in
multiplication and accumulation operations.
Digital Filters
Digital filters can be made without benefit of inductors or capacitors. A
typical digital filter of the so-called FIR type is shown in Fig. 6-8. The
analog signal is applied to the input on the left. The analog-to-digital
(A/D) converter digitizes and quantisizes the analog signal. An oscillator
(clock) determines the number of digitizing pulses per second and
controls all timing of the device. The type of filter is determined by the
program in the read-only memory.
Application of Digital Signal
Processing (DSP)
Digital signal processing has been successfully applied in various
ways, including1:
ANALOG AND DIGITAL SIGNAL PROCESSING 127
 Mixing two signals (convolution)
 Comparing two signals (correlation)
 Changing ac signals to dc (rectification)
 Amplifying signals
 Acting as a transformer
 Spectral analysis
 Speech processing, recognition, etc.
 Noise cancellation
 Music synthesis and processing.
Specific tasks ideally suited to digital signal processing include:
 Subsonic filters
 Ultrasonic filters
 Limiters
 Compressors
 Expanders
 Companders
 Noise gates
 Bass correction
A/D
converter
Clock
and register
Read-write
memory
Programmable
read-only
memory
Multiplier
Accumulator
F I G U R E 6 - 8
Block diagram of a typical digital filter, in this case one of the finite impulse response
(FIR) type. The type of filter is determined by the read-only program.
128 CHAPTER SIX
 Noise reduction
 Image enhancement
 Stereo synthesizer
This very brief treatment of digital signal processing has skipped
over such vital aspects as aliasing, the multiplier/accumulator, sampling
rates, quantization levels, etc.
Application of DSP to Room Equalization
The application of digital signal processing to the loudspeaker-roomlistener
problem is currently being pursued vigorously. Huge errors
are common in both the room and the loudspeakers. The problems of
the listener are primarily the great differences in sensitivity between
listeners.
The basic idea is to measure the frequency and phase response of
the speaker/room combination and apply an equalizer that perfectly
compensates for the defects. This is a very complicated operation, but
theoretically possible through the great potential of digital signal processing.
One problem to be solved is how to generalize the solution
based on data picked up at one point in the room. There are many
other problems but the next decade should bring major advances in
this field.
Endnote
1Burkhard, Mahlon, Filters and Equalizers, Chapter 20, Handbook For Sound Engineers,
Glen Ballou, Ed., Carmel, IN, Howard W. Sams Co., 2nd Edition (1987).
Pressing the gas pedal of an automobile results in acceleration of the
vehicle to a certain speed. If the road is smooth and level, this speed
will remain constant. With this accelerator setting the engine produces
just enough torque to overcome all the frictional losses and a balanced
(steady-state) condition results.
So it is with sound in a room. When the switch is closed, a loudspeaker
arranged to emit random noise into a room will produce a
sound that quickly builds up to a certain level. This is the steady-state
or equilibrium point at which the sound energy radiated from the
loudspeaker is just enough to supply all the losses in the air and at
the boundaries of the room. A greater sound energy radiated from the
loudspeaker will result in a higher equilibrium level, less power to
the loudspeaker will result in a lower equilibrium level.
When the loudspeaker switch is opened, it takes a finite length of
time for the sound level in the room to decay to inaudibility. This
“hanging-on” of the sound in a room after the exciting signal has been
removed is called reverberation and it has a very important bearing on
the acoustic quality of the room.
In England, a symphony orchestra was recorded as it played in a
large anechoic (echo-free) chamber. This music, recorded with almost
no reverberation for research purposes, is of very poor quality for normal
listening. This music is even thinner, weaker, and less resonant
Reverberation
7
129
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
130 CHAPTER SEVEN
than outdoor recordings of symphonic music, which are noted for
their flatness. Clearly, symphonic and other music requires reverberation
and the amount of it is a lively technical topic to be covered.
At one time reverberation was considered the single most important
characteristic of an enclosed space for speech or music. It has
shriveled in importance under the influence of important research on
the acoustics of enclosures. At the present time it is only one of several
measurable parameters that define the quality of a space.
Reverberation and Normal Modes
The natural resonances of a small room are revealed in the various
vibrational modes as described in chapter 15. It is necessary to anticipate
this chapter a bit to understand the relationship of these natural
room resonance frequencies and the reverberation of the room.
Our introduction to the measurement of reverberation will be
through a method wholly unsuited to practical use, but as an investigative
tool, it reveals some very important factors that focus attention
on the normal modes of rooms. Historically, in the Broadcasting House
in New Delhi, India, is (or was) a Studio 10 used for news broadcasts.
Measurements of reverberation time in this studio were reported by
Beranek1 and later analyzed by Schultz. The first set of measurements
were made in the completely bare, untreated room. Knowing construction
in India, it can be safely assumed that concrete and ceramic
tile dominated the room surfaces. The measurements were made with
sine-wave signals, and great patience and care were exercised to
obtain the detailed results.
Starting with the oscillator set to about 20 Hz below the first axial
mode, the acoustics of the room do not load the loudspeaker and a
relatively weak sound is produced with the amplifier gain turned up
full (even assuming the use of a good subwoofer). As the oscillator
frequency is adjusted upward, however, the sound becomes very
loud as the 1, 0, 0 mode (24.18 Hz) is energized (Fig. 7-1). Slowly
adjusting the oscillator upward we go through a weak valley but at
the frequency of the 0, 1, 0 mode (35.27 Hz) there is high level sound
once more. Similar peaks are found at the 1, 1, 0 tangential mode
(42.76 Hz), the 2, 0, 0 axial mode (48.37 Hz), and the 0, 0, 1 axial
mode (56.43 Hz).
REVERBERATION 131
Now that the loudness of peaks and valleys have been explored, let’s
examine the decay of sound. Exciting the 1, 0, 0 mode at 24.18 Hz, the
decay is measured as the source is interrupted and we get a long reverberation
time of 2.3 seconds. Similar slow decays are observed at 35.27
Hz, 42.76, 48.37, and 56.43 Hz with faster decays (shorter reverberation
times) in between. The decays at the modal frequencies are decay rates
characteristic of individual modes, not of the room as a whole.
Long reverberation time implies low absorbance, and short reverberation
time implies high absorbance. It is difficult to believe that the
sound absorbing qualities of the walls, floor, and ceiling vary this much
within a frequency range of a few hertz. For the 1, 0, 0 mode, only the
absorbance of the two ends of the room comes into play; the four other
surfaces are not involved at all. For the 0, 0, 1 mode, only the floor and
2.5
2.0
1.5
1.0
0.5
0
Reverberation time - seconds
1.0.0
0.1.01.1.0
2.0.00.0.1
Untreated
Treated
10 30 100 300 1 kHz 3 kHz 10 kHz
Frequency - Hz
F I G U R E 7 - 1
Reverberation time measured with pure sine signals at low frequencies reveals slow
sound decay (long reverberation time) at the modal frequencies. These peaks apply only
to specific modes and are not representative of the room as a whole. High modal density,
resulting in uniformity of distribution of sound energy and randomizing of directions
of propagation, is necessary for reverberation equations to apply. (Beranek,1 and
Schultz.2)
132 CHAPTER SEVEN
ceiling are involved. All we have done in this low-frequency range is to
measure the decay rate of individual modes, definitely not the average
condition of the room.
We see now why there is that big question mark over applying the
concept of reverberation time to small rooms having dimensions comparable
to the wavelength of sound. Schultz states that reverberation
time is a statistical concept “in which much of the mathematically
awkward details are averaged out.”2 In small rooms these details are
not averaged out.
The reverberation time formulas of Sabine, Eyring, and others are
based on the assumption of an enclosed space in which there is highly
uniform distribution of sound energy and random direction of propagation
of the sound. At the low-frequency points of Fig. 7-1, energy is
distributed very unevenly and direction of propagation is far from random.
After the room was treated, reverberation time measurements
followed the broken line, but statistical randomness still does not prevail
below 200 Hz even though modal frequencies are brought under
some measure of control.
Growth of Sound in a Room
Referring to Fig. 7-2A, let us consider a source S and a human receiver
H in a room. As source S is suddenly energized, sound travels outward
from S in all directions. Sound travels a direct path to H and we shall
consider zero time (see Fig. 7-2B) as that time at which the direct
sound reaches the ears of listener H. The sound pressure at H instantly
jumps to a value less than that which left S due to spherical divergence
and small losses in the air. The sound pressure at H stays at this value
until reflection R1 arrives and then suddenly jumps to the D + R1 value.
Shortly thereafter R2 arrives, causing the sound pressure to increase a
bit more. The arrival of each successive reflected component causes
the level of sound to increase stepwise. These additions are, in reality,
vector additions involving both magnitude and phase, but we are
keeping things simple for the purposes of illustration.
Sound pressure at receiver H grows step by step as one reflected
component after another adds to the direct component. The reason
the sound pressure at H does not instantly go to its final value is that
sound travels by paths of varying length. Although 1,130 ft/sec, the
REVERBERATION 133
speed of sound, is about the muzzle velocity of a .22 caliber rifle,
reflected components are delayed an amount proportional to the difference
in distance between the reflected path and the direct path.
The buildup of sound in a room is thus relatively slow due to finite
transit time.
The ultimate level of sound in the room is determined by the energy
going into the source S. The energy it radiates (less than the input by
S
A
R1
R3
R4
R2
H
Direct
B
Sound pressure
D  R1 R2 R3 R4
D  R1 R2 R3 D  R1 R2
D  R1
D
0
Time
D
R1
R2
R3 R4
Sound pressure
Time
0
C
F I G U R E 7 - 2
The buildup and decay of sound in a room. (A) The direct sound arrives first at time = 0,
reflected components arriving later. (B) The sound pressure at H builds up stepwise.
(C) The sound decays exponentially after the source ceases.
134 CHAPTER SEVEN
the amount of loss in S) is dissipated as heat in wall reflections and
other boundary losses, along with a small amount in the air itself. With
a constant input to S, the sound-pressure level builds up as in Fig. 7-2B
to a steady-state equilibrium, even as an automobile traveling steadily
at 50 miles per hour with the accelerator in a given position. Pushing
down on the accelerator pedal increases the energy to the engine, and
the automobile stabilizes at a new equilibrium point at which the many
frictional losses are just supplied. Increasing the input to the source S
means a new equilibrium of room-sound-pressure level as room losses
are just supplied.
Decay of Sound in a Room
After opening the switch feeding source S, the room is momentarily
still filled with sound, but stability is destroyed because the losses
are no longer supplied with energy from S. Rays of sound, however, are
caught in the act of darting about the room with their support cut off.
What is the fate of the ceiling reflected component R1? As S is cut
off, R1 is on its way to the ceiling. It loses energy at the celling bounce
and heads toward H. After passing H it hits the rear wall, then the
floor, the ceiling, the front wall, the floor again, and so on . . . losing
energy at each reflection and spreading out all the time. Soon it is so
weak it can be considered dead. The same thing happens to R2, R3, R4,
and a multitude of others not shown. Figure 7-2C shows the exponential
decrease of the first bounce components, which would also apply
to the wall reflections not shown and to the many multiple bounce
components. The sound in the room thus dies away, but it takes a
finite time to do so because of the speed of sound, losses at reflections,
the damping effect of the air, and divergence.
Idealized Growth and Decay of Sound
From the view of geometrical (ray) acoustics, the decay of sound in a
room, as well as its growth, is a stepwise phenomenon. However, in
the practical world, the great number of small steps involved result
in smooth growth and decay of sound. In Fig. 7-3A, the idealized
forms of growth and decay of sound in a room are shown. Here the
sound pressure is on a linear scale and is plotted against time. Figure
REVERBERATION 135
7-3B is the same thing except that the vertical
scale is plotted in decibels, i.e., to a
logarithmic scale.
During the growth of sound in the room,
power is being fed to the sound source.
During decay, the power to the source is
cut off, hence the difference in the shapes of
the growth and decay curves. The decay
of Fig. 7-3B is a straight line in this idealized
form, and this becomes the basis for
measuring the reverberation time of an
enclosure.
Reverberation Time
Reverberation time is defined as that time
required for the sound in a room to decay
60 dB. This represents a change in sound
intensity or sound power of 1 million
(10 log 1,000,000 = 60 dB), or a change of
sound pressure or sound-pressure level
of 1,000 (20 log 1,000 = 60 dB). In very
rough human terms, it is the time required
for a sound that is very loud to decay to
inaudibility. W. C. Sabine, the Harvard pioneer in acoustics who introduced
this concept, used a portable wind chest and organ pipes as a
sound source, a stopwatch, and a pair of keen ears to measure the time
from the interruption of the source to inaudibility. Today we have
better technical measuring facilities, but we can only refine our understanding
of the basic concept Sabine gave us.
This approach to measuring reverberation time is illustrated in
Fig. 7-4A. Using a recording device that gives us a hardcopy trace of
the decay, it is a simple step to measuring the time required for the
60-dB decay. At least it is simple in theory. Many problems are
encountered in practice. For example, obtaining a nice, straight decay
spanning 60 dB or more as in Fig. 7-4A is a very difficult practical
problem. Background noise, an inescapable fact of life, suggests that a
higher source level is needed. This may occur if the background noise
Sound source
Off On
Sound pressure
Time
Sound pressurelevel, dB
Time
A
B
F I G U R E 7 - 3
The growth and decay of sound in a room. (A)
Vertical scale in linear sound pressure units. (B) The
vertical scale in logarithmic units (decibels).
136 CHAPTER SEVEN
level is 30 dB (as in Fig. 7-4A), because source levels of 100 dB are
quite attainable. If, however, the noise level is near 60 dB as shown in
Fig. 7-4B, a source level greater than 120 dB is required. If a 100-watt
amplifier driving a certain loudspeaker gives a sound-pressure level
of 100 dB at the required distance, doubling the power of the source
increases the sound-pressure level only 3 dB, hence 200 watts gives
103 dB, 400 watts gives 106 dB, 800 watts gives 109 dB, etc. The limitations
of size and cost can set a ceiling on the maximum levels in a
practical case.
The situation of Fig. 7-4B is the one commonly encountered, a
usable trace less than the desired 60 dB. The solution is simply to
extrapolate the straight portion of the decay.
Actually, it is important to strive for the greatest decay range possible
because we are vitally interested in both ends of the decay. It has
been demonstrated that in evaluating the quality of speech or music,
the first 20 or 30 dB of decay is the most important to the human ear.
On the other hand, the significance of double-slope phenomena is
revealed only near the end of the decay. In practice, the highest level
of sound source reasonably attainable is used, and filters are often
incorporated to improve the signal-to-noise ratio.
100
80
60
40
0
Sound pressure level, dB
60 dB
RT60
Time
100
80
60
40
0
Sound pressure level, dB
RT60
60 dB
Time
A B
F I G U R E 7 - 4
The length of the decay dependent on strength of the source and the noise level. (A)
Rarely do practical circumstances allow a full 60-dB decay. (B) The slope of the limited
decay is extrapolated to determine the reverberation time.
REVERBERATION 137
Measuring Reverberation Time
There are many approaches to measuring the reverberation time of a
room, and many instant-readout devices are on the market to serve
those who have only a casual interest in reverberation effects. For example,
sound contractors need to know the approximate reverberation
time of the spaces in which they are to install a sound-reinforcement
system, and measuring it avoids the tedious process of calculating it.
The measurements can also be more accurate because of uncertainty
in absorption coefficients. Acoustical consultants (at least the oldfashioned
ones) called upon to correct a problem space or verify a carefully
designed and newly constructed space, generally lean toward the
method of recording many sound decays. These sound decays give a
wealth of detail meaningful to the practiced eye.
Impulse Sound Sources
The sound sources used to excite the enclosure must have enough
energy throughout the spectrum to ensure decays sufficiently above the
noise to give the required accuracy. Both impulse sources and those giving
a steady-state output are used. For large spaces, even small cannons
have been used as impulse sources to provide adequate energy, especially
in the lower frequencies. More common impulse sources are
powerful electrical spark discharges and pistols firing blanks. Even
pricked balloons have been used.
The impulse decays of Fig. 7-5 for a small studio have been
included to show their appearance. The sound source was a Japanese
air pistol that ruptures paper discs. This pistol was originally intended
as an athletic starter pistol but failed to find acceptance in that area. As
reported by Sony engineers,3 the peak sound-pressure level at 1 meter
distance is 144 dB, and the duration of the major pulse is less than
1 millisecond. It is ideal for recording echograms, in fact, the decays of
Fig. 7-5 were made from impulses recorded for that very purpose.
In Fig. 7-5, the straight, upward, traveling part on the left is the
same slope for all decays because it is a result of machine limitation
(writing speed 500 mm/sec). The useful measure of reverberation is
the downward traveling, more irregular slope on the right side. This
slope yields a reverberation time after the manner of Fig. 7-4. Notice
138 CHAPTER SEVEN
that the octave-band noise level is higher for the lower frequency
bands. The impulse barely poked its head above noise for the 250 Hz
and lower octaves. This is a major limitation of the method unless the
heavy artillery is rolled out.
Steady-State Sources
As stated, Sabine used a wind chest and organ pipes. Sine-wave sources
providing energy at a single frequency give highly irregular decays that
are difficult to analyze. Warbling a tone, which spreads its energy over a
narrow band, is an improvement over the fixed tone, but random noise
sources have essentially taken over. Bands of random noise give a
steady and dependable indication of the average acoustical effects taking
place within that particular slice of the spectrum. Octave and 13-
octave bands of random noise (white or pink) are most commonly used.
Equipment
The equipment layout of Fig. 7-6 to obtain the following reverberation
decays is quite typical. A wideband pink-noise signal is amplified
0
10
20
30
40
50
Relative octave band level, dB
4 kHz
Octave
2 kHz
Octave
1 kHz
Octave
500 Hz
Octave
0 0.5 1.0
Time - seconds
F I G U R E 7 - 5
Reverberatory decays produced by impulse excitation of a small studio. The upgoing left
side of each trace is recording-machine limited; the downgoing right side is the reverberatory
decay.
REVERBERATION 139
and used to drive a rugged loudspeaker. A switch for interrupting the
noise excitation is provided. By aiming the loudspeaker into a corner
of the room (especially in smaller rooms), all resonant modes are
excited, because all modes terminate in the corners (Chap. 15).
A nondirectional microphone is positioned on a tripod, usually at
ear height for a listening room, or microphone height for a room used
for recording. The smaller the microphone, the less its directional
effects. Some of the larger microphones (e.g., 1-inch-diameter diaphragms)
can be fitted with random incidence correctors, but using a
smaller microphone (e.g., 12-inch-diameter diaphragm) is considered
best for essentially uniform sensitivity to sound arriving from all
angles. In Fig. 7-6, the microphone is a high-quality condenser microphone,
part of the Brüel & Kjaer 2215 sound-level meter, but separated
from it by an extension cable. This provides an excellent preamplifier,
built-in octave filters, a calibrated system, and a line-level output signal
for the tape recorder.
Pink
noise
gen.
Power
amplifier
Tape
recorder
B&K 2215
SLM*
A *With octave filters
Tape
recorder
B&K 2215
SLM*
B&K 2305
Level
recorder
B
F I G U R E 7 - 6
Equipment arrangement for measuring the reverberation time of an enclosure. (A)
Recording decays on tape on location. (B) Later recording decays for analysis.
140 CHAPTER SEVEN
Measurement Procedure
Every time the switch is closed the room is filled with a very loud
wideband pink noise “sh-h-h-h” sound. This is usually loud enough to
require the use of ear protectors for everybody in the room. Opening
the switch, the sound in the room decays. The microphone, at its
selected position, picks up this decay, which is recorded on magnetic
tape for later analysis and study.
Signal-to-noise ratio determines the length of the reverberatory
decay available for study. As mentioned previously, it is rarely possible
to realize the entire 60-dB decay involved in the definition of
RT60, nor is it necessary. It is quite possible, however, to get 45- to
50-dB decays with the equipment shown in Fig. 7-6 by the simple
expedient of double filtering. For example, the octave filter centered
on 500 Hz in the sound level meter is used both in recording and in
later playback for analysis.
The analysis procedure outlined in the lower part of Fig. 7-6 uses the
same magnetic recorder and B&K 2215 sound-level meter, with
the addition of a B&K 2305 graphic-level recorder. The line output of the
tape recorder is connected to the front end of the sound-level meter circuit
through a 40-dB attenuating pad. To do this, the microphone of the
sound-level meter is removed and a special fitting is screwed in its
place. The output of the sound-level meter is connected directly to the
graphic-level recorder input, completing the equipment interconnection.
The appropriate octave filters are switched in as the played-back
decay is recorded on the level recorder. The paper drive provides for
spreading out the time dimension at adjustable rates. The graphic-level
recorder offers a 50-dB recording range for the tracing pen on the paper.
Analysis of Decay Traces
An octave slice of pink noise viewed on a cathode-ray oscilloscope
shows a trace that looks very much like a sine wave except that it is
constantly shifting in amplitude and phase, which was the definition
of random noise in Chap. 5. This characteristic of random noise has its
effect on the shape of the reverberatory decay trace. Consider what this
constantly shifting random noise signal does to the normal modes of a
room (Chap. 15). When the axial, tangential, and oblique resonant
modes are considered, they are quite close together on the frequency
REVERBERATION 141
scale. The number of modes included within an octave band centered
on 63 Hz in the specific case to be elaborated later is as follows: 4 axial,
6 tangential, and 2 oblique modes between the –3-dB points. These are
graphically shown in Fig. 7-7 in which the taller lines represent the
more potent axial modes, the intermediate height the tangential
modes, and the shorter lines the oblique modes.
As the switch of Fig. 7-6 is closed, the high-level random noise from
the loudspeaker energizes the various modes of the room, exciting
mode A, and an instant later exciting mode B. While the shift is being
made in the direction of mode B, mode A begins to decay. Before it
decays very far, however, the random-noise instantaneous frequency is
once more back on A, giving it another boost. All the modes of the room
are in constant agitation, alternating between high and somewhat lower
levels, as they start to decay in between kicks from the loudspeaker.
At what point will this erratic dance of the modes be as the switch
is opened to begin the decay? It is strictly a random situation, but it
can be said with confidence that each time the switch is opened for
five successive decays, the modal excitation pattern will be somewhat
different. The 12 modes in the 63-Hz octave will all be highly
energized, but each to a somewhat different level the instant the
switch is opened.
Octave centered on 63 Hz
0 20 40 60 80 100
Frequency - hertz
F I G U R E 7 - 7
The normal modes included (–3 dB points) in an octave centered on 63 Hz. The tallest
lines axial modes, the intermediate length tangential modes, and the shortest ones
oblique modes.
142 CHAPTER SEVEN
Mode Decay Variations
To make this discussion more meaningful, real-life measurements in a
real room are discussed. The room is a rectangular studio for voice
recording having the dimensions 20'6" × 15'0" × 9'6", with a volume of
2,921 cubic feet. The measuring equipment is exactly that outlined in
Fig. 7-6, and the technique is that described above. Four successive
63-Hz octave decays traced directly from the graphic-level recorder
paper are shown in Fig. 7-8A. These traces are not identical, and any
differences must be attributed to the random nature of the noise signal
because everything else was held constant. The fluctuations in the
decays result from beats between closely spaced modes. Because the excitation
level of the modes is constantly shifting, the form and degree
of the beat pattern shifts from one decay to another depending on
10
20
30
40
50
0
10
20
30
40
50
Relative octave band level, dB
63-Hz octave A
500-Hz octave B
F I G U R E 7 - 8
Actual decays of random noise recorded in a small studio having a volume of 2,921 cu
ft; (A) Four successive 63-Hz octave decays recorded under identical conditions. (B)
Four successive 500-Hz octave decays also recorded under identical conditions. The differences
noted result from the differences in random-noise excitation the instant the
switch is opened to start the decay.
REVERBERATION 143
where the random excitation happens to be the instant the switch is
opened. Even though there is a family resemblance between the four
decays, fitting a straight line to evaluate the reverberation time of each
can be affected by the beat pattern. For this reason, it is good practice to
record five decays for each octave for each microphone position of a
room. With eight octaves (63 Hz–8 kHz), five decays per octave, and
three microphone positions, this means 120 separate decays to fit
and figure for each room, which is laborious. This approach is one way
to get a good, statistically significant view of the variation with frequency.
A hand-held reverberation time measuring device could accomplish
this with less work, but it would not give hard-copy detail of the
shape of each decay. There is much information in each decay, and
acoustical flaws can often be identified from aberrant decay shapes.
Four decays at 500 Hz are also shown in Fig. 7-8B for the same room
and the same microphone position. The 500-Hz octave (354–707 Hz)
embraces about 2,500 room modes. With such a high mode density, the
500-Hz octave decay is much smoother than the 63-Hz octave with only
a dozen. Even so, the irregularities for the 500-Hz decay of Fig. 7-8B
result from the same cause. Remembering that some modes die away
faster than others, the decays in Fig. 7-8 for both octaves are composites
of all modal decays included.
Writing Speed
The B&K 2305 graphic-level recorder has a widely adjustable writing
speed. A sluggish pen response is useful when fast fluctuations need
to be ironed out. When detail is desired, faster writing speeds are
required. A too slow writing speed can affect the rate of decay as it
smooths out the trace, as will be examined.
In Fig. 7-9, the same 63-Hz decay is recorded with five different pen
response speeds ranging from 200 to 1,000 mm/sec. The instrumentlimited
decay for each is indicated by the solid straight lines. A writing
speed of 200 mm/sec smooths the fluctuations very well. The
decay detail increases as the writing speed is increased, suggesting
that a cathode-ray oscilloscope tracing of the decay would show even
more modal interference effects during the decay.
The big question is: Does writing speed affect the decay slope from
which we read the reverberation time values? Obviously, an extremely
slow pen response would record the machine’s decay characteristic
144 CHAPTER SEVEN
rather than that of the room. For every writing speed and paper speed
setting, there is a minimum reverberation time that can be measured.
The broken lines drawn through the decays all have the same slope. In
Fig. 7-9, it would appear that this particular decay is measured equally
well by any of the five traces, although the more detail, the more uncertainty
in fitting a straight line. Writing speed is just one of the several
adjustments that must be carefully monitored to ensure that important
information is not obscured or that errors are not introduced.
Frequency Effect
Typical decays for octave bands of noise from 63 Hz to 8 kHz are
included in Fig. 7-10. The greatest fluctuations are in the two lowest
bands, the least in the two highest. This is what we would expect from
the knowledge that the higher the octave band, the greater the number
of normal modes included, and the greater the statistical smoothing.
We should not necessarily expect the same decay rate because reverberation
time is different for different frequencies. In the particular
voice studio case of Fig. 7-10, a uniform reverberation time with frequency
was the design goal, which was approximated in practice.
200 315 500 800 1000
Writing speed-millimeters/second
0
10
20
30
40
50
Relative sound pressure level, dB
F I G U R E 7 - 9
The same 63-Hz octave decays recorded with five different recorder-pen-response
speeds. The solid straight lines indicate instrument-limited decay. The broken lines all
have the same slope.
REVERBERATION 145
Reverberation Time
Variation with Position
There is enough variation of reverberation time from one position to
another in most rooms to justify taking measurements at several positions.
The average then gives a better statistical picture of the behavior
of the sound field in the room. If the room is symmetrical, it might be
wise to spot all measuring points on one side of the room to increase
the effective coverage with a given effort.
0
10
20
30
40
50
0
10
20
30
40
50
Relative octave band level, dB
63 Hz
125 Hz
250 Hz 500 Hz
0.1 Sec
1 kHz 2 kHz 4 kHz 8 kHz
F I G U R E 7 - 1 0
Decay of octave bands of noise in a small studio of volume 2,921 cu ft. Fluctuations due
to modal interference are greatest for low-frequency octaves containing fewer modes.
146 CHAPTER SEVEN
Acoustically Coupled Spaces
The shape of the reverberation decay can
point to acoustical problems in the space.
One common effect that alters the shape of
the decay is due to acoustically coupled
spaces. This is quite common in large public
gathering spaces, but is also found in
offices, homes, and other small spaces. The
principle involved is illustrated in Fig. 7-
11. The main space, perhaps an auditorium,
is acoustically quite dead and has a
reverberation time corresponding to the
slope A. An adjoining hall with very hard
surfaces and a reverberation time corresponding
to slope B opens into the main
room. A person seated in the main hall near
the hall opening could very well experience
a double-slope reverberation decay.
Not until the sound level in the main room
falls to a fairly low level would the main
room reverberation be dominated by sound fed into it from the slowly
decaying sound in the hall. Assuming slope A is correct for the main
room, persons subjected to slope B would hear inferior sound.
Electroacoustically Coupled Spaces
What is the overall reverberant effect when sound picked up from a
studio having one reverberation time is reproduced in a listening room
having a different reverberation time? Does the listening room reverberation
affect what is heard? The answer is definitely yes. This problem
has been analyzed mathematically by Mankovsky.4 In brief, the
sound in the listening room is affected by the reverberation of both studio
and listening room as follows:
 The combined reverberation time is greater than either alone.
 The combined reverberation time is nearer the longer reverberation
time of the two rooms.
Sound pressure level, dB
Time
B
A
F I G U R E 7 - 1 1
Reverberatory decay with a double slope due to
acoustically coupled spaces. The shorter reverberation
time represented by slope A is that of the main room.
A second, highly reflective space is coupled through
an open doorway. Those seated near the doorway are
subjected first to the main-room response and then
to the decay of the coupled space.
REVERBERATION 147
 The combined decay departs somewhat from a straight line.
 If one room has a very short reverberation time, the combined
reverberation time will be very close to the longer one.
 If the reverberation time of each of the two rooms alone is the
same, the combined reverberation time is 20.8% longer than one
of them.
 The character and quality of the sound field transmitted by a
stereo system conforms more closely to the mathematical assumptions
of the above than does a monaural system.
 The first five items can be applied to the case of a studio linked
to an echo chamber as well as a studio linked to a listening room.
Decay Rate
The definition of reverberation time is based on uniform distribution of
energy and random directions of propagation. Because these conditions
do not exist in small rooms, there is some question as to whether what
we measure should be called reverberation time. It is more properly
termed decay rate. A reverberation time of 0.3 of a second is equivalent
to a decay rate of 60 dB/0.3 sec = 200 dB per second. The use of decay
rate instead of reverberation time would tell the experts that we are
aware of the basic problems. Speech and music sounds in small rooms
do decay even though the modal density is too low to hang the official
“reverberation time” tag on the process.
Eliminating Decay Fluctuations
The measurement of reverberation time by the classical method that
has been described involves the recording of many decays for each condition
and much work in analyzing them. Schroeder has published a
new method by which the equivalent of the average of a great number
of decays can be obtained in a single decay.5 One practical, but clumsy,
method of accomplishing the mathematical steps required is to:
1. Record the decay of an impulse (noise burst or pistol shot) by the
normal method.
2. Play back that decay reversed.
3. Square the voltage of the reversed decay as it builds up.
148 CHAPTER SEVEN
4. Integrate the squared signal with a resistance-capacitance circuit.
5. Record this integrated signal as it builds up during the reversed
decay. Turn it around and this trace will be mathematically identical
to averaging an infinite number of traditional decays. Programming
this operation into a computer would be easier and more
satisfactory.
Influence of Reverberation on Speech
Let us consider what happens to just one tiny word in a reverberant
space. The word is back. It starts abruptly with a “ba...” sound and
ends with the consonant “...ck”, which is much lower in level. As
measured on the graphic-level recorder, the “ck” sound is about 25 dB
below the peak level of the “ba” sound and reaches a peak about 320
milliseconds after the “ba” peak.
Both the “ba” and “ck” sounds are transients that build up and decay
after the manner of Fig. 7-3. Sketching these various factors to scale
yields something like Fig. 7-12. The “ba” sound builds to a peak at an
0.32
Sec
25
dB
“BA...”
0
20
40
60
Relative soundpressure level, dB
“...CK”
RT60=0.5
RT60-1.5 Sec
0 0.5 1.0 1.5
Time - seconds
F I G U R E 7 - 1 2
An illustration of the effects of reverberation on the intelligibility of speech. Understanding
the word “back” depends on apprehending the later, lower level consonant
“....ck,” which is masked by reverberation if the reverberation time is too long.
REVERBERATION 149
arbitrary level of 0 dB at time = 0, after which it decays according to the
reverberation time of the room, which is assumed to be 0.5 seconds (60
dB decay in 0.5 seconds). The “ck” consonant sound, peaking 0.32 seconds
later, is 25 dB below the “ba” sound peak. It too decays at the same
rate as the “ba” sound according to the assumed 0.5-second RT60.
Under the influence of the 0.5-second reverberation time, the “ck” consonant
sound is not masked by the reverberation time of “ba.” If the
reverberation time is increased to 1.5 seconds, as shown by the broken
lines, the consonant “ck” is covered completely by reverberation.
The primary effect of excessive reverberation is to impair the intelligibility
of speech by masking the lower level consonants. In the word
“back,” the word is unintelligible without a clear grasping of the “ck”
part. Understanding the “ck” ending is the only way to distinguish
“back” from bat, bad, bass, ban, or bath. In this oversimplified way, we
can grasp the effect of reverberation on the understandability of
speech and the reason why speech is more intelligible in rooms having
lower reverberation times.
Sound-reinforcement engineers have been assisted greatly by
the work of the Dutch investigators, Peutz6 and Klein.7 Because of their
work, it is a straightforward procedure to predict with reasonable
accuracy the intelligibility of speech in a space from geometrical factors
and a knowledge of reverberation time.
Influence of Reverberation on Music
The effect of hall “resonance” or reverberation on music is intuitively
grasped but is not generally well understood. This subject has received
much attention from scientists as well as musicians, and the final word
has yet to appear. Beranek has made a valiant attempt to summarize
present knowledge and to pinpoint essential features of concert and
opera halls around the world,8,9 but our understanding of the problem
is still quite incomplete. Suffice it to say that the reverberation decay
of a music hall is only one important factor among many, another being
the echo pattern, especially the so-called “early sound.” It is beyond the
scope of this book to treat this subject in any detail, but an interesting
point or two commonly overlooked are discussed briefly.
Normal modes have been considered in some detail because of their
basic importance (see Chap. 15). They are also active in music halls
150 CHAPTER SEVEN
and listening rooms. An interesting phenomenon is pitch change during
reverberant decay. In reverberant churches, organ tones have been
observed to change pitch as much as a semitone during decay. In
searching for an explanation for this phenomenon, two things have
been mentioned: shift of energy between normal modes, and the perceptual
dependence of pitch on sound intensity. There are problems in
both. Balachandran has demonstrated the physical (as opposed to psychophysical)
reality of the effect10 using the Fast Fourier Transform
(FFT) technique on the reverberant field created by 2-kHz pulses. He
revealed the existence of a primary 1,992-Hz spectral peak, and curiously,
another peak at 3,945 Hz. Because a 6-Hz change would be just
perceptible at 2 kHz, and a 12-Hz change at 4 kHz, we see that the 39-
Hz shift from the octave of 1,992 Hz would give a definite impression
of pitch change. The reasons for this are still under study. The reverberation
time of the hall in which this effect was recorded was about 2
seconds.
Optimum Reverberation Time
Considering the full range of possible reverberation times, there must
be an optimum time between the “too dry” condition of the outdoors
and anechoic chambers, and the obvious problems associated with
excessively long reverberation times in a stone cathedral. Such an optimum
does exist, but there is usually great disagreement as to just what
it is because it is a subjective problem and some differences in opinions
must be expected. The optimum value depends not only on the one
making the judgment, but also on the type of sounds being considered.
Reverberation rooms, which are used for measuring absorption coefficients,
are carefully designed for the longest practical RT60 to achieve
the maximum accuracy. The optimum here is the maximum attainable.
The best reverberation time for a space in which music is played
depends on the size of the space and the type of music. Slow, solemn,
melodic music, such as some organ music, is best served by long reverberation
time. Quick rhythmic music requires a different reverberation
time from chamber music. No single optimum universally fits all types
of music, the best that can be done is to establish a range based on subjective
judgments of specialists.
Recording studios present still other problems that do not conform
to simple rules. Separation recording in which musical instruments are
recorded on separate tracks for later mixdown in general require quite
REVERBERATION 151
dead spaces to realize adequate acoustical separation between tracks.
Music directors and band leaders often require different reverberations
for different instruments, hence hard areas and absorptive areas may be
found in the same studio. The range of reverberation realized in this
manner is limited, but proximity to reflective surfaces does affect local
conditions.
Spaces for speech require shorter reverberation times than for music
because of the general interest in direct sound. In general, long reverberation
time tends toward lack of definition and clarity in music and loss
of intelligibility in speech. In dead spaces in which reverberation time is
very short, loudness and tonal balance may suffer. It is not possible to
specify precisely optimum reverberation times for different services, but
Figs. 7-13 through 7-15 show at least a rough indication of recommendations
given by a host of experts in the field who do not always agree with
each other.
The reverberation times for churches in Fig. 7-13 range from highly
reverberant liturgical churches and cathedrals to the shorter ranges of
4
3
2
1
0
Reverberation time - seconds
0 10,000 100,000 1,000,000
Volume - cubic feet
Churches
F I G U R E 7 - 1 3
“Optimum” reverberation time for churches. The upper area applies to the more reverberant
liturgical churches and cathedrals, the lower to churches having services more oriented
to speech. A compromise between music and speech is required in most churches.
152 CHAPTER SEVEN
the lower shaded area characteristic of the more talk-oriented churches.
Churches generally represent a compromise between music and speech.
Figure 7-14 represents the range of recommended reverberation
times for different concert halls. Symphony orchestras are near the
top, lighter music somewhat lower. The lower shaded area applies for
opera and chamber music.
Those spaces used primarily for speech and recording require close
to the same reverberation times as shown in Fig. 7-15. Television studios
have even shorter reverberation times to deaden the sounds associated
with rolling cameras, dragging cables, and other production
noises. It should also be remembered that acoustics in television are
dominated by the setting and local furnishings. In many of the spaces
represented in Fig. 7-15, speech reinforcement is employed.
Bass Rise of Reverberation Time
The goal in voice studios is to achieve a reverberation time that is the
same throughout the audible spectrum. This can be difficult to realize,
4
3
2
1
0
Reverberation time - seconds
0 10,000 100,000 1,000,000
Volume - cubic feet
Concert halls
F I G U R E 7 - 1 4
“Optimum” reverberation time for concert halls. Symphony orchestras are near the top
of the shaded areas; lighter music is lower. The lower shaded area applies to opera and
chamber music.
REVERBERATION 153
especially at low frequencies. Adjustment of reverberation time at high
frequencies is easily accomplished by adding or removing relatively
inexpensive absorbers. At low frequencies, the situation is quite different
as absorbers are bulky, difficult to install, and sometimes unpredictable.
Researchers at the British Broadcasting Corporation observed that
subjective judgments seemed to indicate a tolerance for a certain amount
of bass rise of reverberation time. Investigating this in controlled tests,
Spring and Randall11 found that bass rise to the extent indicated in
Fig. 7-16 was tolerated by the test subjects for voice signals. Taking the
1 kHz value as reference, rises of 80% at 63 Hz and 20% at 125 Hz were
found to be acceptable. These tests were made in a studio 22 × 16 × 11
feet (volume about 3,900 cu ft) for which the midband reverberation
time was 0.4 second (which agrees fairly well with Fig. 7-15).
Bass rise in reverberation time for music has traditionally been
accepted to give “sonority” to the music in music halls. Presumably,
somewhat greater bass rise than that for speech would be desirable in
listening rooms designed for classical music.
4
3
2
1
0
Reverberation time - seconds
Speech and
recording
Auditoriums
for speech
Recording studios
Television studios
0 10,000 100,000 1,000,000
Volume - cubic feet
F I G U R E 7 - 1 5
Spaces designed for speech and music recording require shorter reverberation times.
154 CHAPTER SEVEN
Living Room Reverberation Time
The reverberation characteristic of the average living room is of interest
to the high-fidelity enthusiast, the broadcaster, and the recording
specialist. This living room is where the high-fidelity recordings are
to be played. Further, the quality control monitoring room of the
broadcast and recording studio must have a reverberation time not
too far from that of the living room in which the final product will be
heard. Generally, such rooms should be “deader” than the living
room, which will add its own reverberation to that of recording or
broadcast studio.
Figure 7-17 shows the average reverberation time of 50 British living
rooms measured by Jackson and Leventhall12 using octave bands of
noise. The average reverberation time decreases from 0.69 second at
125 Hz to 0.4 second at 8 kHz. This is considerably higher than earlier
measurements of 16 living rooms made by BBC engineers in which
reverberation times between 0.35 and 0.45 were found on the average.
Apparently, the living rooms measured by the BBC engineers were better
furnished than those measured by Jackson and Leventhall and, presumably,
would agree better with living rooms in the United States.
The 50 living rooms of the Jackson-Leventhall study were of varying
sizes, shapes, and degree of furnishing. The sizes varied from 880
to 2,680 cu ft, averaging 1550 cu ft. Figure 7-15 shows an optimum
100%
50%
0
Permissible bass risein reverberation time
63 125 250 500 1 kHz
Frequency - Hz
F I G U R E 7 - 1 6
Permissible bass rise of reverberation time for voice studios derived by subjective evaluation
in controlled tests by BBC researchers. (After Spring and Randall.11)
REVERBERATION 155
reverberation time for speech for rooms of this size to be about 0.3 second.
Only those living rooms near the lower limit approach this, and
in them we would expect to find much heavy carpet and overstuffed
furniture. These reverberation measurements tell us little or nothing
about the possible presence of colorations. The BBC engineers checked
for colorations and reported serious ones in a number of the living
rooms studied.
Artificial Reverberation: The Past
Artificial reverberation is considered a necessity in audio-signal processing.
Recordings of music played in “dry” (nonreverberant) studios
lack the richness of the room effect contributed by the music hall. The
addition of artificial reverberation to such recordings has become standard
practice, and there is great demand for equipment that will provide
natural-sounding artificial reverberation at a reasonable cost.
Upper limit of spread
Average of 50 living rooms
1.0
0.8
0.6
0.4
0.2
0
Reverberation time - seconds
Lower limit of spread
100 1,000 10,000
Frequency - Hz
F I G U R E 7 - 1 7
Average reverberation time for 50 living rooms. (After Jackson and Leventhall.12)
156 CHAPTER SEVEN
There are many ways of generating artificial reverberation, but the
challenge is finding that method which mimics actual music halls and
does not introduce colorations (frequency-response aberrations) into
the signal. Historically, a dedicated reverberation room has been
employed by the larger organizations. The program is played into this
room, picked up by a microphone, and the reverberated signal mixed
back into the original in the amount to achieve the desired effect.
Small reverberation rooms are afflicted with serious coloration problems
because of widely spaced modes. Large rooms are expensive.
Even though the three-dimensional reverberation room approach has
certain desirable qualities, the problems outweigh the advantages, and
they are now a thing of the past.
Spring reverberators have been widely used in semiprofessional
recording because of their modest cost. In this form of reverberator, the
signal is coupled to one end of a spring, the sound traveling down
the spring being picked up at the other end. Because of quality problems,
spring reverberators are also rapidly passing from the scene.
The reverberation plate, such as the Gotham Audio Corporation’s
EMT-140, has been a professional standard for many years. It, too, is
slipping into oblivion because of the more favorable cost/performance
ratio of the newer digital devices.
Artificial Reverberation: The Future
An understanding of the digital reverberators now dominating the
field can be achieved best by studying the basic principles of the old
abandoned methods. What audiophile has not fed a signal from the
playback head of his or her magnetic tape recorder back into the record
head, and was enthralled by the repetitive “echo” effect? The secret
lies in the delay resulting from the travel of the tape from one head to
the other. Delay is the secret ingredient of every form of reverberation
device. The delays associated with the return of successive echoes in
a space is the secret ingredient of the natural reverberation.
This principle is illustrated in the simple signal flow schematic of
Fig. 7-18. The incoming signal is delayed, and a portion of the delayed
signal is fed back and mixed with the incoming signal, the mixture
being delayed again, and so on.
Schroeder has found that approximately 1,000 echoes per second
are required to avoid the flutter effect that dominates the above tapeREVERBERATION
157
recorder experiment and to sound natural to the ear. With a 40-msec
delay, only 1/0.04 = 25 echoes are produced each second, a far cry from
the 1,000 per second desired. One solution is to arrange many of the
simple reverberators of Fig. 7-18 in parallel. Four such simple reverberators,
arranged in parallel as in Fig. 7-19, produce only 4 × 25 = 100
echoes per second. It would require 40 such reverberators in parallel to
achieve the required echo density, and this is impractical.
One approach to producing the necessary echo density, and simultaneously,
a flat frequency response is illustrated in Fig. 7-20. Here numerous
delays feed back on themselves, combining to feed other delays,
which in turn recirculate back to the first delay. The + signs in Fig. 7-20
indicate mixing (addition), and the × signs indicate gain (multiplication).
Remember that multiplying by a fraction less than unity gives a gain of
less than unity—in other words, attenuation. The digital reverberator of
Fig. 7-20 only suggests how greater echo density along with good frequency
response might be achieved. Actually, the better digital reverberators
in use today are far more complicated than this.13 The resulting
artificial reverberation available today from the top of the line has far
higher echo density, flatter frequency response, and a more natural
sound than the best of the old mechanical devices. The less expensive
digital units also show great improvement over those of the past.
Arrival Time Gap
There is one characteristic of natural reverberation in music halls
that was revealed by Beranek’s careful study of music halls around
Delay Analog
in
Analog
out
F I G U R E 7 - 1 8
The simplest artificial reverberation device. The technique of utilizing the time delay
between the record and playback heads of a magnetic tape recorder is of this type.
158 CHAPTER SEVEN
Delay
Analog
in
Analog
out
Delay
Delay
Delay
F I G U R E 7 - 1 9
An artificial-reverberation device incorporating several different delays produces an
effect far superior to that of the single delay in Fig. 7-18.

  
Analog
in A/D Delay

Delay D/A
Delay 

F I G U R E 7 - 2 0
The required echo density is achieved in a reverberation device that incorporates numerous
delays and recirculation of the signal. Actual artificial-reverberation devices available
today are far more complicated than this one.
REVERBERATION 159
the world.8,9 At a given seat, the direct
sound arrives first because it follows the
shortest path. Shortly after the direct
sound, the reverberant sound arrives.
The time between the two is called the
arrival time gap or early time gap as
shown in Fig. 7-21. If this gap is less than
40 or 50 msec, the ear integrates the
direct and the reverberant sound successfully.
This gap is important in
recorded music because it is the cue that
gives the ear information on the size of
the hall. In addition to all of the delays
responsible for achieving echo density,
the initial time-delay gap is yet another
important delay that must be included in
digital reverberators.
The Sabine Equation
Sabine’s reverberation equation was developed at the turn of the century
in a strictly empirical fashion. He had several rooms at his disposal
and by adding or removing seat cushions of a uniform kind he
established the following relationship (adapted from the metric units
he used):
RT60  (7-1)
where
RT60  reverberation time, seconds
V  volume of room, cu ft
S  total surface area of room, sq ft
a  average absorption coefficient of room surfaces
Sa  total absorption, sabins
Young14 has pointed out that the absorption coefficients published
by materials manufacturers (such as the list in the appendix) are
Sabine coefficients and can be applied directly in the Sabine equation.
After a thorough study of the historical development of the offshoots
of Sabine’s work, he recommends that Equation 7-1 be used for all
0.049 V

Sa
Arrival time
gap
Direct
Rel. level - dB
Reflected
Time
F I G U R E 7 - 2 1
The introduction of an initial time-delay gap in the
artificial reverberation adds to its lifelike character.
160 CHAPTER SEVEN
engineering computations rather than the Eyring equation or the
several derivatives thereof. Two unassailable reasons for this are
simplicity and consistency. In spite of the fact that this simpler procedure
was suggested as early as 1932, and Young’s convincing arguments
for it were given in 1959, many technical writings have
continued to put forth the Eyring or other equations for studio use.
Even though there was authoritative backing for using Eyring for
more absorbent spaces, why continue if the commonly available coefficients
apply only to Sabine? These are the reasons why we use only
Equation 7-1 in this volume.
The total Sabine absorption in a room would be easy to get if all surfaces
of the room were uniformly absorbing, but this condition rarely
exists. Walls, floor, and ceiling may well be covered with quite different
materials, and then there are the doors and windows. The total absorption,
Sa, of Equation 7-1, can be found by considering the absorption
contributed by each type of surface. For example, in our imaginary
room, let us say that an area S1 is covered with a material having an
absorption coefficient a1 as obtained from the table in the appendix.
This area then contributes (S1) (a1) absorption units, called sabins, to the
room. Likewise, another area S2 is covered with another kind of material
with absorption coefficient a2, and it contributes (S2) (a2) sabins
of absorption to the room. The total absorption in the room is Sa =
S1a1  S2a2  S3a3 .... etc. With a figure for Sa in hand, it is a simple
matter to go back to Equation 7-1 and calculate the reverberation time.
Reverberation Calculation: Example 1
A completely untreated room will first be taken to illustrate the implementation
of Sabine’s equation (Eq. 7-1). The dimensions of the room
are assumed to be 23.3 × 16 × 10 ft. Other assumptions are that the
room has a concrete floor and that the walls and the ceiling are of
frame construction with 1⁄2 in gypsum board (drywall) covering. As a
simplification the door and a window will be neglected as having
minor effect. The tabulation of Fig. 7-22 illustrates the untreated condition.
The concrete floor area of 373 sq ft and the gypsum board area
of 1,159 sq ft are entered in the table. The appropriate absorption coefficients
are entered from the table in the appendix for each material
and for the six frequencies. Multiplying the concrete floor area of S =
373 sq ft by the coefficient a = 0.01 gives Sa = 3.7 sabins. This is
REVERBERATION 161
entered under Sa for 125 and 250 Hz. The absorption units (sabins) are
then figured for both materials and for each frequency. The total number
of sabins at each frequency is obtained by adding that of the concrete
floor to that of the gypsum board. The reverberation time for each
frequency is obtained by dividing 0.049 V = 182.7 by the total Sa product
for each frequency.
To visualize the variation of reverberation time with frequency, the
values are plotted in Fig. 7-23A. The peak reverberation time of 3.39
seconds at 1 kHz is excessive and would ensure quite poor sound conditions.
Two persons separated 10 ft would have difficulty understanding
each other as the reverberation of one word covers up the
next word.
S
sq ft
Size
Treatment
Floor
Walls
Ceiling
Volume
Material
Concrete
gypsum board
373
1,159
125 Hz
a Sa
0.01
0.29
3.7
336.1
250 Hz
a Sa
0.01
0.10
3.7
115.9
500 Hz
a Sa
0.015
0.05
5.6
58.0
1 kHz
a Sa
0.02
0.04
7.5
46.4
2 kHz
a Sa
0.02
0.07
7.5
81.1
4 kHz
a Sa
0.02
0.09
7.5
104.3
Total sabins 339.8 119.6 63.6 53.9 88.6 111.8
Reverberation time
(seconds)
0.54 1.53 2.87 3.39 2.06 1.63
a = absorption coefficient for that material
and for that frequency (See Appendix)
Sa = S times a, absorption units, sabins
RT60 = (0.049)(3728)
= 182.7
Sa Sa
Example: For 125 Hz, RT60 = 182.7
= 0.54 second
339.8
23.3 16 10 ft
None
Concrete
Gypsum board, 1/2", on frame construction
Ditto
(23.3) (16) (10) = 3,728 cu ft
F I G U R E 7 - 2 2
Room conditions and calculations for Example 1.
162 CHAPTER SEVEN
Reverberation Calculation:
Example 2
The goal now is to correct the reverberation
of curve A of Fig. 7-23. It is evident that
much absorption is needed at midband
frequencies, a modest amount at higher frequencies,
and very little at lower frequencies.
The need is for a material having an
absorption characteristic shaped more or
less like the reverberation curve A. Skipping
the laborious thumbing through of
handbooks, 34-in acoustical tile seems to
have the right shape. Giving no thought at
this point to how it is to be distributed,
what area of this tile is required to correct
for Fig. 7-23A?
A new table, Fig. 7-24, is set up to organize
the calculations. Everything is identical
to Fig. 7-22 except that the 3⁄4-in acoustical tile has been added with
coefficients from the appendix. What area of tile is required? In Fig.
7-22 a total of 53.9 sabins is noted at the peak reverberation time at
1 kHz and 339.8 sabins at 125 Hz at which the reverberation time is a
reasonable 0.54 sec. How much 3⁄4-in acoustical tile would be required
to add 286 sabins at 1 kHz? The absorption coefficient of this material
is 0.84 at 1 kHz. To get 286 sabins at 1 kHz with this material would
require 286/0.84 = 340 sq ft of the material. This is entered in Fig. 7-24
and the calculations extended. Plotting these reverberation time points
gives the curve of Fig. 7-23B. Obtaining reverberation time this uniform
across the band on the first trial is unusual, but satisfying. The overall
precision of coefficients and measurements is so poor that the deviations
of curve B from flatness are insignificant.
No carpet has been specified in this example and people usually
demand it. The use of carpet would change everything as much
absorption in the high frequencies is added.
Reverberant Field
In our 23.3 × 16 × 10 ft room the volume is 3,728 cu ft. The inner surface
area is 1,533 sq ft. By statistical theory (geometrical ray acoustics
4
3
2
1
0
Reverberation time - seconds
125 250 500 1 k 2 k 4 k
Frequency - Hz
B
A
F I G U R E 7 - 2 3
The calculated reverberation characteristics of a 23.3 ×
16 × 10 ft room: (A) the “as found,” untreated condition
of Example 1, (B) treated condition of Example 2.
REVERBERATION 163
on which the Sabine equation is based) the mean free path (the
average distance sound travels between reflections) is 4V/S or (4)
(3,728)/1,533 = 9.7 ft. If the reverberation time is 0.3 second, there
would be at least 35 reflections during the 60 dB decay. This would
appear to be a fair involvement of all room surfaces.
In a small, relatively dead room such as the average studio, control
room, and listening room, one never gets very far away from the direct
influence of the source. A true reverberant field is often below the
ambient noise level. The reverberation time equations have been
derived for conditions that exist only in the reverberant field. In this
sense, then, the concept of reverberation time is inapplicable to small,
relatively dead rooms. And yet we measure something that looks very
much like what is measured in large, more live spaces. What is it?
What we measure is the decay rate of the normal modes of the room.
Each axial mode decays at its own rate determined by the absorbance
of a pair of walls and their spacing. Each tangential and oblique mode
S
sq ft
Material
Concrete
Gypsum board
Acoustical tile
373
1,159
340
125 Hz
a Sa
0.01
0.29
0.09
3.7
336.1
30.6
250 Hz
a Sa
0.01
0.10
0.28
3.7
115.9
95.2
500 Hz
a Sa
0.015
0.05
0.78
5.6
58.0
265.2
1 kHz
a Sa
0.02
0.04
0.84
7.5
46.4
285.6
2 kHz
a Sa
0.02
0.07
0.73
7.5
81.1
248.2
4 kHz
a Sa
0.02
0.09
0.64
7.5
104.3
217.6
Total sabins 370.4 214.8 328.8 339.5 336.8 329.4
Reverberation time
(seconds)
0.49 0.85 0.56 0.54 0.54 0.55
S  area of material
a  absorption coefficient for that material
and for that frequency (See Appendix)
Sa  S times a, absorption units, sabins
RT60  (0.049)(3728)
 182.7
Sa Sa
Size
Treatment
Floor
Walls
Ceiling
Volume
23.3 16 10 ft
Acoustical tile
Concrete
Gypsum board, 1/2", on frame construction
Ditto
(23.3) (16) (10)  3,728 cu ft
F I G U R E 7 - 2 4
Room conditions and calculations for Example 2.
164 CHAPTER SEVEN
has its own decay rate determined by distance traveled, the number of
surfaces involved, the variation of the absorption coefficient of the surfaces
with angle of incidence, etc. Whatever average decay rate is measured
for an octave of random noise will surely be representative of the
average decay rate at which that octave of speech or music signals
would die away. Although the applicability of computing reverberation
time from the equations based on reverberant field conditions might be
questioned because of the lack of reverberant field, the measured decay
rates (by whatever name you call them) most certainly apply to this
space and to these signals.
Endnotes
1Beranek, Leo L., Broadcast Studio Design, J. Soc. Motion Picture and Television Eng., 64,
Oct., 1955, 550-559.
2Schultz, Theodore J., Problems in the Measurement of Reverberation Time, J. Audio Eng.
Soc., 11, 4, 1963, 307-317.
3Matsudaira, T. Ken, et al., Fast Room Acoustic Analyzer (FRA) Using Public Telephone Line
and Computer, J. Audio Eng. Soc., 25, 3, 1977, 82-94.
4Mankovsky, V.S., Acoustics of Studios and Auditoria, London, Focal Press, 1971.
5Schroeder, M.R., New Method of Measuring Reverberation Time, J. Acous. Soc. Am., 37,
1965, 409-412.
6Peutz, V.M.A., Articulation Loss of Consonants as a Criterion for Speech Transmission in a
Room, J. Audio Eng. Soc., 19, 11, 1971, 915-919.
7Klein, W., Articulation Loss of Consonants as a Basis for the Design and Judgement of
Sound Reinforcement Systems, J. Audio Eng. Soc., 19, 11, 1971, 920-922.
8Beranek, L.L., Music, Acoustics, and Architecture, New York, John Wiley & Sons, 1962.
9Beranek, L.L., Concert and Opera Halls—How They Sound, Woodbury, N.Y., Acoustical
Society of America, 1996.
10Balachandran, C.G., Pitch Changes During Reverberation Decay, J. Sound and Vibration,
48, 4, 1976, 559-560
11Spring, N.F. and K.E. Randall, Permissible Bass Rise in Talks Studios, BBC Engineering, 83,
1970, 29-34.
12Jackson, G.M. and H.G. Leventhall, The Acoustics of Domestic Rooms, Applied Acoustics,
5, 1972, 265-277
13Hall, Gary, Digital Reverb—How It Works, MIX, 9, 6, 1985, 32.
14Young, Robert W., Sabine Equation and Sound Power Calculations, J. Acous. Soc. Am., 31,
12, 1959, 1681.
There are four basic approaches to reducing noise in a listening room
or a recording studio:
 Locating the room in a quiet place
 Reducing the noise energy within the room
 Reducing the noise output of the offending source
 Interposing an insulating barrier between the noise and the
room
Locating a sound-sensitive area away from outside interfering
sounds is a luxury few can enjoy because of the many factors (other
than acoustical) involved in site selection. If the site is a listening
room, which is part of a residence, due consideration must be given
to serving the other needs of the family—at least if some degree of
peace is to prevail. If the room in question is a recording or broadcast
studio, it is probably a part of a multipurpose complex and the noises
originating from business machines, air conditioning equipment, or
foot traffic within the same building, or even sounds from other studios,
may dominate the situation.
Control of
Interfering Noise
8
165
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
166 CHAPTER EIGHT
Noise Sources and Some Solutions
Protecting a room from street traffic noise is becoming more difficult
all the time. It is useful to remember that doubling the distance from a
noisy street or other sound source reduces the level of airborne noise
approximately 6 dB. Shrubbery and trees can help in shielding from
street sounds; a cypress hedge 2 ft thick gives about a 4 dB reduction.
The level of noise that has invaded a room by one means or another
can be reduced by introducing sound-absorbing material into the studio.
For example, if a sound level meter registers a noise level of 45 dB
inside a studio, this level might be reduced to 40 dB by covering the
walls with great quantities of absorbing materials. Going far enough in
this direction to reduce the noise significantly, however, would probably
make the reverberation time too short. The control of reverberation
must take priority. The amount of absorbent installed in the control of
reverberation will reduce the noise level only slightly, and beyond this
we must look to other methods for further noise reduction.
Reducing the noise output of the offending source, if accessible
and if possible, is the most logical and profitable approach. Traffic
noise on a nearby street or airplanes overhead may be beyond control,
but the noise output of a ventilating fan might be reduced 20 dB
by the installation of a pliant mounting or the separation of a metal
air duct with a simple canvas collar. Installing a carpet in a hall
might solve a foot traffic noise problem, or a felt pad might reduce a
typewriter noise problem. In most cases working on the offending
source and thus reducing its noise output is far more productive
than corrective measures at or within the room in question.
As for terminology, a wall, for example, must offer a given transmission
loss to sound transmitted through it, as shown in Fig. 8-1. An
outside noise level of 80 dB would be reduced to 35 dB by a wall having
a transmission loss of 45 dB. A 60 dB wall would reduce the same
noise level to 20 dB if no “flanking” or bypassing of the wall by other
paths is present. The wall “attenuates” the sound or it “insulates” the
interior from the outside noise. The walls, floor, and ceiling of the
sound-sensitive area must give the required transmission loss to outside
noises, reducing them to tolerable levels inside the room.
Noise can invade a studio or other room in the following ways: airborne,
transmitted by diaphragm action of large surfaces, transmitted
through solid structures, or a combination of all three.
CONTROL OF INTERFERING NOISE 167
Airborne Noise
A heavy metal plate with holes to the extent of 13 percent of the total
area can transmit as much as 97 percent of the sound impinging on it.
The amount of sound that can pass through a small crack or aperture
in an otherwise solid wall is astounding. A crack under a door or
loosely fitting electrical service box can compromise the insulating
properties of an otherwise excellent structure. Air-tightness is especially
necessary to insulate against airborne noises.
Noise Carried by Structure
Unwanted sounds can invade an enclosure by mechanical transmission
through solid structural members of wood, steel, concrete, or
masonry. Air conditioner noises can be transmitted to a room by the air
in the ducts, by the metal of the ducts themselves, or both. Water pipes
and plumbing fixtures have excellent sound-carrying capabilities.
It is very difficult to make a solid structure vibrate by airborne
noise falling upon it because of the inefficient transfer of energy from
tenuous air to a dense solid. On the other hand, a motor bolted to a
floor, a slammed door, or an office machine on a table with legs on the
bare floor can cause the structure to vibrate very significantly. These
vibrations can travel great distances through solid structure with little
loss. With wood, concrete, or brick beams, longitudinal vibrations are
80 dB
Outside
noise
level 45 dB T.L.
35 dB
In studio
80 dB
Outside
noise
level
60 dB T.L.
20 dB
In studio
T.L. is transmission loss
F I G U R E 8 - 1
The difference between the outside noise level and the desired noise level inside determines
the required transmission loss of the wall.
168 CHAPTER EIGHT
attenuated only about 2 dB in 100 ft. Sound travels in steel about 20
times as far for the same loss! Although joints and cross-bracing members
increase the transmission loss, it is still very low in common
structural configurations.
Noise Transmitted by Diaphragm Action
Although very little airborne sound energy is transmitted directly to a
rigid structure, airborne sound can set a wall to vibrating as a
diaphragm and the wall, in turn, can transmit the sound through the
interconnected solid structure. Such structure-borne sound might then
cause another wall at some distance to vibrate, radiating noise into the
space we are interested in protecting. Thus two walls interconnected
by solid structure can serve as a coupling agent between exterior airborne
noise and the interior of the listening room or studio itself.
Sound-Insulating Walls
For insulating against outside airborne sounds, the general rule is the
heavier the wall the better. The more massive the wall, the more difficult
it is for sound waves in air to move it to and fro. Figure 8-2 shows
how the transmission loss of a rigid, solid wall is related to the density
of the wall. The wall weight in Fig. 8-2 is expressed as so many pounds
per square foot of surface, sometimes called the surface density. For
example, if a 10 × 10 ft concrete block wall weighs 2,000 lb, the “wall
weight” would be 2,000 lb per 100 sq ft, or 20 lb per sq ft. The thickness
of the wall is not directly considered.
From Fig. 8-2 you can see that the higher the frequency, the greater
the transmission loss, or in other words, the better the wall is as a barrier
to outside noises. The line for 500 Hz is made heavier than the
lines for other frequencies as it is common to use this frequency for
casual comparisons of walls of different materials. However, don’t forget
that below 500 Hz the wall is less effective and for frequencies
greater than 500 Hz it is more effective as a sound barrier.
The transmission losses indicated in Fig. 8-2 are based on the mass
of the material rather than the kind of material. The transmission loss
through a layer of lead of certain thickness can be matched by a plywood
layer about 95 times thicker. But doubling the thickness of a concrete
wall, for instance, would increase the transmission loss only
about 5 dB.
CONTROL OF INTERFERING NOISE 169
A discontinuous structure such as bricks set in lime mortar conducts
sound less efficiently than a more homogeneous material like
concrete or steel. Unbridged air cavities between walls are very effective
in sound reduction, but completely unbridged cavities are unattainable.
Only in the case of two separate structures, each on its own
foundation, is this unbridged condition approached.
Porous Materials
Porous materials such as fiberglass (rock wool, mineral fiber) are
excellent sound absorbers and good heat insulators, but they are of
limited value in insulating against sound. Using fiberglass to reduce
sound transmission will help to a certain extent, but only moderately.
The transmission loss for porous materials is directly proportional to
the thickness traversed by the sound. This loss is about 1 dB (100 Hz)
to 4 dB (3,000 Hz) per inch of thickness for a dense, porous material
(rock wool, density 5 lb/cu ft) and less for lighter material. This direct
dependence of transmission loss on thickness for porous materials is
50
40
30
20
Transmission loss - dB
4,000 Hz 2,000 Hz 1,000 Hz 500 Hz
250 Hz
100 Hz
1 2 5 10 20 50 80
Wall weight - lb/sq ft
F I G U R E 8 - 2
The mass of the material in a barrier rather than the kind of material determines the transmission loss of sound
going through the barrier. The transmission loss is also dependent on frequency although values at 500 Hz are
commonly used in casual estimates. The wall weight is expressed in pounds per square foot of wall surface.
170 CHAPTER EIGHT
in contrast to the transmission loss for solid, rigid walls, which is
approximately 5 dB for each doubling of the thickness.
Sound Transmission Classification (STC)
The solid line of Fig. 8-3 is simply a replotting of data from the mass
law graphs of Fig. 8-2 for a wall weight of 10 lb per sq ft. If the mass
law were perfectly followed, we would expect the transmission loss of
a practical wall of this density to vary with frequency, as shown by the
solid line. Unfortunately, things are not this simple and actual measurements
of transmission loss on this wall might be more like the broken
line of Fig. 8-3. These deviations reflect resonance and other
effects in the wall panel, which are not included in the simple mass
law concept.
Because of such commonly occurring irregularities, it would be of
great practical value to agree on some arbitrary procedure of arriving
60
50
40
30
20
Transmission loss - dB
Actual
Mass law
100 200 500 1,000 2,000 5,000
Frequency - Hz
F I G U R E 8 - 3
Actual measurements of transmission loss in walls often deviate considerably from
the mass law (Fig. 8-2) because of resonances and other effects.
CONTROL OF INTERFERING NOISE 171
at a single number that would give a reasonably accurate indication
of the sound transmission loss characteristics of a wall. This has
been done in a procedure specified by the American Society for Testing
and Materials in which the measured graph of a wall would be
placed in a certain Sound Transmission Class (STC) by comparison
to a reference graph (STC contour). The details of this procedure are
beyond the scope of this book, but the results of such classification
have been applied to walls of various types to be described for ready
comparison. An STC rating of 50 dB for a wall would mean that it is
better in insulating against sound than a wall of STC 40 dB. It is not
proper to call STC ratings “averages” but the whole procedure is to
escape the pitfalls of averaging dB transmission losses at various frequencies.
1
Comparison of Wall Structures
Figure 8-4 gives the measured performance of a 4 in concrete block wall
as a sound barrier. It is interesting to note that plastering both sides
increases the transmission loss of the wall from STC 40 to 48. Figure 8-
5 shows a considerable improvement in doubling the thickness of the
concrete block wall. In this case the STC 45 is improved 11 dB by plastering
both sides. In Fig. 8-6 is illustrated the very common 2 × 4 frame
construction with 5⁄8-in gypsum board covering. The STC 34 without
fiberglass between is improved only 2 dB by filling the cavity with
fiberglass material, a meager improvement that would probably not justify
the added cost.
Figure 8-7 describes a very useful and inexpensive type of wall of
staggered stud construction. Here the inherently low coupling
between the two independent wall diaphragms is further reduced by
filling the space with fiberglass building material. Attaining the full
STC 52 rating would require careful construction to ensure that the
two wall surfaces are truly independent and not “shorted out” by electrical
conduits, outlet boxes, etc.
The last wall structure to be described is the double wall construction
of Fig. 8-8. The two walls are entirely separate, each having its
own 2 × 4 plate.2 Without fiberglass this wall is only 1 dB better than
the staggered stud wall of Fig. 8-7 but by filling the inner space with
building insulation, STC ratings up to 58 dB are possible.
172 CHAPTER EIGHT
It was stated earlier that porous sound absorbing materials are
of limited value in insulating against sound. This is true when normal
transmission loss is considered, but in structures as those in
Figs. 8-7 and 8-8, such porous materials have a new contribution to
make in absorbing sound energy in the cavity. This can improve
the transmission loss in some wall structures by as much as 15 dB,
principally by reducing resonances in the space between the walls,
while in others the effect is negligible. The low-density mineral
fiber batts commonly used in building construction are as effective
as the high-density boards, and they are much cheaper. Mineral
fiber batts within a wall may also meet certain fire-blocking
requirements in building codes.
16"
4"
8"
Construction - 4" concrete block
Weight - 18 lb/sq ft (plain)
30 lb/sq ft (plastered both sides)
Sound transmission class -
STC 40 dB plain
STC 48 dB plastered both sides
(Solite Corporation)
F I G U R E 8 - 4
Four-inch concrete block construction.
CONTROL OF INTERFERING NOISE 173
The staggered stud wall and the double wall, on the basis of mass
alone, would yield a transmission loss of only about 35 dB (Fig. 8-2). The
isolation of the inner and the outer walls from each other and the use of
insulation within have increased the wall effectiveness by 10 or 15 dB.
Double Windows
Between the control room and the studio a window is quite necessary,
and its sound transmission loss should be comparable to that of the
wall itself. A well-built staggered stud or double wall might have an
STC of 50 dB. To approach this performance with a window requires
very careful design and installation.3
16"
8"
8"
Construction - 8" concrete block
Weight - 30 lb/sq ft (plain)
67 lb/sq ft (plastered both sides)
Sound transmission class -
STC 45 dB plain
STC 56 dB plastered both sides
(Solite Corporation, and LECA, Norway)
F I G U R E 8 - 5
Eight-inch concrete block construction.
174 CHAPTER EIGHT
A double window is most certainly
indicated; a triple window adds little
more. The mounting must minimize coupling
from one wall to the other. One
source of coupling is the window frame,
another is the stiffness of the air between
the glass panels. The plan of Fig. 8-9 is a
practical solution to the double-window
problem for concrete block walls. Figure
8-9B is an adaptation to the staggered
stud construction. In the latter there are,
in effect, two entirely separate frames—
one fixed to the inner and the other to the
outer staggered stud walls. A felt strip
may be inserted between them to ensure
against accidental contact.
Heavy plate glass should be used, the
heavier the better. There is a slight advantage
in having two panes of different thickness. If
desired, one glass can be inclined to the
other to control light or external sound
reflections, but this will have negligible
effect on the transmission loss of the window
itself. The glass should be isolated from
the frame by rubber or other pliable strips.
The spacing between the two glass panels
has its effect—the greater the spacing the
greater the loss—but there is little gain in going beyond 8 inches, nor serious
loss in dropping down to 4 or 5 inches.
The absorbent material between the panes in the design of Fig. 8-9
discourages resonances in the air space. This adds significantly to the
overall insulation efficiency of the double window, and it should
extend completely around the periphery of the window. If the double
window of Fig. 8-9 is carefully constructed, sound insulation should
approach that of an STC 50 wall but will probably not quite reach it.
For the staggered stud wall in which a double window is to be placed,
the use of a 2 × 8 plate instead of the 2 × 6 plate will simplify mounting
of the inner and outer window frames.
5/8" Gypsum bd.
2 4
2 4
16"
Construction - Standard stud partition
Weight - 7.3 lb/sq ft
Sound transmission class
STC 34 dB without fiberglass
STC 36 dB with 31/2" fiberglass
(Owens-Corning Fiberglas Corporation)
F I G U R E 8 - 6
Standard stud partition.
CONTROL OF INTERFERING NOISE 175
Prefabricated double glass windows are available commercially,
one of which is rated at STC 49 dB.
Sound-Insulating Doors
The transmission loss of a door is determined by its mass, stiffness,
and air-tightness. An ordinary household panel door hung in the usual
way might offer less than 20 dB sound insulation. Increasing the
weight and taking reasonable precautions on seals might gain another
10 dB, but a door to match a 50 dB wall requires great care in design,
construction, and maintenance. Steel doors or patented acoustical
doors giving specified values of transmission
loss are available commercially but
they are quite expensive. To avoid the
expense of doors having high transmission
loss, sound locks are commonly used.
These small vestibules with two doors of
medium transmission loss are very effective
and convenient.
Doors with good insulating properties
can be constructed if the requirements of
mass, stiffness, and airtightness are met.
Figure 8-10 suggests one inexpensive
approach to the mass requirement, filling a
hollow door with sand. Heavy plywood (3⁄4
in) is used for the door panels.
Achieving a good seal around a “soundproof”
door can be very difficult. Great
force is necessary to seal a heavy door.
Wear and tear on pliant sealing strips can
destroy their effectiveness, especially at
the floor where foot-wear is a problem.
The detail of Fig. 8-10 shows one approach
to the sound leakage problem in which a
very absorbent edge built around the
periphery of the door serves as a trap for
sound traversing the crack between door
and jamb. This absorbent trap could also
5/8" Gypsum bd.
2 6
2 4
16"
Construction - Staggered stud partition
Weight - 7.2 lb/sq ft
Sound transmission class
STC 42 dB without fiberglass
STC 46 to 52 dB with fiberglass
(Owens-Corning Fiberglas Corporation)
F I G U R E 8 - 7
Staggered stud partition.
176 CHAPTER EIGHT
be embedded in the door jamb. Such a soft
trap could also be used in conjunction
with one of the several types of seals.
Figure 8-11 shows a do-it-yourself door
seal that has proved reasonably satisfactory.
The heart of this seal is a rubber or
plastic tubing an inch or less in outside
diameter with a wall thickness of about 3⁄32
in. The wooden nailing strips hold the
tubing to the door frame by means of a
canvas wrapper. A raised sill is required
at the floor if the tubing method is to be
used all around the door (or another type
of seal such as weatherstripping could be
utilized at the bottom of the door). An
advantage of tubing seal is that the degree
of compression of the tubing upon which
the sealing properties depend is available
for inspection.
A complete door plan patterned after
BBC practice is shown in Fig. 8-12. It is
based upon a 2-in-thick solid slab door and
utilizes a magnetic seal such as used on
refrigerator doors. The magnetic material is
barium ferrite in a PVC (polyvinyl chloride)
rod. In pulling toward the mild steel strip, a
good seal is achieved. The aluminum strip “C” decreases sound leakage
around the periphery of the door.
It is possible to obtain a very slight acoustical improvement and, to
some at least, an improvement in appearance by padding both sides of
a door. A plastic fabric over 1-in foam rubber sheet can be “quilted”
with upholstery tacks.
Noise and Room Resonances
Room resonances can affect the problem of outside noise in a studio.
Any prominent modes persisting in spite of acoustic treatment make
a room very susceptible to interfering noises having appreciable
energy at these frequencies. In such a case a feeble interfering sound
could be augmented by the resonance effect to a very disturbing level.
5/8" Gypsum bd.
2 4
2 4
16"
Construction - Double wall
Weight - 7.1 lb/sq ft
Sound transmission class
STC 43 dB without fiberglass
STC 55 dB with 31/2" fiberglass
STC 58 dB with 9" fiberglass
(Owens-Corning Fiberglas Corporation)
F I G U R E 8 - 8
Double wall partition.
CONTROL OF INTERFERING NOISE 177
Active Noise Control
Many have had the idea of canceling
noise by radiating a replica of the noise
in inverse phase. It sounds simple but is
very difficult to implement. It has been
made to work fairly well in the immediate
vicinity of a telephone in an industrial
area of heavy noise. Very serious
investigations are in progress to apply
the principle to automobiles and other
such controlled spaces. The prospects of
active noise control being applied in
sound-sensitive areas such as home listening
rooms, recording studios, or control
rooms are rather remote but new
digital sound processing techniques
might change that.
Heavy plywood
Framing
Sand
Perforated hardboard
Mineral
fiber
1/16" lead
F I G U R E 8 - 1 0
A reasonably effective and inexpensive “acoustic”
door. Dry sand between the plywood faces adds to the
mass and thus the transmission loss. Sound traveling
between the door and jamb tends to be absorbed by
the absorbent door edge.
A
1/4" Rubber or neoprene strips
B
1/4" Polished plate glass 3/8" Polished plate glass
Mineral fiber absorbent
Cover cloth
F I G U R E 8 - 9
Four-inch concrete block construction.
178 CHAPTER EIGHT
Endnotes
1Jones, R.E., How to Design Walls of Desired STC
Ratings, Sound and Vibration, 12, 8, 1978, 14-17.
2Green, D.W. and C.W. Sherry, Sound Transmission Loss
of Gypsum Wallboard Partitions, Report No. 3, 2x4
Wood Stud Partitions, J. Acous. Soc. Am., 71, 4, 1982,
908-914.
3Everest, F.A., Glass in the Studio, dB The Sound. Eng.
Mag. Part I, 18, 3 April 1984, 28-33. Part II, 18, 4 May
1984, 41-44.
Door
F I G U R E 8 - 1 1
A door can be sealed by compressible rubber or plastic
tubing held in place by a fabric wrapper.
A
B
A - PVC magnetic seal
B - Mild steel 3/4" 1/8" flat
C - Aluminum cover strip 11/2" 1/8"
D - Aluminum glazing beads 3/4" 3/8"
E - Polished plate glass - 5/8"
C
B
E
D
C
B
A A
F I G U R E 8 - 1 2
A BBC door design utilizing magnetic seals of the
type used on refrigerator doors.
The law of the conservation of energy states that energy can neither
be created nor destroyed, but it can be changed from one form to
another. If we have some sound energy in a room to get rid of, how
can it be done? Sound is the vibratory energy of air particles, and it
can be dissipated in the form of heat. If it takes the sound energy of
a million people talking to brew a cup of tea, we must give up any
idea of heating our home with sound from the high-fidelity loud
speakers.
Dissipation of Sound Energy
When sound wave S hits a wall (such as in Fig. 9-1), what happens to
the energy it contains? If the sound wave is traveling in air and it
strikes a concrete block wall covered with an acoustical material, there
is first a reflected component A returned to the air from the surface of
the acoustical material. Of course, there is a certain heat loss E in the
air that is appreciable only at the higher audio frequencies.
Some of the sound penetrates the acoustical material represented
by the shaded layer in Fig. 9-1. The direction of travel of the sound is
refracted downward because the acoustical material is denser than
air. There is heat lost (F) by the frictional resistance the acoustical
material offers to the vibration of air particles. As the sound ray
Absorption of Sound
9
179
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
180 CHAPTER NINE
strikes the surface of the concrete blocks, two things happen: a component
is reflected (B), and the ray is bent strongly downward as it
enters the much denser concrete blocks. Of course, there is further
heat loss (G) within the concrete blocks. As the ray travels on, getting
weaker all the time, it strikes the concrete-air boundary and undergoes
another reflection (C) and refraction (D) with heat lost (I, J, and
K) in three media.
S
E
F
G
H
A
B
K
I
C
D
J
F I G U R E 9 - 1
A sound ray impinging on an acoustical material on a masonry wall undergoes reflection
from three different surfaces and absorption in the air and two different materials, with
different degrees of refraction at each interface. In this chapter, the absorbed component
is of chief interest.
ABSORPTION OF SOUND 181
The sound ray S of Fig. 9-1 experiences many rather complex
events during its odyssey through this barrier, and every reflection and
passage through air or acoustical material dissipates some of its original
energy. The refractions bend the ray but do not necessarily dissipate
heat. Fortunately, all this minutia is not involved in practical
absorption problems.
Evaluation of Sound Absorption
The absorption coefficient is a measure of the efficiency of a surface
or material in absorbing sound. If 55 percent of the incident sound
energy is absorbed, the absorption coefficient is said to be 0.55. One
square foot of this material gives 0.55 absorption units (sabins). An
open window is considered a perfect absorber because sound passing
through it never returns to the room. It would have an absorption
coefficient of 1.0. Ten square feet of open window would give 10
sabins of absorbance.
The absorption coefficient of a material varies with frequency and
with the angle at which the sound wave or ray impinges upon the
material. In an established sound field in a room, sound is traveling in
every imaginable direction. What we need in our calculations are
sound absorption coefficients averaged over all possible angles of
incidence.
Young has pointed out a long-standing and widespread confusion
in the field of acoustics concerning the sound absorption coefficient.
1 There really are two kinds, one based on the arithmetic mean
reflection coefficient of the several sound absorbing surfaces, a, and
the other the geometric mean reflection coefficient,
, which are
related by:
a loge (1 
) (9-1)
in which
a  Sabine absorption coefficient

 energy absorption coefficient
We can skirt this problem by concentrating our attention on the
Sabine coefficient, a, which is actually what is measured and published
in various tables.
182 CHAPTER NINE
Reverberation Chamber Method
The reverberation chamber method of determining the absorption
coefficients of absorbing materials automatically measures the average
value. This chamber is a large room with highly reflective walls, ceiling,
and floor. The reverberation time of such a room is very long, and
the longer it is, the more accurate the measurement. A standard sample
of the material to be tested, 8 9 ft in size, is laid on the floor and
the reverberation time measured. Comparing this time with the known
reverberation time of the empty room yields the number of absorption
units the sample adds to the room. From this the absorption attributed
to each square foot of material is determined, giving the equivalent of
the absorption coefficient.
This description is a highly simplified view of the reverberation
chamber method. The construction of the chamber is very important to
ensure many modal frequencies and to equalize mode spacing as
much as possible. The position of the sound source and the number
and position of the measuring microphones must be carefully worked
out. It is common to use large rotating vanes to ensure adequate diffusion
of sound. All absorption coefficients supplied by manufacturers
for use in architectural acoustic calculations are measured by the
reverberation chamber method.
If the open window is the perfect absorber, what happens if chamber
measurements show absorption coefficients greater than 1? This is
a regular occurrence. The diffraction of sound from the edges of the
standard sample makes the sample appear, acoustically, of greater area
than it really is. There is no standard method of making adjustments
for this artifact. Some manufacturers publish the actual measured values
if greater than unity; others arbitrarily adjust the values down to
unity or to 0.99.
Sound absorption coefficients vary with frequency. It is standard
practice to publish coefficients and make calculations for the following
six frequencies: 125, 250, 500, 1,000, 2,000, and 4,000 Hz.
Impedance Tube Method
The Kundt tube has been applied to the measurement of the absorption
coefficient of materials. Used in this way it is commonly a standingABSORPTION
OF SOUND 183
wave tube or an impedance tube. No matter what name it bears, it is a
very handy device for quickly and accurately determining coefficients.
It also has the advantage of small size, modest demands in terms of supporting
equipment, and it requires only a small sample. This method is
primarily used for porous absorbers because it is not suited to those
absorbers that depend on area for their effect such as vibrating panels
and large slat absorbers.
The construction and operation of the impedance tube are illustrated
in Fig. 9-2. The tube usually has a circular cross section with
rigid walls. The sample to be tested is cut to fit snugly into the tube. If
the sample is intended to be used while mounted on a solid surface, it
is placed in contact with the heavy backing plate. If the material is to
be used with a space behind it, it is mounted an appropriate distance
from the backing plate.
At the other end of the tube is a small loudspeaker with a hole drilled
through its magnet to accommodate a long, slender probe tube coupled
to a microphone. Energizing the loudspeaker at a given frequency sets
up standing waves due to the interaction of the outgoing wave with the
wave reflected from the sample. The form of this standing wave gives
important information on the absorbance of the material under test.
Pmax
Pmin
Pressure
Sample
Backing plate
Z tube
Probe tube
Loudspeaker
Mic
Filter
F I G U R E 9 - 2
The standing-wave tube method of measuring the absorption coefficient of absorbing
materials at normal incidence.
184 CHAPTER NINE
The sound pressure is maximal at the surface of the sample. As
the microphone probe tube is moved away from the sample, the
sound pressure falls to the first minimum. Successive, alternating
maxima and minima will be detected as the probe tube is further
withdrawn. If n is the ratio of the maximum sound pressure to its
adjacent minimum, the normal absorption coefficient an is equal to:
an  (9-2)
which is plotted in Fig. 9-3.2
4

n  
n
1
  2
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient - an
1 2 3 5 10 20 30 50 100
Standing-wave ratio - n
F I G U R E 9 - 3
A graph for interpreting the standing-wave ratio in terms of absorption coefficient. The
standing-wave ratio can be found by dividing any pressure maximum by its adjacent
pressure minimum (see Fig. 9-2).
ABSORPTION OF SOUND 185
The advantages of the impedance tube method are offset by the disadvantage
that the absorption coefficient so determined is truly only for normal
(perpendicular) incidence. In a room, sound impinges on the surface
of a material at all angles. Figure 9-4 is a graph for obtaining the randomincidence
coefficient from the absorption coefficient for normal incidence
as measured by the standing-wave method. The random-incidence
coefficients are always higher than the coefficients for normal incidence.
Tone-Burst Method
The utilization of short pulses of sound has made it possible to perform
anechoic acoustical measurements in ordinary rooms. It takes time for
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient - random incidence
0 0.2 0.4 0.6 0.8 1.0
Absorption coefficient - normal incidence
F I G U R E 9 - 4
An approximate relationship between the absorption coefficients at normal incidence
and those with random incidence.
186 CHAPTER NINE
bothersome reflections from walls and other surfaces to arrive at the
measuring position. If the pulse is short enough, the time gate can be
opened only for the desired sound pulse, shutting out the interfering
pulses. This tone-burst method can be used to measure the soundabsorption
coefficient of a material at any desired angle of incidence.
Such an arrangement is illustrated in principle in Fig. 9-5. The
source-microphone system is calibrated at distance x as shown in Fig.
9-5A. The geometry of Fig. 9-5B is then arranged so that the total path
of the pulse reflected from the material to be tested is equal to this
same distance x. The strength of the reflected pulse is then compared
to that of the unreflected pulse at distance x to determine the absorption
coefficient of the sample.
A recent surge of interest in the influence of individual reflections
on the timbre of sound is a new and promising development in
acoustics. In this new field specific normal reflections, called “early
sound,” are of special interest. Although random-incidence coefficients
are still of interest in room reverberation calculations, for these
image control problems normal-incident reflection coefficients are
generally required. Thus, interest may be returning to normal (rightangle)
coefficients obtained by the resonance-
tube method. There may even be a
renewed interest in the old “quarter-wavelength
rule” in which the porous absorber
for normal incidence must be at least a
quarter wavelength thick at the frequency
of interest. For example, for a frequency of
1 kHz, the minimum absorber thickness
should be about 3.4″.
Mounting of Absorbents
The method of mounting the test sample
on the reverberation chamber floor is
intended to mimic the way the material is
actually used in practice. Table 9-1 lists
the standard mountings, both in the old
form and in the ASTM form that will be
used in the future.
x
x2
x2
Barrier
Sample

A
B
F I G U R E 9 - 5
Determining the absorption coefficients of materials
by a tone-burst method. The source-microphone system
is calibrated at distance X as shown in (A).
ABSORPTION OF SOUND 187
The mounting has a major effect on the absorption characteristics
of the material. For example, the absorption of porous materials is
much greater with an airspace between the material and the wall.
Tables of absorption coefficients should always identify the standard
mounting or include a description of the way the material was
mounted during the measurements, or the coefficients are of little
value. Mounting A with no air space between the sound absorbing
material and the wall is widely used. Another one commonly used is
mounting E-405, which is at least an approximation to the varying
spaces encountered in suspended ceilings (Fig. 9-6).
Mid/High-Frequency Absorption by Porosity
The key word in this discussion of porous sound absorbers is interstices.
It is simply the space between two things. If a sound wave
strikes a wad of cotton batting, the sound energy sets the cotton fibers
vibrating. The fiber amplitude will never be as great as the air particle
amplitude of the sound wave because of frictional resistance. Some
sound energy is changed to frictional heat as fibers are set in motion,
restricted as this motion is. The sound penetrates more and more into
the interstices of the cotton, losing more and more energy as more and
Table 9-1 Mountings commonly used in sound-absorption measurements.
New mounting Old mounting
designation* designation**
A Material directly on hard surface #4
B Material cemented to plasterboard #1
C-20 Material with perforated, expanded or other
open facing furred out 20 mm (3/4″) #5
C-40 Ditto, furred out 40 mm (112″) #8
D-20 Material furred out 20 mm (34″) #2
E-405 Material spaced 405 mm (16″) from hard surface #7
*ASTM designation: E 795-83.
**Mountings formerly listed by Acoustical and Board Products Manufacturers Association, ABPMA (formerly
the Acoustical and Insulating Materials Association, AIMA).
The cooperation of Riverbank Acoustical Laboratory in providing information on current practice is gratefully
acknowledged.
188 CHAPTER NINE
more fibers are vibrated. Cotton is an excellent sound absorber that has
been specified in studio treatment in Africa where it was plentiful and
cheap, and because imported materials were out of the question.
Porous absorptive materials most commonly used as sound
absorbers are usually fuzzy, fibrous materials in the form of boards,
foams, fabrics, carpets, cushions, etc. If the fibers are too loosely
packed, there will be little energy lost as heat. On the other hand, if
they are packed too densely, penetration suffers and the air motion
cannot generate enough friction to be effective. Between these
extremes are many materials that are very good absorbers of sound.
These are commonly composed of cellulose or mineral fiber. Their
effectiveness depends on the thickness of the material, the airspace,
and the density of the material.
The absorption efficiency of materials depending on the trapping
and dissipating of sound energy in tiny pores can be seriously
impaired if the surface pores are filled so that penetration is limited.
Coarse concrete block, for example, has many such pores and is a fair
absorber of sound. Painting that block fills the surface pores and
greatly reduces sound penetration, and thus absorption. However, if
spray painted with a nonbridging paint, the absorption may be
reduced very modestly. Acoustical tile painted at the factory mini-
Mounting A
Mounting E-405
Air space
Mounting
depth
16"
F I G U R E 9 - 6
Commonly used standard mountings associated with listings of absorption coefficients. With Mounting A the
material is flat against the backing. Mounting B (not shown) is similar to Mounting A but with a 116″ airspace
resulting when acoustical materials are cemented to a surface. Mountings B and A are essentially the
same. Mounting E-405 applies to suspended ceilings with lay-in panels. (See Table 9-1.)
ABSORPTION OF SOUND 189
mizes the problem of reduced absorption. Under certain conditions, a
painted surface can reduce porosity but act as a diaphragm that might
actually become a fair absorber on a different principle, that of a
damped vibrating diaphragm.
In the first radio broadcasting studios, the acoustical treatment was
an overuse of carpeted floors and drapes, which emphasized a serious
shortcoming of most porous absorbers—that of poor low frequency
absorption. Tiles of cellulose fiber with perforated faces became the
next style of treatment, but they were also deficient in low-frequency
absorption. Overly enthusiastic use of porous absorbers, not only during
the early days but even today, causes overabsorption of the highfrequency
sound energy, without touching a major problem of room
acoustics, low-frequency standing waves.
Glass Fiber: Building Insulation
Great quantities of glass fiber materials are used in the acoustical
treatment of recording studios, control rooms, and public gathering
spaces. These glass fibers can consist of both special, high-density
materials, and ordinary building insulation. In wood or steel stud single
frame walls, double walls, and staggered stud walls thermal insulating
batts are commonly used. This material usually has a density of
about 1 lb/cu ft. Such material is often identified as R-11, R-19, or
other such numbers. These R-prefix designations have to do with
thermal insulating qualities, but are related to thickness. Thus R-8 is
2.5″ thick, R-11 is 3.5″, and R-19 is 6″.3
Building insulation installed within a wall increases its transmission
loss a modest amount, primarily by reducing cavity resonance
that would tend to couple the two wall faces at the resonance frequency
of the cavity. A certain increase in the transmission loss of the
wall can also be attributed to attenuation of sound in passing through
the glass fiber material, but this loss is small because of the low density
of the material. Considering all mechanisms, the transmission loss
of a staggered stud wall with a layer of gypsum board on each side can
be increased about 7 dB by adding 3.5″ of building insulation. A double
wall might show as much as a 12 dB increase by adding 3.5″ and
15 dB with 9″ of insulation. As far as wall transmission loss is concerned,
using the denser, more expensive glass fiber between wall
faces offers only a slight advantage over ordinary building insulation.
190 CHAPTER NINE
Glass Fiber: Boards
This type of glass fiber usually used in the acoustical treatment of
audio rooms is in the form of semi-rigid boards of greater density than
building insulation. Typical of such materials are Owens-Corning
Type 703 Fiberglas and Johns-Manville 1000 Series Spin-Glass, both of
3 lb/cu ft density. Other densities are available, for example, Type 701
has a density of 1.58 lb/cu ft and Type 705 a density of 6 lb/cu ft. The
Type 703 density, however, is widely applied in studios.
These semi-rigid boards of glass fiber do not excel cosmetically,
hence they are usually covered with fabric. They do excel in sound
absorption.
Acoustical Tile
During the 1960s and 1970s many top-line manufacturers of acoustical
materials offered their competitive lines of 12″- -12″-acoustical tiles.
Surface treatments of the tiles included even-spaced holes, random
holes, slots, or fissured or other special textures. They continue to be
available from local building material suppliers. Such tiles are reputable
products for noise and reverberation control as long as they are
used with full knowledge of their limitations. One of the problems of
using acoustical tile in critical situations is that absorption coefficients
are rarely available for the specific tile obtainable. Going back into the
earlier literature, the average of the coefficients for eight cellulose and
mineral fiber tiles of 34″ thickness is shown in Fig. 9-7. The range of
the coefficients is indicated by the vertical lines. The average points
could be used for 34″ tile for which no coefficients are available. Coefficients
20% lower would be a fair estimate for 12″ tiles.
Effect of Thickness of Absorbent
It is logical to expect greater sound absorption from thicker materials,
but this logic holds primarily for the lower frequencies. Figure 9-8
shows the effect of varying absorbent thickness where the absorbent is
mounted directly on a solid surface (mounting A). In Fig. 9-8, there is
little difference above 500 Hz in increasing the absorbent thickness
from 2 inches to 4 inches, but there is considerable improvement
below 500 Hz as thickness is increased. There is also a proportionally
greater gain in overall absorption in a 1-inch increase of thickness in
ABSORPTION OF SOUND 191
going from 1 to 2 inches than going from 2 to 3 inches or 3 to 4 inches.
A 4-inch thickness of glass fiber material of 3-lb/cu ft density has
essentially perfect absorption over the 125-Hz to 4-kHz region.
Effect of Airspace Behind Absorbent
Low-frequency absorption can also be improved by spacing the
absorbent out from the wall. This is an inexpensive way to get
improved performance—within limits. Figure 9-9 shows the effect on
the absorption coefficient of furring 1-inch glass-fiber wallboard out
from a solid wall. Spacing 1-inch material out 3 inches makes its
absorption approach that of the 2-inch material of Fig. 9-8 mounted
directly on the wall.
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
F I G U R E 9 - 7
The average absorption characteristics of 8 acoustical tile brands of 34″ thickness. The
vertical lines show the spread of the data.
192 CHAPTER NINE
Effect of Density of Absorbent
Glass fiber and other materials come in various
densities from the flimsy thermal insulation
batts to the semi-rigid and rigid
boards used widely in industry. All of these
have their proper place in acoustical treatment
of spaces, but the question right now
is, “What effect does density—the packing
of the fibers—have on sound absorption
coefficient?” In other words, is the sound
able to penetrate the interstices of the highdensity,
harder surface material as well as
one of the flimsy kind? The answer appears
in Fig. 9-10, which shows relatively little
difference in absorption coefficient as the
density is varied over a range of almost 4 to
1. For very low densities the fibers are so
widely spaced that absorption suffers. For
extremely dense boards, the surface reflection
is high and sound penetration low.
Open-Cell Foams
Flexible polyurethane foams are widely
used in noise quieting of automobiles,
machinery, aircraft, and in various industrial
applications. After a slow start, foams
are finding some application as sound
absorbers in architectural applications,
including recording studios and home listening
rooms. Figure 9-11 is a photograph
of one form of Sonex,4 a foam product contoured
to simulate the wedges used in anechoic
rooms. These are shaped in male
and female molds and come in meshed
pairs. This material can be cemented or
stapled to the surface to be treated.
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
4"
3"
2"
1"
Thickness of
material
F I G U R E 9 - 8
The thickness of glass-fiber sound-absorbing material
determines the low-frequency absorption (density,
3 lb/cu ft). The material is mounted directly on
a hard surface.
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
3"
2"
1"
0"
Air space
Thickness of
material = 1"
F I G U R E 9 - 9
The low-frequency absorption of 1-inch glass fiberboard
is improved materially by spacing it out from
the solid wall.
ABSORPTION OF SOUND 193
The sound absorption coefficients of
Sonex for thicknesses of 2″, 3″, and 4″ are
shown in Fig. 9-12 for Mounting A. The 2″
glass fiber of Fig. 9-8 is considerably superior
acoustically to the 2″ Sonex but a few
things should be considered in this comparison.
These are:
 The Type 703 has a density of 3
lb/cu ft while Sonex is 2 lb/cu ft.
 The 2″ Sonex is the wedge height
and the average thickness is far less,
while the 703 thickness prevails
throughout.
 Comparing the two products is, in a
sense, specious because the much
higher cost of Sonex is justified in
the minds of many by appearance
and ease of mounting rather than
straight acoustical considerations.
Drapes as Sound Absorbers
Drapes are a porous type of sound absorber because air can flow
through the fabric under pressure. Variables affecting absorbency
include weight of material, degree of drape, and distance from the
wall. Data are scarce, but Fig. 9-13 compares the absorption of 10, 14,
and 18 oz/sq yd velour hung straight and presumably at some distance
from the wall. One intuitively expects greater absorption with
heavier material. However, the greater absorption in going from 14
to 18 oz/sq yd than in going from 10 to 14 oz/sq yd is difficult
to explain. The effect, whatever it is, is concentrated in the 500- to
1-kHz region.
The amount of fullness of the drape has a great effect as shown in
Fig. 9-14. The “draped to 78 area” means that the entire 8/8 area is
drawn in only slightly (18) from the flat condition. The deeper the
drape fold, the greater the absorption.
1.0
0.8
0.6
0.5
0.2
0
Absorption coefficient
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
Density
1.6 lb/cu ft
3.0 lb/cu ft
6.0 lb/cu ft
F I G U R E 9 - 1 0
The density of glass-fiber absorbing material has relatively
little effect on absorption in the range of 1.6
to 6 lb/cu ft. The material is mounted directly on
solid wall.
194 CHAPTER NINE
The distance a drape is hung from a reflecting surface can have a
great effect on its absorption efficiency. This is best explained by Fig.
9-15. In Fig. 9-15A a drape or other porous material is hung parallel to
a solid wall, and the distance d between the two is varied. The frequency
of the sound impinging on the porous material is held constant
at 1,000 Hz. If the sound absorption provided by the porous material is
measured, we find that it varies greatly as the distance d from the wall
is changed. Looking at the situation closely reveals that the wavelength
of the sound is related to maxima and minima of absorption.
The wavelength of sound is the speed of sound divided by frequency,
which in the case of 1,000 Hz, is 1,130/1,000  1.13 feet or about 13.6
inches. A quarter wavelength is 3.4 inches, and a half wavelength is
6.8 inches. We note that there are absorption peaks at 14 wavelength,
F I G U R E 9 - 1 1
Sonex contoured acoustical foam simulating anechoic wedges. This is an open-cell type of foam.
ABSORPTION OF SOUND 195
and if we carry it further than indicated in Fig. 9-15A at each odd multiple
of quarter wavelengths. Absorption minima occur at even multiples
of quarter wavelengths.
These effects are explained by reflections of the sound from the
solid wall. At the wall surface, pressure
will be highest, but air particle velocity
will be zero because the sound waves cannot
supply enough energy to shake the
wall. At a quarter wavelength from the
wall, however, pressure is zero, and air
particle velocity is maximum. By placing
the porous material, such as a drape, a
quarter wavelength from the wall, it will
have maximum absorbing effect because
the particle velocity is maximum. At half
wavelength, particle velocity is at a minimum,
hence absorption is minimum.
In Fig. 9-15B, the spacing of the drape
from the wall is held constant at 12 inches
as the absorption is measured at different
frequencies. The same variation of absorption
is observed, maximum when the spacing
is at odd quarter wavelengths and
minimum at even quarter wavelengths
from the wall. At this particular spacing of
12 inches, a wavelength of spacing occurs
at 1,130/1  1,130 Hz, a quarter wavelength
at 276 Hz, and a half wavelength at
565 Hz, etc.
Figure 9-16 shows actual reverberationchamber
measurements of the absorption
of 19 oz/sq yd velour. The solid graph is
presumably for a drape well removed from
all walls. The other graphs, very close
together, are for the same material spaced
10 cm (about 4″) and 20 cm (about 8″) from
the wall. The 10-cm distance is one wavelength
at 3,444 Hz, the 20-cm distance is at
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
4"
3"
2"
F I G U R E 9 - 1 2
Absorption coefficients of Sonex contoured foam of
various thicknesses.4
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
18 oz/sq yd
14 oz/sq yd
10 oz/sq yd
F I G U R E 9 - 1 3
The sound absorption of velour hung straight for
three different weights of fabric. (After Beranek.18)
196 CHAPTER NINE
1,722 Hz. The odd multiples of both the
10-cm and the 20-cm quarter wavelengths
are spotted on the upper part of Fig. 9-16.
The absorption of the velour is greater
when spaced from the wall, and the effect
is greatest in the 250-Hz to 1-kHz region.
At 125 Hz, the 10 and 20 cm spacing adds
practically nothing to the drape absorption
because at 125 Hz, the quarter wavelength
spacing is 2.26 feet. When referring to
quarter wavelengths, sine waves are
inferred. Absorption measurements are
invariably made with bands of random
noise. Hence we must expect the wiggles
of Fig. 9-15B to be averaged out by the use
of such bands.
Carpet as Sound Absorber
Carpet commonly dominates the acoustical picture in spaces as
diverse as living rooms, recording studios, and churches. It is the one
amenity the owner often specifies in advance and the reason is more
often comfort and appearance than acoustic. For example, the owner
of a recording studio with a floor area of 1,000 square feet specified
carpet. He was also interested in a reverberation time of about 0.5 second,
which requires 1,060 sabins of absorption. At the higher audio
frequencies, this heavy carpet and pad with an absorption coefficient
of around 0.6 gives 600 sabins of absorption at 4 kHz or 57% of the
required absorption for the entire room before the absorption needs of
walls and ceiling are even considered. The acoustical design is almost
frozen before it is started.
There is another, more serious problem. This high absorbance of
carpet is only at the higher audio frequencies. Carpet having an
absorption coefficient of 0.60 at 4 kHz offers only 0.05 at 125 Hz. In
other words, the 1,000 sq ft of carpet introduces 600 sabins at 4 kHz
but only 50 sabins at 125 Hz! This is a major problem encountered in
many acoustical treatment jobs. The unbalanced absorption of carpet
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
Draped to 1/2 area
Draped to 3/4 area
Draped to 7/8 area
F I G U R E 9 - 1 4
The effect of draping on the sound absorption of
drapes. “Draped to 12 area” means that folds are
introduced until the resulting drape area is half that
of the straight fabric. (After Mankovsky.9)
ABSORPTION OF SOUND 197
1
0
Absorption coefficient
0 2 6 4 8
Freq. = 1,000 Hz
d = Distance from wall-inches
4
2
d
Variable
A
Porous
materials
Solid
wall
1
0
Absorption coefficient
100 1 kHz 2 kHz 3 kHz
Frequency - Hz
4
B
Porous
materials
Solid
wall
3
4
5
4
7
4
12"
Fixed
F I G U R E 9 - 1 5
(A) The sound absorption of porous material such as a drape varies with the distance from
a hard wall. The maximum absorption is achieved when the drape is one-quarter wavelength
from the wall, the minimum at a half wavelength. (B) The sound absorption of
porous material hung at a fixed distance from a wall will show maxima at spacings of a
quarter wavelength and odd multiples of a quarter wavelength as the frequency is varied.
198 CHAPTER NINE
can be compensated in other ways, principally with resonant-type,
low-frequency absorbers.
To compound the problem of unbalanced absorption of carpet,
dependable absorption coefficients are hard to come by. A bewildering
assortment of types of carpet and variables in underlay add to the
uncertainty. Unfortunately, reverberation chamber measurements of
random-incidence absorption coefficients for specific samples of carpet
are involved and expensive, and generally unavailable to the
acoustical designer. Therefore, it is well to be informed on the factors
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
20 cm
spacing
4
3
4
4
10 cm
spacing
20 cm
from wall
5
4
7
4
3
4
10 cm from wall
Free space
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
F I G U R E 9 - 1 6
Actual measurements of sound-absorption coefficients of a velvet material (19
oz/sq yd) in free space and 10 cm and 20 cm from a solid wall. The point at which
the increase in absorption due to wall reflection is to be expected are indicated.
(After Mankovsky.9)
ABSORPTION OF SOUND 199
affecting the absorption of carpets and make judgments on what coefficients
to use for the specific carpet at hand.
Effect of Carpet Type on Absorbance
What variation in sound absorption should one expect between types
of carpet? Figure 9-17 shows the difference between a heavy Wilton
carpet and a velvet carpet with and without a latex back. The latex
backing seems to increase absorption materially above 500 Hz and to
decrease it a modest amount below 500 Hz.
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
B
A
C
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
F I G U R E 9 - 1 7
A comparison of the sound-absorption characteristics of three different types of carpet.
(A) Wilton, pile height 0.29″, 92.6 oz/sq yd. (B) Velvet, latex backed, pile height
0.25″, 76.2 oz/sq yd. (C) The same velvet without latex backing, 37.3 oz/sq yd, all with
40-oz hair felt underlay. (After Harris.5)
200 CHAPTER NINE
Effect of Carpet Underlay on Absorbance
Hair felt was formerly used almost exclusively as the padding under the
carpet. It is interesting that in the early days of this century hair felt was
used for general acoustical treatment until it was displaced by numerous
proprietary materials. Today foam rubber, sponge rubber, felts,
polyurethane, or combinations have replaced hair felt. Foam rubber is
made by whipping a latex water dispersion, adding a gelling agent, and
pouring into molds. The result is always open-celled. Sponge rubber, on
the other hand, formed by chemically generated gas bubbles, can yield
either open or closed cells. Open cells provide the interstices required
for good sound absorption while closed cells do not.
The influence of underlay on carpet absorption is very great. Figure
9-18 shows chamber measurements of absorption coefficients for a single
Axminster type of carpet with different underlay conditions.
Graphs A and C show the effect of hair felt of 80 and 40 oz/sq yd
weight. Graph B shows an intermediate combination of hair felt and
foam. While these three graphs differ considerably, they all stand in
stark contrast to graph D for the carpet laid directly on bare concrete.
Conclusion: the padding underneath the carpet contributes markedly
to overall carpet absorption.
Carpet Absorption Coefficients
The absorption coefficients plotted in Figs. 9-17 and 9-18 are taken
from Harris’s 1957 paper,5 perhaps the most exhaustive study of carpet
characteristics available. In general, these coefficients are higher than
those in currently published tables. In Fig. 9-19 the coefficients listed
in a widely used publication6 and included in the appendix are plotted
for comparison with Figs. 9-17 and 9-18. Carpets vary widely,
which can account for some of the great variability with which any
designer of acoustical systems is confronted.
Sound Absorption by People
People absorb sound too. Just how much of this is due to absorption by
flesh and how much by hair and clothing has yet to be reported. The
important thing is that the people making up an audience account for
a significant part of the sound absorption of the room. It also makes an
ABSORPTION OF SOUND 201
acoustical difference whether one or ten people are in a small monitoring
room. The problem is how to rate human absorption and how to
involve it in calculations. The usual method of multiplying a human
absorption coefficient by the area of a human has its problems. The
easy way is to determine the absorption units (sabins) a human presents
at each frequency and add them to the sabins of the carpet,
drapes, and other absorbers in the room at each frequency. Table 9-2
lists the absorption of informally dressed college students in a classroom
along with a range of absorption for more formally dressed people
in an auditorium environment.
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
A
B
C
D
F I G U R E 9 - 1 8
Sound-absorption characteristics of the same Axminster carpet with different underlay.
(A) 80-oz hair felt. (B) Hair felt and foam. (C) 40-oz hair felt. (D) No underlay, on bare
concrete. (After Harris.5)
202 CHAPTER NINE
For 1 kHz and higher, the absorption offered by college students in
informal attire in the Spartan furnishings of a classroom falls at the
lower edge of the range of a more average audience. The low-frequency
absorption of the students, however, is considerably lower than that of
the more formally dressed people.
Sound propagated across rows of people, as in an auditorium or
music hall, is subjected to an unusual type of attenuation. In addition
to the normal decrease in sound with distance from the stage, there is
an additional dip of up to 15 or 20 dB around 150 Hz and spreading
over the 100- to 400-Hz region.7 In fact, this is not strictly an audience
effect because it prevails even when the seats are empty. A similar dip
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
D
B
C
A
F I G U R E 9 - 1 9
Carpet absorption coefficients from a commonly used table. (A) 18″ pile height. (B)
14″ pile height. (C) 316″ combined pile and foam. (D) 516″ combined pile and foam.
Compare these graphs with those of Figs. 9-16 and 9-17.6
ABSORPTION OF SOUND 203
in sound-pressure level affects those most important first reflections
from the sidewalls. All of this apparently results from interference.
Absorption of Sound in Air
For frequencies 1 kHz and above and for very large auditoriums, the
absorption of sound by the air in the space becomes important. For
example, a church seating 2,000 has a volume of about 500,000 cubic
feet.
Frequency (Hz) Absorption (sabins per 1,000 cu ft)
1,000 0.9
2,000 2.3
4,000 7.2
Notice that for 50 percent relative humidity the absorption is 7.2
sabins per 1,000 cubic feet or a total of (500) (7.2)  3,600 sabins at 4
kHz. This is equivalent to 3,600 square feet of perfect absorber.
This could be 20 percent to 25 percent of the total absorption in the
space, and there is nothing that can be done about it other than to take
it into consideration and taking consolation in at least knowing why
the treble reverberation time falls off so much!
Low-Frequency Absorption by Resonance
The concept of wall reflection, graphically portrayed for drapes in Fig.
9-15, applies as well to bass traps. This phrase is applied to many
Table 9-2 Sound absorption by people (Sabins per person).
Frequency, Hz
125 250 500 1 kHz 2 kHz 4 kHz
College students informally dressed — 2.5 2.9 5.0 5.2 5.0
seated in tablet arm chairs6
Audience seated, depending on 2.5– 3.5– 4.0– 4.5– 5.0– 4.5–
spacing and upholstery of seats22 4.0 5.0 5.5 6.5 7.0 7.0
204 CHAPTER NINE
kinds of low-frequency sound absorbers, such as panel absorbers, but
perhaps it should really be reserved for a special type of reactive cavity
absorber that has been widely used in sound recording circles.8 A
true bass trap is shown in Fig. 9-20. It is simply a box or cavity of critical
depth but with a mouth opening of size to suit particular purposes.
This is a tuned cavity with a depth of a quarter wavelength at
the design frequency at which maximum absorption is desired. Sound
absorption at the lowest octave or two of the audible spectrum is often
difficult to achieve. The bass trap is commonly used in recording studio
control rooms to reduce standing waves at these bass frequencies.
The sound pressure at the bottom of the cavity is maximum at the
quarter-wavelength design frequency. The air particle velocity is zero
at the bottom. At the mouth the pressure is zero and the particle velocity
is maximum, which results in two interesting phenomena. First, a
glass fiber semi-rigid board across the opening offers great friction to
the rapidly vibrating air particles resulting in maximum absorption at
this frequency. In addition, the zero pressure at the opening consti-
Glass fiber
Pressure
Particle
velocity
F I G U R E 9 - 2 0
The bass trap depends upon reflections of sound from the bottom for its action. The
pressure for the frequency at which the depth is a quarter wavelength is maximum at
the bottom and the particle velocity is zero at the bottom. At the mouth, the pressure
is zero (or very low) and the particle velocity is maximum. Absorbent placed where the
particle velocity is maximum will absorb sound very effectively. The same action occurs
at odd multiples of the quarter wavelength.
ABSORPTION OF SOUND 205
tutes a vacuum that tends to suck sound energy into this “sound sink”
from surrounding areas. The bass trap’s effect, then, is greater than its
opening area would suggest because of this “vacuum cleaner” effect.
The bass trap effect, like the drape spaced from a reflective wall,
occurs not only at a quarter-wavelength depth, but also at odd multiples
of a quarter wavelength. Great trap depths are required for very
low bass frequencies. For example, a quarter wavelength for 40 Hz is
7 ft. Unused spaces above control room ceilings and between inner
walls and outer shells are often used for trap space.
Diaphragmatic Absorbers
The absorption of sound at the lower audible frequencies can be
achieved by porous absorbers or by resonant (or reactive) absorbers.
Glass fiber and acoustical tiles are common forms of porous absorption
in which the sound energy is dissipated as heat in the interstices of the
fibers. The absorption of commercial forms of glass fiber and other
fibrous absorbers at low audio frequencies, however, is quite poor. To
absorb well, the thickness of the porous material must be comparable
to the wavelength of the sound. At 100 Hz, the wavelength is 11.3 ft,
and using any porous absorber approaching this thickness would be
impractical. For this reason we turn our attention to the resonant type
of absorber to obtain absorption at low frequencies.
Some of the great chamber music rooms owe their acoustical excellence
to the low-frequency absorption offered by extensive paneled
walls. Plywood or tongue-and-groove flooring or subflooring vibrates
as a diaphragm and contributes to low-frequency absorption. Drywall
construction on walls and the ceiling does the same thing. All such
components of absorption must be included in the acoustical design of
a room, large or small.
Drywall or gypsum board plays a very important part in the construction
of homes, studios, control rooms, and other spaces. It also
plays an important part in the absorption of low-frequency sound in
these spaces. Usually, such low-frequency absorption is welcome, but in
larger spaces designed for music, drywall surfaces can absorb so much
low-frequency sound as to prevent the achievement of the desired reverberatory
conditions. Drywall of 12-inch thickness on studs spaced 16
inches offers an absorption coefficient of 0.29 at 125 Hz and even higher
206 CHAPTER NINE
at 63 Hz (which would be of interest in music recording studios). Drywall
absorption in small audio rooms is free; you just have to recognize
its existence and remember to include its low-frequency absorption in
calculations.
The simplest resonant type of absorber utilizes a diaphragm vibrating
in response to sound and absorbing some of that sound by frictional
heat losses in the fibers as it flexes.
A piece of 14″ plywood is an excellent example. Assume that it is
spaced out from the wall on two-by-fours, which gives close to 334″
airspace behind. The frequency of resonance of this structure can be
calculated from the expression:
fo  (9-3)
where
fo  frequency of resonance, Hz
m  surface density of the panel, lb/sq ft of panel surface
d  depth of airspace, inches.
The surface density of 14″ plywood, 0.74 lb/sq ft, can be measured
or found in the books. Substituting in Eq. 9-3 we get:
f
o

fo  102 Hz
Figure 9-21 is a graphical solution of Eq. 9-3 for maximum convenience.
Knowing only the thickness of the plywood and the depth of
the space behind the plywood, the frequency of resonance can be read
off the diagonal lines. Equation 9-3 applies to membranes and
diaphragms of materials other than plywood such as masonite, fiberboard,
or even Kraft paper. For other than plywood, the surface density
must be determined. The surface density is easily found by weighing a
piece of the material of known area.
How accurate are Eq. 9-3 and Fig. 9-21? Actual measurements on
three plywood membrane absorbers are shown in Fig. 9-22. Such calculations
of the frequency of peak absorption at resonance are not perfect,
but they are a good first approximation of sufficient accuracy for
most purposes.
170

(0.74) (3.75 )
170

(m) (d )
ABSORPTION OF SOUND 207
Plywood: 1/16" 1/8" 3/16" 1/4" 1/2"
60
50
30
40
20
10
8
6
5
4
3
2
1
5/16" 3/8"
Depth - inches
1 2 3 4 6 8 10 20 30
Surface density - Oz/sq. ft.
fo = 40 Hz
50 Hz
60 Hz
70 Hz
85 Hz
100 Hz
125 Hz
150 Hz
175 Hz
200 Hz
250 Hz
300 Hz
350 Hz
400 Hz
F I G U R E 9 - 2 1
Convenient design chart for resonant panel absorbers. (See also Fig. 9-34.)
208 CHAPTER NINE
Panel sound absorbers are quite simple to build. An example of a
panel absorber to be mounted on a flat wall or ceiling surface is shown
in Fig. 9-23. A 14″ or 116″ plywood panel is fastened to a wooden
framework to give the desired spacing from the wall. A glass or mineral
fiber blanket of 1″ to 112″ is glued to the wall surface. An airspace
of 14″ or 12″ should be maintained between the absorbent and the rear
surface of the plywood panel.
A corner panel absorber is shown in Fig. 9-24. For computations,
an average depth is used. Depths greater and smaller than the average
simply mean that the peak of absorbance is broader than that of an
absorber with uniform depth. Spacing the absorber 14″ to 12″ from the
rear of the plywood panel is simple if a mineral fiber board such as
Tectum is used. Using a flexible blanket of glass fiber requires support
by hardware cloth, open-weave fabric, or even expanded metal. For
0.8
0.6
0.4
0.2
0
Absorption coefficient
100 1,000 4,000
Frequency - Hz
C
B
A
F I G U R E 9 - 2 2
Actual absorption measurements of three panel absorbers. (A) 316″ plywood with 2″ air
space.19 (B) 116″ plywood with 1″ mineral wool and 14″ air space. (C) The same as (B)
but for 18″ panel.20
ABSORPTION OF SOUND 209
applications in which reflected mid/high-frequency sound from the
panel absorber might create problems, a facing of glass fiber board
would not interfere with the low-frequency absorbing action if it was
spaced to avoid damping of the vibration of the plywood panel. Chapter
15 emphasizes that all room modes terminate in the corners of a
room. A corner panel absorber could be used to control such modes.
Polycylindrical Absorbers
Flat paneling in a room might brighten an
interior decorator’s eye and do some good
acoustically, but wrapping a plywood or
hardboard skin around some semicylindrical
bulkheads can provide some very
attractive features. These polycylindrical
elements (polys) are coming back into
fashion. A few are used in top-flight
recording studios today. Visually, they are
rather overpowering, which can be good or
bad depending on the effect one wants to
achieve. With polys it is acoustically possible
to achieve a good diffuse field along
with liveness and brilliance, factors tending
to oppose each other in rooms with flat
surfaces.
One of the problems of using polys has
been the scarcity of published absorption
coefficients. The Russian acoustician, V. S.
Mankovsky, has taken care of that in his
book.9 As expected, the larger the chord
dimension, the better the bass absorption.
Above 500 Hz there is little significant difference
between the polys of different
sizes.
The overall length of polys is rather
immaterial, ranging in actual installations
from the length of a sheet of plywood to
the entire length, width, or height of a
1/4"–1/2" Airspace
1"–11/2" Glass or
mineral fiber
1/4"–1/16" Plywood
F I G U R E 9 - 2 3
Typical resonant panel absorber for wall mounting.
210 CHAPTER NINE
studio. It is advisable, however, to break up the cavity behind the
poly skin with randomly spaced bulkheads. The polys of Fig. 9-25
incorporate such bulkheads.
Should the polys be empty or filled with something? Mankovsky
again comes to our rescue and shows us the effect of filling the cavities
with absorbent material. Figures 9-25D and 9-25C show the increase in
bass absorption resulting from filling the cavities with absorbent. If
needed, this increased bass absorption can be easily achieved by simply
filling the polys with glass fiber. If the bass absorption is not
needed the polys can be used empty. The great value of this adjustable
1"–11/2" Glass or
mineral fiber
1/4"–1/16" Plywood
1/4"–1/2" Airspace
F I G U R E 9 - 2 4
Typical resonant panel absorber for either vertical or horizontal corner mounting.
45"
16"
A
0.6
0.4
0.2
0
Absorption coeff.
Empty
(Estimated)
100 1,000 4,000
Frequency - Hz
35"
12"
B
0.6
0.4
0.2
0
Absorption coeff.
100 1,000 4,000
Frequency - Hz
Empty
(Estimated)
C
28"
10"
0.4
0.2
0
Absorption coeff.
100 1,000 4,000
Frequency - Hz
Empty
Filled with mineral wool
20"
8"
D
0.4
0.2
0
Absorption coeff.
100 1,000 4,000
Frequency - Hz
Empty
Filled with mineral wool
Empty
F I G U R E 9 - 2 5
Measured absorption of polycylindrical diffusors of various chord and height dimensions.
In C and D, graphs are shown for both empty polys and for polys filled with mineral
wool. In A and B only empty poly data are available; the broken lines are estimated
absorption when filled with mineral wool. (After Mankovsky.9)
212 CHAPTER NINE
feature will become more apparent as the actual acoustical design of
listening rooms and studios is approached.
Poly Construction
The construction of polycylindrical diffusers is reasonably simple. A
framework for vertical polys is shown in Fig. 9-26 mounted above a
structure intended for a low-frequency slat absorber. In this photograph
the variable chord dimensions are apparent, and also the random
placement of bulkheads so that cavities will be of various
volumes resulting in different natural cavity frequencies. It is desirable
that each cavity be essentially airtight, isolated from adjoining
cavities by well-fitted bulkheads and framework. Irregularities in the
wall can be sealed with a nonhardening acoustical sealant. The bulkheads
of each poly are carefully cut to the same radius on a bandsaw.
Sponge rubber weatherstripping with an adhesive on one side is
struck to the edge of each bulkhead to ensure a tight seal against the
plywood or hardboard cover. If such precautions are not taken, annoying
rattles and coupling between cavities can result.
The polys of Fig. 9-26 use 18″ tempered
Masonite as the poly skin. A few hints can
simplify the job of stretching this skin. In
Fig. 9-27 slots of a width to fit the
Masonite snugly are carefully cut along
the entire length of strips 1 and 2 with a
radial saw. Let us assume that poly A is
already mounted and held in place by
strip 1, which is nailed or screwed to the
wall. Working from left to right, the next
job is to mount poly B. First the left edge of
Masonite sheet B is inserted in the remaining
slot of strip 1. The right edge of
Masonite sheet B is then inserted in the
left slot of strip 2. If all measurements and
cuts have been accurately made, swinging
strip 2 against the wall should make a
tight seal over the bulkheads 3 and weatherstripping
4. Securing strip 2 to the wall
F I G U R E 9 - 2 6
The construction of polys in a motion picture sound
mixing studio. Note the foam rubber anti-rattle strip
on the edge of each bulkhead. Also note the random
spacing of the bulkheads. (Moody Institute of Science
photo, reprinted with permission of the Journal of the
Audio Engineering Society.21)
ABSORPTION OF SOUND 213
completes poly B. Poly C is mounted in a similar fashion and so on to
the end of the series of polys. The end result is shown in Fig. 9-28.
Notice that the axes of symmetry of the polys on the side wall are perpendicular
to those on the rear wall. If polys were used on the ceiling,
their axes should be perpendicular to both the others.
It is quite practical and acceptable to construct each poly as an
entirely independent structure rather than building them on the wall.
Such independent polys can be spaced at will.
Membrane Absorbers
Building insulation commonly comes
with a kraft paper backing. Between walls
this paper has no significant effect, but if
building insulation is to be used as a
sound absorber on walls, perhaps behind a
fabric facing, the paper becomes significant.
Figure 9-29 compares the sound
absorption efficiency of R-19 (6 inch) and
R-11 (3.5 inch) with the kraft paper backing
exposed and with the glass fiber
exposed to the incident sound. When the
paper is exposed it shields the glass fiber
from sound above 500 Hz but has little
effect below 500 Hz. The net effect is an
absorption peak at 250 Hz (R-19) and 500
Hz (R-11), which may be important in
A
1
3
4
B 2
C
F I G U R E 9 - 2 7
The method of stretching the plywood or hardboard skin over the poly bulkheads shown
in Fig. 9-26.
F I G U R E 9 - 2 8
Finished poly array of Fig. 9-26 mounted on the wall
above a low-frequency absorber structure. Note that
the axes of the polys on the rear wall are perpendicular
to the axes of the polys on the other wall. (Moody
institute of Science photo, reprinted with permission of
Journal of the Audio Engineering Society.21)
214 CHAPTER NINE
room treatment. With insulation exposed there is essentially perfect
absorption above 250 Hz (R-19) or 500 Hz (R-11).
Building insulation has not caught on as an inexpensive absorbing
material. One reason is that some sort of cosmetic and protective cover
is required, but this is often true of denser materials as well. Fabric,
1.2
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
A
R-19
Kraft paper toward sound
Insulation toward sound
125 250 500 1 kHz 2 kHz 4 kHz
Insulation toward sound
Kraft
paper
toward
sound
R-11
1.2
1.0
0.8
0.6
0.4
0.2
0 B 125 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
Frequency - Hz
F I G U R E 9 - 2 9
When ordinary building insulation is used as a wall treatment (perhaps with fabric facing)
the position of the kraft paper backing becomes important. (A) R-19 Fiberglass
building insulation, (B) R-11 Fiberglass building insulation, Mounting A.
ABSORPTION OF SOUND 215
expanded metal, metal lath, hardware cloth, or even perforated vinyl
wall covering can be used as a cover. Do not be surprised by absorption
coefficients greater than 1.0. This manufacturer elected to publish the
coefficients actually obtained from laboratory measurements rather
than arbitrarily reducing those over 1.0 to 0.99 or 1.0. The greater
absorption of the standard 8 9 ft sample results from edge diffraction
and other effects that make the sample appear larger acoustically than
it really is.
Helmholtz Resonators
The Helmholtz type of resonator is widely used to achieve adequate
absorption at lower audio frequencies. There is nothing particularly
mysterious about such resonators; in fact they pop up in various forms
in everyday life. Blowing across the mouth of any bottle or jug produces
a tone at its natural frequency of resonance. The air in the cavity
is springy, and the mass of the air in the neck of the jug reacts with
this springiness to form a resonating system, much as a weight on a
spring vibrating at its natural period. Change the volume of the air cavity,
or the length or diameter of the neck, and you change the frequency
of resonance. Such a Helmholtz resonator has some very
interesting characteristics. For instance, sound is absorbed at the frequency
of resonance and at nearby frequencies. The width of this
absorption band depends on the friction of the system. A glass jug
offers little friction to the vibrating air and would have a very narrow
absorption band. Adding a bit of gauze across the mouth of the jug or
stuffing a wisp of cotton into the neck, the amplitude of vibration is
reduced and the width of the absorption band is increased.
The sound impinging on a Helmholtz resonator that is not absorbed
is reradiated. As the sound is reradiated from the resonator opening, it
tends to be radiated in a hemisphere. This means that unabsorbed
energy is diffused, and diffusion of sound is a very desirable thing in a
studio or listening room.
Bottles and jugs are not appropriate forms of a Helmholtz resonator
with which to apply the resonance principle in studios. An interesting
experiment conducted many years ago at Riverbank Acoustical Laboratories
bears this out.10 To demonstrate the effectiveness of a continuously
swept narrow-band technique of measuring sound absorption
216 CHAPTER NINE
coefficients, the idea was conceived to measure the absorption of Coca
Cola bottles. A tight array of 1,152 empty 10-oz bottles was arranged in
a standard 8 9 ft space on the concrete floor of the reverberation
chamber. It was determined that a single, well-isolated bottle has an
absorption of 5.9 sabins at its resonance frequency of 185 Hz, but with
a bandwidth (between 3dB points) of only 0.67 Hz! Absorption of 5.9
sabins is an astounding amount of absorption for a Coke bottle! This is
about what a person, normally clothed, would absorb at 1,000 Hz, or
what 5.9 sq ft of glass fiber (2 in thick, 3 lb/cu ft density) would absorb
at midband. The sharpness of this absorption characteristic is even
more amazing. This would correspond to a Q of 185/0.67  276! As
interesting as these data are, they tell us that leaving an empty Coke
bottle in a studio will not devastate the acoustics of the room, but it
might have a tiny effect at 185 Hz.
In Helmholtz resonators, we have acoustical artifacts that far antedate
Helmholtz himself. Resonators in the form of large pots were
used in ancient times by the Greeks and Romans in their open-air
theaters. Apparently they were used to provide some reverberation
in this nonreverberant outdoor setting. Some of the larger pots that
have survived to modern times have reverberation times of from 0.5
to 2 seconds. These would also absorb sound at the lower frequencies.
Groupings of smaller pots supplied sound absorption at the
higher frequencies.
More recently (that is, in medieval times) such resonators were
used in a number of churches in Sweden and Denmark.11 Pots like
those of Fig. 9-30 were embedded in the walls, presumably to reduce
low-frequency reverberation that is often a problem in churches.
Ashes have been found in some of the pots, undoubtedly introduced to
“kill the Q” of the ceramic pot and to broaden the frequency of its
effectiveness.
If bottles and ceramic pots are not suitable forms of Helmholtz
resonators for a studio, what is? Figure 9-31 shows a conveniently
idealized square bottle with a tubular neck. This bottle alone would
produce its characteristic tone if one were to blow across the opening.
Stacking these bottles does not detract from the resonator
action, but rather enhances it. It is a small step to a box of length L,
width W, and depth H that has a lid of thickness equal to the length
of necks of the bottles. In this lid are drilled holes having the same
ABSORPTION OF SOUND 217
diameter as the holes in the neck. It is just a bit harder to realize that
partitions between each segment can be removed without greatly
affecting the Helmholtz action. In this
way, a Helmholtz resonator of the perforated
face type can be related to oddly
shaped bottles, giving something of a
visual picture of how perforated face resonators
perform.
In a similar way, Fig. 9-32 illustrates
another bottle with an elongated slit
neck. These, too, can be stacked, even in
multiple rows. It is but a short step to a
slot-type resonator. The separating walls
in the air cavity can also be eliminated
without destroying the resonator action.
A word of caution is in order, however.
Subdividing the airspace can improve
the action of perforated face or slit resonators
but only because this reduces
F I G U R E 9 - 3 0
Pots embedded in the walls of medieval churches in Sweden and Denmark served as
Helmholtz resonators, absorbing sound. Ashes, found in some of the pots, apparently
served as a dissipative agent. (After Brüel.11)
H
W
L
F I G U R E 9 - 3 1
Development of perforated-face Helmholtz resonator
from a single rectangular-battle resonator.
218 CHAPTER NINE
spurious, unwanted modes of vibration being set up within the air
cavity.
Perforated Panel Absorbers
Perforated hardboard or plywood panels spaced from the wall constitute
a resonant type of sound absorber.12–15 Each hole acts as the
neck of a Helmholtz resonator, and the share of the cavity behind
“belonging” to that hole is comparable to the cavity of the
Helmholtz resonator. In fact, we can view this structure as a host of
coupled resonators. If sound arrives perpendicular to the face of the
perforated panel, all the tiny resonators are in phase. For sound
waves striking the perforated board at an angle, the absorption efficiency
is somewhat decreased. This loss can be minimized by sectionalizing
the cavity behind the perforated face with an egg crate
type of divider of wood or corrugated paper.
F I G U R E 9 - 3 2
Development of a slot-type Helmholtz resonator from a single rectangular-bottle resonator.
ABSORPTION OF SOUND 219
The frequency of resonance of perforated panel absorbers backed
by a subdivided air space is given approximately by:
fo 200 (9-4)
in which
fo  frequency, Hz
p  perforation percentage,
 hole area divided by panel area 100
t  effective hole length, inches, with correction factor applied,
 (panel thickness)  (0.8) (hole diameter)
d  depth of air space, inches
There is a certain amount of confusion in the literature concerning
p, the perforation percentage. Some writers use the decimal ratio of
hole area to panel area, rather than the percentage of hole area to panel
area, introducing an uncertainty factor of 100. This perforation percentage
is easily calculated by reference to Fig. 9-33.
Equation 9-4 is true only for circular holes. This information is presented
in graphical form in Fig. 9-34 for a panel thickness of 316″.
Common pegboard with holes 316″ in diameter spaced 1″ on centers
with the square configuration of Fig. 9-33 has 2.75% of the area in
holes. If this pegboard is spaced out from the wall by 2 4s on edge,
the system resonates at about 420 Hz and the peak absorption appears
near this frequency.
p(
d) (t)
S
S d
PERF % = 78.5 ( d )2
s
S
d
S
PERF % = 90.6 ( d )2
s
r
w
PERF % = 100 ( r ) wr
F I G U R E 9 - 3 3
Formulas for calculating perforation percentage for Helmholtz resonators.
220 CHAPTER NINE
E D C B A
10
7
5
3
2
1
0.7
0.5
0.3
0.2
0.1
Perforation, percent
8"
6"
4"
2"
1"
Panel
thickness 3/16"
50 100 200 500 1,000 2,000
Frequency, Hz
F I G U R E 9 - 3 4
A graphic presentation of Eq. 9-4 relating percent perforation of perforated panels, the
depth of air space, and the frequency of resonance. The graphs are for a panel of 316″
thickness. (See also Fig. 9-21.) (A) For 1″ furring lumber. The lines are drawn to correspond
to furring lumber which is furnished, e.g., the line for 8″ is actually 7 34 airspace.
(B) For 2″ furring lumber. (C) For 4″ furring lumber. (D) For 6″ furring lumber.
(E) For 8″ furring lumber.
ABSORPTION OF SOUND 221
In commonly available perforated materials, such as pegboard, the
holes are so numerous that resonances at only the higher frequencies
can be obtained with practical air spaces. To obtain much needed lowfrequency
absorption, the holes can be drilled by hand. Drilling 732″
holes 6″ on centers gives a perforation percentage of about 1.0%.
Figure 9-35 shows the effect of varying hole area from 0.18% to
8.7% in a structure of otherwise fixed dimensions. The plywood is
532″ thick perforated with 316″ holes, except for the 8.7% case in
which the hole diameter is about 34″. The perforated plywood sheet is
spaced 4″ from the wall and the cavity is half filled with glass fiber and
half is air space.
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
A
B
C
D
2"
4"
100 1,000 4,000
Frequency - Hz
F I G U R E 9 - 3 5
Actual absorption measurements of perforated panel absorbers of 4″ air space, half
filled with mineral wool and for panel thickness of 532″. (A) Perforation 0.18%. (B)
Perforation 0.79%. (C) Perforation 1.4%. (D) Perforation 8.7%. Note that the presence
of the mineral wool shifts the frequency of resonance considerably from the theoretical
values of Eq. 9-4 and Fig. 9-34. (Data from Mankovsky.9)
222 CHAPTER NINE
Figure 9-36 is identical to Fig. 9-35 except that the perforated plywood
is spaced 8″ and glass fiber of 4″ thickness is mounted in the cavity.
The general effect of these changes is a substantial broadening of
the absorption curve.
It would be unusual to employ such perforated panel absorbers
without acoustic resistance in the cavity in the form of glass fiber batts
or boards. Without such resistance the graph is very sharp. One possible
use of such sharply tuned absorbers would be to control specific
troublesome room modes or isolated groups of modes with otherwise
minimum effect on the signal and overall room acoustics.
Table 9-3 includes the calculated frequency of resonance of 48 different
combinations of airspace depth, hole diameter, panel thickness,
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
4"
100 1,000 4,000
Frequency - Hz
8"
A
B
C
D
F I G U R E 9 - 3 6
Actual absorption measurements on the same perforated panel absorbers of Fig. 9-35
except that the air space is increased to 8″, half of which is taken up with mineral fiber.
Panel thickness is 532″. (A) Perforation 0.18%. (B) Perforation 0.79%. (C) Perforation
1.4%. (D) Perforation 8.7%. (Data from Mankovsky.9)
ABSORPTION OF SOUND 223
Table 9-3 Helmholtz low-frequency absorber perforated-face type.
Depth of Hole Panel x Hole Freq. of
airspace dia. dia. % Perf. spacing resonance
358″ 18″ 18″ 0.25% 2.22″ 110 Hz
0.50 1.57 157
0.75 1.28 192
1.00 1.11 221
1.25 0.991 248
1.50 0.905 271
2.00 0.783 313
3.00 0.640 384
358″ 18″ 14″ 0.25% 2.22″ 89 Hz
0.50 1.57 126
0.75 1.28 154
1.00 1.11 178
1.25 0.991 199
1.50 0.905 217
2.00 0.783 251
3.00 0.640 308
358″ 14″ 14″ 0.25% 4.43″ 89 Hz
0.50 3.13 126
0.75 2.56 154
1.00 2.22 178
1.25 1.98 199
1.50 1.81 217
2.00 1.57 251
3.00 1.28 308
558″ 18″ 18″ 0.25% 2.22″ 89 Hz
0.50 1.57 126
0.75 1.28 154
1.00 1.11 178
1.25 0.991 199
1.50 0.905 218
2.00 0.783 251
3.00 0.640 308
224 CHAPTER NINE
and hole spacing. This convenient listing should assist in approximating
the desired condition.
Slat Absorbers
Another type of resonant absorber is that utilizing closely spaced slats
over a cavity. The mass of the air in the slots between the slats reacts
with the springiness of air in the cavity to form a resonant system,
again comparable to the Helmholtz resonator. The glass fiber board
usually introduced behind the slots acts as a resistance, broadening
the peak of absorption. The narrower the slots and the deeper the cavity,
the lower the frequency of maximum absorption.
The resonance frequency of the slat absorber can be estimated from
the statement16:
fo  216 (9-5)
p(
d) (D)
Table 9-3 Helmholtz low-frequency absorber perforated-face type. (Cont.)
Depth of Hole Panel x Hole Freq. of
airspace dia. dia. % Perf. spacing resonance
558″ 18″ 14″ 0.25% 2.22″ 74 Hz
0.50 1.57 105
0.75 1.28 128
1.00 1.11 148
1.25 0.991 165
1.50 0.905 181
2.00 0.783 209
3.00 0.640 256
558″ 14″ 14″ 0.25% 4.43″ 63 Hz
0.50 3.13 89
0.75 2.56 109
1.00 2.22 126
1.25 1.98 141
1.50 1.81 154
2.00 1.57 178
3.00 1.28 218
ABSORPTION OF SOUND 225
in which
fo  frequency of resonance, Hz
p  percent perforation (see Fig. 9-33)
D  airspace depth, inches
d  thickness of slat, inches
Suggestion: If slats are mounted vertically, it is recommended that
they be finished in a dark color conforming to the shadows of the slots
to avoid some very disturbing “picket fence” optical effects!
Placement of Materials
The application of sound-absorbing materials in random patches has
already been mentioned as an important contribution to diffusion.
Other factors than diffusion might influence placement. If several
types of absorbers are used, it is desirable to place some of each type
on ends, sides, and ceiling so that all three axial modes (longitudinal,
transverse, and vertical) will come under their influence. In rectangular
rooms it has been demonstrated that absorbing material placed near
corners and along edges of room surfaces is most effective. In speech
studios, some absorbent effective at the higher audio frequencies
should be applied at head height on the walls. In fact, material applied
to the lower portions of high walls can be as much as twice as effective
as the same material placed elsewhere. Untreated surfaces should
never face each other.
Winston Churchill once remarked that as long as he had to wear
spectacles he intended to get maximum cosmetic benefit from them.
So it is with placement of acoustical materials. After the demands of
acoustical function have been met, every effort should be made to
arrange the resulting patterns, textures, and protuberances into esthetically
pleasing arrangements, but do not reverse priorities!
Reverberation Time
of Helmholtz Resonators
Some concern has been expressed about the possibility of acoustically
resonant devices, such as Helmholtz absorbers, “ringing” with a
“reverberation time” of their very own adding coloration to the voice
226 CHAPTER NINE
and music signals. It is true that any resonant
system, electronic or acoustical, has a
certain time constant associated with it.
The Q-factor (quality factor) describes the
sharpness of tuning of the Helmholtz resonator
as shown in Fig. 9-37. Once the
tuning curve has been obtained experimentally,
the width of the tuning curve at
the 3 dB points gives f. The Q of the
system is then Q  fo/f, where fo is the
frequency to which the system is tuned.
Measurements on a number of perforated
and slat Helmholtz absorbers gave Qs
around 1 or 2 but some as high as 5. Table
9-4 shows how the decay rate of resonant
absorbers of several Qs relates to reverberation
time.
With resonant absorber Qs of 100, real
problems would be encountered in a room
having a reverberation time of, say, 0.5 second as the absorbers tailed
off sound for several seconds. However, Helmholtz absorbers with
such Qs would be very special devices, made of ceramic, perhaps.
Absorbers made of wood with glass fiber to broaden the absorption
curve have Qs so low that their sound dies away much faster than the
studio or listening room itself.
Taming Room Modes
The following example of taming a troublesome room mode is based
on Acoustisoft’s ETF 5 room acoustics analyzer program.23 The ETF
3dB
 f
fo
F I G U R E 9 - 3 7
Once the tuning curve of a Helmholtz type resonant
absorber has been determined, its Q-factor can be
found from the expression, f0/f. The “reverberation
time” of such absorbers is very short for Qs normally
encountered.
Table 9-4 Sound decay in resonant absorbers.
“Reverberation time”
Q fo (seconds)
100 100 Hz 2.2
5 100 0.11
1 100 0.022
ABSORPTION OF SOUND 227
stands for energy, time, frequency. By means of this program the
detailed low-frequency modal structure of the room is revealed in Fig.
9-38A and B. The mode that caused an audible distortion in room
sound is the one with the pronounced reverberant “tail” at 47 Hz on
the extreme left of Fig. 9-38A. Once the culprit is identified, steps can
be taken to reduce its activity so that it will behave as the other modes
of the room.
The solution rests in the placement of a highly tuned absorber in
the room at some point of high pressure of the 47-Hz mode. The
20 40 60 80 100 120 140 160 180 200
0
-4
-8
-12
-16
-20
-24
-28
-32
-36
-40
0
300
600
900
1200
1500
1800
Time (ms)
200 180 160 140 120 100 80 60 40 20
Level (dB)
Frequency (Hz)
Before
A
F I G U R E 9 - 3 8 A
Low-frequency model structure of the sound field of a small room before introduction of the tuned Helmholtz
resonator absorber.
228 CHAPTER NINE
locating of high-pressure points of the 47-Hz mode is found simply
by energizing the room with a 47-Hz sine wave on the loudspeaker
and exploring with a sound-level meter. A spot both convenient and
effective will probably be found in a corner.
The finished Helmholtz resonator is shown in Fig. 9-39A. Constructional
details are obtained from resonator design instruction
included in the ETF 5 program. The resonator is made from a concreteforming
tube available in hardware stores. Laminated wood covers are
tightly fitted into both ends of the tube.
20 40 60 80 100 120 140 160 180 200
0
-4
-8
-12
-16
-20
-24
-28
-32
-36
-40
0
300
600
900
1200
1500
1800
Time (ms)
200 180 160 140 120 100 80 60 40 20
Level (dB)
Frequency (Hz)
After
B
F I G U R E 9 - 3 8 B
Low-frequency model structure of the sound field of the same small room after the introduction of the tuned
Helmholtz resonator absorber.
ABSORPTION OF SOUND 229
The length of the PVC pipe, Fig. 9-
39(A), is varied to tune the resonator to
specific frequencies. An absorbent partially
fills the resonator. The Q-factor of the
finished resonator was measured at Q  1.
Increasing Reverberation Time
Low-Q Helmholtz resonators are capable
of shortening reverberation time by
increasing absorption. High-Q resonators
can increase reverberation time through
storage of energy as described by Gilford.2
To achieve the high Qs necessary, plywood,
particleboard, masonite, and other
such materials must be abandoned and
ceramics, plaster, concrete, etc., used in
resonator construction. By proper tuning
of the resonators, the increase in reverberation
time can be placed where needed in
regard to frequency.
Modules
The British Broadcasting Corporation has
pioneered a modular approach to the
acoustical treatment of their numerous
small voice studios, which is very interesting.
17 Because they have applied it in several
hundred such studios economically
and with very satisfactory acoustical
results, it deserves our critical attention.
Basically, the idea is to cover the walls with
standard-sized modules, say 2 3 ft, having
a maximum depth of perhaps 8″. These can
be framed on the walls to give a flush surface
appearing very much like an ordinary
room, or they can be made into boxes with
A
F I G U R E 9 - 3 9 A
Typical Helmholtz resonator made of readily
available materials.
Wood top
Tube
Duct
Laminated
wood bottom
Helmholtz Resonator Design
B
F I G U R E 9 - 3 9 B
Details of Helmholtz resonator design.
230 CHAPTER NINE
grill cloth covers mounted on the walls in regular patterns. All modules
can be made to appear identical, but the similarity is only skin deep.
There are commonly three, or perhaps four, different types of modules,
each having its own distinctive contribution to make acoustically.
Figure 9-40 shows the radically different absorption characteristics
obtained by merely changing the covers of the standard module. This is
for a 2 3 ft module having a 7″ air space and a 1″ semirigid glass fiber
board of 3 lb/cu ft density inside. The wideband absorber has a highly
perforated cover (25% or more perforation percentage) or no cover at
all, yielding essentially complete absorption down to about 200 Hz.
Even better low-frequency absorption is possible by breaking up the air
space with egg-crate type dividers of corrugated paper to discourage
unwanted resonance modes. A cover 14″ thick with a 5% perforation
percentage peaks in the 300–400 Hz range. A true bass absorber is
obtained with a low-perforation cover (0.5% perforation). If essentially
1.0
0.8
0.6
0.4
0.2
0
Absorption coefficient
A
B
C
D
100 1,000 10,000
Frequency - Hz
F I G U R E 9 - 4 0
Modular absorber having a 7″ air space and 1″ semirigid glass fiber board of 9 to 10
lb/cu ft density behind the perforated cover. (A) No perforated cover at all, or at least
more than 25% perforation. (B) 5% perforated cover. (C) 0.5% perforated cover. (D)
34″ plywood cover, essentially to neutralize the module. (Data from Brown.17)
ABSORPTION OF SOUND 231
neutral modules are desired, they can be covered with 38″ and 14″ plywood,
which would give relatively low absorption with a peak around
70 Hz. Using these three or four modules as acoustical building blocks,
the desired effect can be designed into a studio by specifying the number
and distribution of each of the basic types.
Figure 9-41 shows an adaptation from BBC practice where the wall
is used as the “bottom” of the module box. In this case the module
size is 2 4 ft. The modules are fastened to the 2 2″ mounting
strips, that in turn are fastened to the wall. A studio wall 10 ft high
4'-0" 8"
2'-0"
Hi perf Grill cloth
Lo perf2
(A) Bass absorber (B) Wideband absorber
Air space
6–7" depth Air space
6–7" depth
0.5 Percent perf
(26 3/16" holes
per sq. ft.)
25 Percent perf
or more or
no perf cover
1" Fiberglass
5–8 lb./cu. ft.
density
1"
2 2
1 1
1" Fiberglass
5-8 lb./cu. ft.
density
F I G U R E 9 - 4 1
Plan for a practical module absorber utilizing the wall as the bottom of the module. (Left) Bass absorber.
(Right) Wideband absorber.
232 CHAPTER NINE
and 23 or 24 ft long might use 20 modules of distributed types, four
modules high and five long. It is good practice to have acoustically
dissimilar modules opposing each other on opposite walls.
The question that comes to mind is, “How about diffusion of sound
with such modular treatment?” BBC experience has shown that careful
distribution of the different types of modules results in adequate
diffusion.
Endnotes
1Young, Robert W., Sabine Reverberation and Sound Power Calculations, J. Acous. Soc. Am.,
31, 7 (July 1959) 912–921. See also by the same author, On Naming Reverberation Equations,
31, 12 (Dec 1959) 1681.
2Gilford, Christofer, Acoustics for Radio and Television Studios, London, Peter Peregrinus,
Ltd. (1972), Chapter 8.
3Anon., Noise Control Manual, Owens-Corning Fiberglas Corp. (1980), publication No. 5-
BMG-8277-A.
4Sonex is manufactured by Illbruck USA, 3600 Washington Ave., North, Minneapolis, MN
55412.
5Harris, Cyril M., Acoustical Properties of Carpet, J. Acous. Soc. Am., 27, 6 (Nov 1955)
1077–1082.
6Anon., Acoustical Ceilings: Use and Practice, Published by the Ceiling and Interior Systems
Contractors Association (1978). See Appendix.
7Schultz, Theodore J. and B. G. Watters, Propagation of Sound Across Audience Seating, J.
Acous. Soc. Am. 36, 5 (May 1964) 885–896.
8Rettinger, Michael, Bass Traps, Recording Eng./Prod., 11, 4 (Aug 1980) 46–51.
9Mankovsky, V. S., Acoustics of Studios and Auditoria, London, Focal Press, Ltd., (1971).
10Siekman, William, private communication. (Mr. Seikman was manager of Riverbank
Acoustical Laboratories at time of these measurements, which were reported to the Acous.
Soc. Am. April 1969.)
11Brüel, Per V., Sound Insulation and Room Acoustics, London, Chapman and Hall (1951).
12Callaway, D. B. and L. G. Ramer, The Use of Perforated Facings in Designing Low-
Frequency Resonant Absorbers, J. Acous. Soc. Am., 24, 3 (May 1952), 309–312.
13Ingard, U. and R. H. Bolt, Absorption Characteristics of Acoustic Materials with Perforated
Facings, J. Acous. Soc. Am., 23, 5 (Sept 1951), 533–540.
14Davern, W. A., Perforated Facings Backed with Porous Materials as Sound Absorbers—An
Experimental Study, Applied Acoustics, 10 (1977), 85–112.
15Rettinger, Michael, Low Frequency Sound Absorbers, db the Sound Engineering Magazine,
4, 4 (April 1970), 44–46.
ABSORPTION OF SOUND 233
16Rettinger, Michael, Low-Frequency Slot Absorbers, db the Sound Engineering Magazine,
10, 6 (June 1976) 40–43.
17Brown, Sandy, Acoustic Design of Broadcasting Studios, J. Sound and Vibration, I (3)
(1964) 239–257.
18Beranek, Leo L., Acoustics, New York, McGraw-Hill Book Co. (1954).
19Evans, E. J. and E. N. Bazley, Sound Absorbing Materials, London, National Physical
Laboratories (1960).
20Sabine, Paul E. and L. G. Ramer, Absorption-Frequency Characteristics of Plywood Panels,
J. Acous. Soc. Am., 20, 3 (May 1948) 267–270.
21Everest, F. Alton, The Acoustic Treatment of Three Small Studios, J. Audio Eng. Soc., 15,
3 (July 1968) 307–313.
22Kingsbury, H. F. and W. J. Wallace, Acoustic Absorption Characteristics of People, Sound
and Vibration, 2, 2 (Dec. 1968) 15,16.
23Acoustisoft, Inc., 53 King St., Peterborough, Ontario, Canada K9J-2T1. e-mail:
doug@etfacoustic.com, telephone 800-301-1429, fax (705) 745-4955. Internet site,
www.etfacoustic.com.
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If a sound is activated in a room, sound travels radially in all directions.
As the sound waves encounter obstacles or surfaces, such as
walls, their direction of travel is changed, i.e., they are reflected.
Reflections from Flat Surfaces
Figure 10-1 illustrates the reflection of waves from a sound source
from a rigid, plane wall surface. The spherical wavefronts (solid lines)
strike the wall and the reflected wavefronts (broken lines) are returned
toward the source.
Like the light/mirror analogy, the reflected wavefronts act as
though they originated from a sound image. This image source is
located the same distance behind the wall as the real source is in front
of the wall. This is the simple case—a single reflecting surface. In a
rectangular room, there are six surfaces and the source has an image
in all six sending energy back to the receiver. In addition to this,
images of the images exist, and so on, resulting in a more complex situation.
However, in computing the total sound intensity at a given
receiving point, the contributions of all these images must be taken
into consideration.
Sound is reflected from objects that are large compared to the
wavelength of the impinging sound. This book would be a good
Reflection of Sound
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236 CHAPTER TEN
reflector for 10 kHz sound (wavelength about an inch). At the low end
of the audible spectrum, 20 Hz sound (wavelength about 56 ft) would
sweep past the book and the person holding it as though they did not
exist, and without appreciable shadows.
Below 300–400 Hz, sound is best considered as waves (chapter 15
expounds on this). Sound above 300–400 Hz is best considered as traveling
in rays. A ray of sound may undergo many reflections as it
bounces around a room. The energy lost at each reflection results in
the eventual demise of that ray. Even the ray concept is an oversimplification:
Each ray should really be considered as a “pencil” of diverging
sound with a spherical wavefront to which the inverse square law
applies.
The mid/high audible frequencies have been called the specular
frequencies because sound in this range acts like light rays on a mirror.
Sound follows the same rule as light: The angle of incidence is
equal to the angle of reflection, as in Fig. 10-2.
Source
Wall
Image
F I G U R E 1 0 - 1
Reflection of sound from a point source from a flat surface (incident sound, solid
lines; reflected sound, broken lines). The reflected sound appears to be from a virtual
image source.
REFLECTION OF SOUND 237
Doubling of Pressure at
Reflection
The sound pressure on a surface normal to
the incident waves is equal to the energydensity
of the radiation in front of the surface.
If the surface is a perfect absorber, the
pressure equals the energy-density of the
incident radiation. If the surface is a perfect
reflector, the pressure equals the
energy-density of both the incident and
the reflected radiation. Thus the pressure
at the face of a perfectly reflecting surface
is twice that of a perfectly absorbing surface.
At this point, this is only an interesting
sidelight. In the study of standing
waves in Chap. 15, however, this pressure
doubling takes on greater significance.
Reflections from Convex
Surfaces
Spherical wavefronts from a point source tend to become plane waves
at greater distance from the source. For this reason impinging sound
on the various surfaces to be considered will be thought of as plane
wavefronts. Reflection of plane wavefronts of sound from a solid convex
surface tends to scatter the sound energy in many directions as
shown in Fig. 10-3. This amounts to a diffusion of the impinging
sound.
The polycylindrical sound-absorbing system described in the previous
chapter both absorbs sound and contributes to much-needed diffusion
in the room by reflection from the cylindrically shaped surface.
Reflections from Concave Surfaces
Plane wavefronts of sound striking a concave surface tend to be
focussed to a point as illustrated on Fig. 10-4. The precision with
which sound is focussed to a point is determined by the shape of the
ai
ar
ar
ai
F I G U R E 1 0 - 2
The angle of incidence, ai, is equal to the angle of
reflection, ar.
238 CHAPTER TEN
concave surface. Spherical concave surfaces
are common because they are readily
formed. They are often used to make a
microphone highly directional by placing it
at the focal point. Such microphones are frequently
used to pick up field sounds at
sporting events or in recording songbirds or
other animal sounds in nature. In the early
days of broadcasting sporting events in
Hong Kong, a resourceful technician saved
the day by using an ordinary Chinese wok,
or cooking pan, as a reflector. Aiming the
microphone into the reflector at the focal
point provided an emergency directional
pickup. Concave surfaces in churches or
auditoriums can be the source of serious
problems as they produce concentrations of
sound in direct opposition to the goal of
uniform distribution of sound.
The effectiveness of reflectors for
microphones depends on the size of the
reflector with respect to the wavelength of
sound. A 3-ft-diameter spherical reflector
will give good directivity at 1 kHz (wavelength
about 1 ft), but it is practically
nondirectional at 200 Hz (wavelength
about 5.5 ft).
Reflections from Parabolic
Surfaces
A parabola has the characteristic of
focusing sound precisely to a point (Fig.
10-5). It is generated by the simple equation
y = x2. A very “deep” parabolic surface,
such as that of Fig. 10-5, exhibits far
better directional properties than a shallow
one. Again, the directional properties
A
F I G U R E 1 0 - 3
Plane sound waves impinging on a convex irregularity
tend to be dispersed through a wide angle if the size
of the irregularity is large compared to the wavelength
of the sound.
B
F I G U R E 1 0 - 4
Plane sound waves impinging on a concave irregularity
tend to be focussed if the size of the irregularity
is large compared to the wavelength of the sound.
REFLECTION OF SOUND 239
depend on the size of the opening in terms of wavelengths. Figure 10-5
shows the parabola used as a directional sound source with a small,
ultrasonic Galton Whistle pointed inward at the focal point.
Plane waves striking such a reflector would be brought to a focus at
the focal point. Conversely, sound emitted at the focal point of the parabolic
reflector generates plane wavefronts. This is demonstrated in
the photographs of Figs. 10-6 and 10-7 in which standing waves are
produced by reflections from a heavy glass plate. The force exerted by
the vibration of the air particles on either side of a node is sufficient to
hold slivers of cork in levitation.
Galton
ultrasonic
whistle
Parabolic
surface
Plane
wavefronts
Glass reflector
F I G U R E 1 0 - 5
A parabolic surface can focus sound precisely at a focal point or, the converse, a
sound source placed at the focal point can produce plane, parallel wavefronts. In this
case, the source is an ultrasonic Galton Whistle blown by compressed air with the
results shown in Figs. 10-6 and 10-7.
240 CHAPTER TEN
Reflections Inside a Cylinder
St. Paul’s Cathedral in London boasts a whispering gallery. The way this
whispering gallery works is explained in the diagram of Fig. 10-8. Reflections
from the exterior surfaces of cylindrical shapes have been mentioned
in the treatment of “polys.” In this case the source and receiver are
both inside a mammoth, hard-surfaced cylindrical room.
At the source, a whisper directed tangentially to the surface is
clearly heard on the receiver side. The phenomenon is assisted by the
fact that the walls are dome-shaped. This means that upward-directed
components of the whispered sounds tend to be reflected downward
and conserved rather than lost above.
Standing Waves
The concept of standing waves is directly dependent on the reflection
of sound as emphasized in Chap. 15. Assume two flat, solid parallel
walls separated a given distance. A sound source between them radiates
sound of a specific frequency. The wavefront striking the right
wall is reflected back toward the source, striking the left wall where it
is again reflected back toward the right wall, and so on. One wave travels
to the right, the other toward the left. The two traveling waves
interact to form a standing wave. Only the standing wave, the interaction
of the two, is stationary. The frequency of the radiated sound is
such as to establish this resonant condition between the wavelength of
the sound and the distance between the two surfaces. The pertinent
point at the moment is that this phenomenon is entirely dependent on
the reflection of sound at the two parallel surfaces.
Reflection of Sound
from Impedance Irregularities
The television repairman is concerned about matching the electrical
impedance of the television receiver to that of the transmission line,
and matching the transmission line to the impedance of the antenna
(or cable). Mismatches of impedance give rise to reflections, which
cause numerous undesirable effects.
REFLECTION OF SOUND 241
F I G U R E 1 0 - 6
A parabolic reflector following the equation y=x2 reflects sound from an ultrasonic
Galton Whistle to form a stable standing wave system capable of levitating
bits of cork.
242 CHAPTER TEN
A similar situation prevails in an air-conditioning duct. A sound
wave (noise) traveling in the duct suddenly encounters the large open
space of the room. This discontinuity (impedance mismatch) reflects a
significant portion of the sound (fan noise, etc.) back toward the
F I G U R E 1 0 - 7
Close-up of the levitated cork chips of Fig. 10-6. If the cork chips are spaced about
1⁄2 inch, the frequency of the sounds emitted by the Galton Whistle is about 30 kHz.
REFLECTION OF SOUND 243
source. This is an example of a benevolent
mismatch as the air-conditioner noise is
reduced in the room.
The Corner Reflector
In an art museum with large Dutch paintings
on display, the eyes of certain subjects
seem to follow as one walks by. Corner
reflectors are like that. There seems to be
no way of escaping their pernicious effect.
The corner reflector of Fig. 10-9, receiving
sound from the source S, sends a reflection
directly back toward the source. If the
angles of incidence and reflection are carefully
noted, a source at B will also send a
direct, double-surface reflection returning
to the source. A source at C, on the opposite
of B, is subject to the same effect.
You might be instinctively aware of perpendicular
(normal) reflections from surrounding walls, but now consider
reflections from the four corners of the room that follow the source around
the room. Corner reflections suffer losses at two surfaces, tending to make
them somewhat less intense than normal reflections at the same distance.
The corner reflector of Fig. 10-9 involves only two surfaces. How
about the four upper tri-corners of the room formed by ceiling and
walls and another four formed by floor and wall surfaces? The same
follow-the-source principle applies. In fact, sonar and radar people
have long employed targets made of three circular plates of reflecting
material assembled so that each is perpendicular to the others.
Echo-Sounding
Objects can be located by sending out a pulse of sound and noting the
time it takes for the reflected echo to return. Directional sources of
sound make possible the determination of both the azimuth angle and
the distance to the reflecting object. This principle has been widely
applied in water depth sounders, sonar on submarines, etc. All
Receiver
Source
F I G U R E 1 0 - 8
Graphic explanation of the “Whispering Gallery” of
St. Paul’s Cathedral, London. A whisper directed tangentially
to the cylinderical surface is readily heard
by the receiver on the far side of the room.
244 CHAPTER TEN
depend on the reflection of sound from the
bottom of the ocean, or enemy target.
Perceptive Effects of
Reflections
In the reproduction of sound in a highfidelity
listening room or control room of
a recording studio, the sound of the loudspeakers
falling on the ear of the listener
is very much affected by reflections from
the surfaces of the room. This is another
manifestation of sound reflection. A comprehensive
consideration of human perception
of such reflections is included in
Chap. 16.
S
B
C
F I G U R E 1 0 - 9
The corner reflector has the property of reflecting
sound back toward the source from any direction.
It is well known that sound travels around corners and around obstacles.
Music reproduced in one room of a home can be heard down
the hall and in other rooms. Diffraction is one of the mechanisms
involved in this. The character of the music heard in distant parts of
the house is different. In distant rooms the bass notes are more prominent
because their longer wavelengths are readily diffracted around
corners and obstacles.
Rectilinear Propagation
Wavefronts of sound travel in straight lines. Sound rays, a concept
applicable at mid/high audible frequencies, are considered to be pencils
of sound that travel in straight lines perpendicular to the wavefront.
Sound wavefronts and sound rays travel in straight lines, except
when something gets in the way. Obstacles can cause sound to be
changed in its direction from its original rectilinear path. The process
by which this change of direction takes place is called diffraction.
Alexander Wood, the early Cambridge acoustician, recalled Newton’s
pondering over the relative merits of the corpuscular and wave
theories of light. Newton finally decided that the corpuscular theory
was the correct one because light is propagated rectilinearly. Later it
was demonstrated that light is not always propagated rectilinearly,
Diffraction of Sound
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246 CHAPTER ELEVEN
that diffraction can cause light to change its direction of travel. In fact,
all types of wave motion, including sound, are subject to diffraction.
The shorter the wavelength (the higher the frequency), the less
dominant is the phenomenon of diffraction. Diffraction is less noticeable
for light than it is for sound because of the extremely short wavelengths
of light. Obstacles capable of diffracting (bending) sound must
be large compared to the wavelength of the sound involved. The wellworn
example of ocean waves is still one
of the best. Ocean waves sweep past a piling
of a dock with scarcely a disturbance.
Ocean waves, however, are bent around an
end of an island.
Diffraction and Wavelength
The effectiveness of an obstacle in diffracting
sound is determined by the
acoustical size of the obstacle. Acoustical
size is measured in terms of the wavelength
of the sound. One way of looking
at the illustration shown later, Fig. 11-3,
is that the obstacle in B is the same physical
size as that of A, but the frequency of
the sound of A is one tenth that of B. If
the obstacle in B is 1 ft long and that of A
0.1 ft long, the frequency of the sound in
A could well be 1,000 Hz (wavelength
1.13 ft), and that of B could be 100 Hz
(wavelength 11.3 ft). The same drawing
could be used if the obstacle of A were
0.01 ft long with a frequency of 10,000 Hz
(wavelength 0.113 ft) and the obstacle of
B were 0.1 ft long with a frequency of
1,000 Hz (wavelength 1.13 ft).
In Fig. 11-1, two types of obstructions to
plane wavefronts of sound are depicted. In
Fig. 11-1A a heavy brick wall is the obstacle.
The sound waves are reflected from the
A
New
Source
New Source
B


F I G U R E 1 1 - 1
(A) If the brick wall is large in terms of the wavelength
of the sound, the edge acts as a new source,
radiating sound into the shadow zone. (B) Plane
waves of sound impinging on the heavy plate with a
small hole in it sets up spherical wavefronts on the
other side due to diffraction of sound.
DIFFRACTION OF SOUND 247
face of the wall, as expected. The upper edge of the wall acts as a new,
virtual source sending sound energy into the “shadow” zone behind the
wall by diffraction. The mechanism of this effect will be considered in
more detail later in this chapter.
In Fig. 11-1B the plane wavefronts of sound strike a solid barrier
with a small hole in it. Most of the sound energy is reflected from the
wall surface, but that tiny portion going through the hole acts as a virtual
point source, radiating a hemisphere of sound into the “shadow”
zone on the other side.
Diffraction of Sound
by Large and Small Apertures
Figure 11-2A illustrates the diffraction of sound by an aperture that is
many wavelengths wide. The wavefronts of sound strike the heavy
obstacle: some of it is reflected, some goes right on through the wide
aperture. The arrows indicate that some of the energy in the main
beam is diverted into the shadow zone. By what mechanism is this
diversion accomplished?
A B
F I G U R E 1 1 - 2
(A) An aperature large in terms of wavelength of sound allows wavefronts to go
through with little disturbance. These wavefronts act as lines of new sources radiating
sound energy into the shadow zone. (B) If the aperature is small compared to the
wavelength of the sound, the small wavefronts which do penetrate the hole act almost
as point sources, radiating a hemispherical field of sound into the shadow zone.
248 CHAPTER ELEVEN
For an answer, the work of Huygens is consulted.1 He enunciated a
principle that is the basis of very difficult mathematical analyses of
diffraction. The same principle also gives a simple explanation of how
sound energy is diverted from the main beam into the shadow zone.
Huygens’ principle can be paraphrased as:
Every point on the wavefronts of sound that has passed through an
aperture or passed a diffracting edge is considered a point source radiating
energy back into the shadow zone.
The sound energy at any point in the shadow zone can mathematically
be obtained by summing the contributions of all of these point
sources on the wavefronts.
In Fig. 11-2A, each wavefront passing through the aperture
becomes a row of point sources radiating diffracted sound into the
shadow zone. The same principle holds for Fig. 11-2B except that the
aperture is very small and only a small amount of energy passes
through it. The points on the limited wavefront going through the hole
are so close together that their radiations take the form of a hemisphere.
Diffraction of Sound by Obstacles
In Fig. 11-3A the obstacle is so small compared to the wavelength of
the sound that it has no appreciable effect on the passage of sound. In
Fig. 11-3B, however, the obstacle is many wavelengths long and it has
a definite effect in casting a shadow behind the obstacle. Each wavefront
passing the obstacle becomes a line of new point sources radiating
sound into the shadow zone.
A very common example of an obstacle large compared to the
wavelength of the impinging sound is the highway noise barrier
shown in Fig. 11-4. If the wavelength of the impinging sound is indicated
by the spacing of the spherical wavefronts hitting the barrier, the
barrier size is acoustically great. At higher frequencies the barrier
becomes even larger, and at lower frequencies it becomes acoustically
smaller. First, the sound reflected from the wall must be noted. It is as
though the sound were radiated from a virtual image on the far side of
the wall. The wavefronts passing the top edge of the wall can be considered
as lines of point sources radiating sound. This is the source of
the sound penetrating the shadow zone.
DIFFRACTION OF SOUND 249
Figure 11-5 gives some idea of the effectiveness of highway barriers
and of the intensity of the sound in the shadow of a high, massive
wall. The center of the highway is taken to be 30 ft from the wall
on one side, and the home or other sensitive area is considered to be
30 ft on the other side of the wall (the shadow side). A wall 20 ft
high yields something like 25 dB of protection from the highway
noise at 1,000 Hz. At 100 Hz, the attenuation of the highway noise
is only about 15 dB. At the higher audible frequencies, the wall is
more effective than at lower frequencies. The shadow zone behind
the wall tends to be shielded from the high-frequency components
of the highway noise. The low-frequency components penetrate
the shadow zone by diffraction.
Diffraction of Sound by a Slit
Figure 11-6 diagrams a classical experiment performed by Pohl in
acoustical antiquity and described by Wood1 in somewhat more recent
antiquity. One must admire the precise results obtained with crude
measuring instruments (high-pitched whistle, sound radiometer). The
A B
F I G U R E 1 1 - 3
(A) An obstacle very much smaller than the wavelength of sound allows the wavefronts
to pass essentially undisturbed. (B) An obstacle large compared to the wavelength
of sound casts a shadow that tends to be irradiated from sources on the
wavefronts of sound that go past the obstacle.
250 CHAPTER ELEVEN
equipment layout of Fig. 11-6A is very approximate. Actually the
source/slit arrangement rotated about the center of the slit and
the measuring radiometer was at a distance of 8 meters. The slit
width was 11.5 cm wide, the wavelength of the measuring sound
was 1.45 cm (23.7 kHz). The graph of Fig. 11-6B shows the intensity
of the sound versus the angle of deviation. The dimension B indicates
the geometrical boundaries of the ray. Anything wider that B is
caused by diffraction of the beam by the slit. A narrower slit would
yield correspondingly more diffraction and a greater width of the
beam. The increase in width of the beam is the striking feature of
this experiment.
Diffraction by the Zone Plate
The zone plate can be considered an acoustic lens. It consists of a
circular plate with a set of concentric, annular slits of cunningly
devised radii. If the focal point is at a distance of r from the plate,
Image of source
P
Source
F I G U R E 1 1 - 4
The classic sound barrier case. The sound striking the wall is reflected as though the
sound is radiated from a virtual image of the source. That sound passing the top edge
of the wall acts as though the wavefronts are lines of sources radiating sound energy
into the shadow zone.
DIFFRACTION OF SOUND 251
the next longer path must be r + λ/2 where λ is the wavelength of the
sound falling on the plate from the source. Successive path lengths
are r + λ, r + 3/2λ, and r + 2λ. These path lengths differ by λ/2, which
means that the sound through all the slits will arrive at the focal
point in phase which, in turn, means that they add constructively,
intensifying the sound.2 See Fig. 11-7.
Diffraction around the Human Head
Figure 11-8 illustrates the diffraction caused by a sphere roughly the
size of the human head. This diffraction by the head as well as reflections
and diffractions from the shoulders and the upper torso influences
human perception of sound. In general, for sound of frequency
1–6 kHz arriving from the front, head diffraction tends to increase
the sound pressure in front and decrease it behind the head. For frequencies
in the lower range the directional pattern tends to become
circular.2,3
30
20
10
Attenuation - dB
0
50 100 300 500 1kHz 3kHz 5kHz 10kHz
Frequency - Hz
h=20'
h=15'
h=10'
h
30' 30'
F I G U R E 1 1 - 5
An estimation of the effectiveness of a sound barrier in terms of sound (or noise) attentuation as a function
of frequency and barrier height. (After Rettinger.4)
252 CHAPTER ELEVEN
B
Source
Slit
20
10
10
10
20
0
10
20
Angle of deviation
100
80
60
40
20
0
Percentage of peak
B
A
F I G U R E 1 1 - 6
A consideration of Pohl’s classic experiment in diffraction. (A) A very approximate suggestion
of the equipment arrangement (see text). (B) The broadening of the beam B by
diffraction. The narrower the slit the greater this broadening of the beam. (After Wood.1)
DIFFRACTION OF SOUND 253
Diffraction by Loudspeaker Cabinet Edges
Loudspeaker cabinets are notorious for diffraction effects. If a loudspeaker
is mounted near a wall and aimed away from the wall, the
wall is still illuminated with sound diffracted from the corners of
the box. Reflections of this sound can affect the quality of the sound
at the listener’s position. Measurements of this effect have been
scarce, but Vanderkooy5 and Kessel6 have recently computed the
magnitude of loudspeaker cabinet edge diffraction. The computations
were made on a box loudspeaker with front baffle having the
dimensions 15.7 × 25.2 in and depth of 12.6 in (Fig. 11-9). A point
source of sound was located symmetrically at the top of the baffle.
The sound from this point source was computed at a distance from
the box. The sound arriving at the observation point is the combination
of the direct sound plus the edge diffraction. This combination
is shown in Fig. 11-10. Fluctuations due to edge diffraction for
this particular typical situation approached plus or minus 5 dB.
This is a significant change in overall frequency response of a
reproduction system.
Source
Focal
point
r
r/z
r3/z
r
r2
F I G U R E 1 1 - 7
The zone plate or acoustic lens. The slits are so arranged that the several path lengths differ by multiples of
a half wavelength of the sound so that all diffracted rays arrive at the focal point in phase, combining constructively.
(After Olson.2)
254 CHAPTER ELEVEN
This effect can be controlled (eliminated?) by setting the loudspeaker
box face flush in a much larger baffling surface. There is also
the possibility of rounding edges and the use of foam or “fuzz”.7
Diffraction by Various Objects
Sound level meters were, in early days, boxes with a microphone protruding.
Diffraction from the edges and corners of the box seriously
affected the calibration of the microphones. Modern sound level
meters have carefully rounded contours with the microphone
mounted on a smooth, slender, rounded neck.
Diffraction from the casing of a microphone can cause deviations
from the desired flat sensitivity.
In the measurement of sound absorption in large reverberation
chambers, the common practice is to place the material to be measured
6 kHz 1 kHz
5
0
5
Direction of
plane wave

F I G U R E 1 1 - 8
Diffraction around a solid sphere about the size of a human head. For sound in the
1-6 kHz range, sound pressure is generally increased in the front hemisphere and
generally reduced in the rear. (After Muller, Black and Davis, as reported by Olson.2)
DIFFRACTION OF SOUND 255
Point
source
Observing point
Diffracted
Direct
F I G U R E 1 1 - 9
Arrangement for Vanderkooy’s calculation of loudspeaker cabinet edge diffraction,
shown in Fig. 11-10.
Amplitude - dB15
25
20
10
5
20 50 100 200 500 1 kHz 2 kHz 5 kHz 10 kHz
Frequency - Hz
F I G U R E 1 1 - 1 0
The calculated effects of loudspeaker edge diffraction on the direct signal in the
arrangement of Fig.11-9. (After Vanderkooy,5 and Kessel.6)
256 CHAPTER ELEVEN
in a 8 × 9-ft frame on the floor. Diffraction from the edges of this frame
often result in absorption coefficients greater than unity. In other
words, diffraction of sound makes the sample appear larger than it
really is.
Small cracks around observation windows or back-to-back microphone
or electrical service boxes in partitions can destroy the hopedfor
isolation between studios or between studio and control room. The
sound emerging on the other side of the hole or slit is spread in all
directions by diffraction.
In summary, diffraction causes sound, which normally travels rectilinearly,
to travel in other directions.
Endnotes
1Wood, Alexander, Acoustics, New York, Interscience Publishers, Inc. (1941).
2Olson, Harry F., Elements of Acoustical Engineering, New York, D. Van Nostrand Co. (1940).
3Muller, C.G., R. Black, and T.E. Davis, The Diffraction Produced by Cylinders and Cubical
Obstacles and by Circular and Square Plates, J. Acous. Soc. Am., 10, 1 (1938) p. 6.
4Rettinger, M., Acoustic Design and Noise Control, Chemical Publishing Co. (1973).
5Vanderkooy, John, A Simple Theory of Cabinet Edge Diffraction, J. Audio Eng. Soc., 39, 12
(1991) 923-933.
6Kessel, R.T., Predicting Far-Field Pressures from Near-Field Loudspeaker Measurements, J.
Audio Eng. Soc., Abstract, Vol. 36, p.1,026 (Dec 1988), preprint 2729.
7Kaufman, Richard J., With a Little Help from My Friends, AUDIO, 76, 9 (Sept 1992) 42-46.
About the turn of this century Lord Rayleigh was puzzled because some
very powerful sound sources, such as cannon fire, could be heard
only short distances some times and very great distances at other
times. He set up a powerful siren that required 600 hp to maintain it.
He calculated that if all this power were converted into energy as
sound waves and spread uniformly over a hemisphere, how far could
it be heard? Knowing the minimum audible intensity (10–16 watts per
sq cm), his calculations indicated that the sound should be audible to
a distance of 166,000 miles, more than 6 times the circumference of
the earth!
It is indeed fortunate that such sound propagation is never experienced
and that a range of a few miles is considered tops. There are
numerous reasons why sound is not heard over greater distances. For
one thing, the efficiency of sound radiators is usually quite low; not
much of that 600 hp was actually radiated as sound. Energy is also lost
as wavefronts drag across the rough surface of the earth. Another loss
is dissipation in the atmosphere, but this is known to be very small.
The result of such calculations and early experiments that fell far
short of expectations served only to accelerate research on the effects
of temperature and wind gradients on the transmission of sound.
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258 CHAPTER TWELVE
Refraction of Sound
Refraction changes the direction of travel of the sound by differences in
the velocity of propagation. Diffraction is changing the direction of travel
of sound by encountering sharp edges and physical obstructions (chapter
11). Most people find it easy to distinguish between absorption and
reflection of sound, but there is often confusion between diffraction and
refraction (and possibly diffusion, the subject of the next chapter). The
similarity of the sound of the words might be one cause for this confusion,
but the major reason is the perceived greater difficulty of understanding
diffraction, refraction, and diffusion compared to absorption and reflection.
Hopefully Chaps. 9, 10, 11, 13, and this chapter will help to equalize
and advance understanding of these five important effects.
Figure 12-1 recalls a very common observation of the apparent
bending of a stick as one end touches the water surface or is actually
immersed. This is an illustration of refraction of light. As the present
subject is refraction of sound, which is another wave phenomenon, the
relative refractive indices of air and water will be passed over.
Refraction of Sound in Solids
Figure 12-2 illustrates sound passing from a dense solid medium to a
less dense medium. The sound speed in the denser medium is greater
F I G U R E 1 2 - 1
Touching a stick to the water surface illustrates refraction of light. Sound is another
wave phenomenon that is also refracted by changes in media sound speed.
REFRACTION OF SOUND 259
than that in the less dense one (Table 12-1). As one ray reaches the
boundary between the two media at A, the other still has some distance
to go. In the time it takes one ray to travel from B to C, the other
ray has traveled a shorter distance from A to D in the new medium.
Wavefront A-B represents one instant of time as does wavefront D-C an
instant later. But these two wavefronts are no longer parallel. The rays
of sound have been refracted at the interface of the two media having
unlike sound speeds.
An analogy may assist memory and logic. Assume that the shaded
area is paved and that the lower-density area is ploughed. Assume also
that the wavefront A-B is a line of soldiers. The line of soldiers A-B,
marching in military order, has been making good progress on the
A
D
C
B Denser medium
Higher sound
speed
Less dense medium
Lower sound speed
F I G U R E 1 2 - 2
Rays of sound traveling from a denser medium having a certain sound speed into a less
dense medium having a lower sound speed. The wavefront AB is not parallel to wavefront
DC because the direction of the wave is changed due to refraction.
260 CHAPTER TWELVE
pavement. As soldier A reaches the
ploughed ground he or she slows down
and begins plodding over the rough surface.
Soldier A travels to D on the
ploughed surface in the same time that
soldier B travels the distance BC on the
pavement. This tilts the wavefront off in a
new direction, which is the definition of
refraction. In any homogeneous medium,
sound travels rectilinearly (in the same
direction). If a medium of another density
is encountered, the sound is refracted.
Refraction of Sound in the Atmosphere
The atmosphere is anything but a stable, uniform medium for the
propagation of sound. Sometimes the air near the earth is warmer than
the air at greater heights, sometimes it is colder. Horizontal changes
are taking place at the same time this vertical layering exists. All is a
wondrously intricate and dynamic system, challenging the meteorologists
(as well as acousticians) to make sense of it.
In the absence of thermal gradients, a sound ray may be propagated
rectilinearly as shown in Fig. 12-3A. The sound ray concept is helpful
in considering direction of propagation. Rays of sound are always perpendicular
to sound wavefronts.
In Fig. 12-3B a thermal gradient exists between the cool air near the
surface of the earth and the warmer air above. This affects the wavefronts
of the sound. Sound travels faster in warm air than in cool air
causing the tops of the wavefronts to go faster than the lower parts. The
tilting of the wavefronts is such as to direct the sound rays downward.
Under such conditions, sound from the source is bent down toward the
surface of the earth and can be heard at relatively great distances.
The thermal gradient of Fig. 12-3C is reversed from that of Fig. 12-
3B as the air near the surface of the earth is warmer than the air higher
up. In this case the bottom parts of the wavefronts travel faster than
the tops, resulting in an upward refraction of the sound rays. The
same sound energy from the source S would now be dissipated in the
upper reaches of the atmosphere, reducing the chances of it being
heard at any great distance at the surface of the earth.
Table 12-1. Speed of sound.
Speed of sound
Medium Ft/sec Meters/sec
Air 1,130 344
Sea water 4,900 1,500
Wood, fir 12,500 3,800
Steel bar 16,600 5,050
Gypsum board 22,300 6,800
REFRACTION OF SOUND 261
Figure 12-4A presents a distant view of the downward refraction situation
of Fig. 12-3B. Sound traveling directly upward from the source S
penetrates the temperature gradient at right angles and would not be
refracted. It would speed up and slow down slightly as it penetrates the
warmer and cooler layers, but would still travel in the vertical direction.
All rays of sound except the vertical would be refracted downward.
The amount of this refraction varies materially: the rays closer
to the vertical are refracted much less than those more or less parallel
to the surface of the earth.
Figure 12-4B is a distant view of the upward refraction situation of
Fig.12-3C. Shadow zones are to be expected in this case. Again, the
vertical ray is the only one escaping refractive effects.
It is a common experience to hear sound better downwind than
upwind. Air is the medium for the sound. If wind moves the air at a
S
S
S
Uniform
Air temperature
Warm
Warm
Cool
Cool
Wavefronts
Ray
A
B
C
F I G U R E 1 2 - 3
Refraction of sound paths resulting from temperature gradients in the atmosphere; (A)
air temperature constant with height, (B) cool air near the surface of the earth and
warmer air above, (C) warm air near the earth and cooler air above.
262 CHAPTER TWELVE
certain speed, it is to be expected that the speed of sound will
be affected. If sound travels 1,130 ft/sec and a 10 mi/hour (about
15 ft/sec) wind prevails, what will be the effect of the wind on the
sound? Upwind the sound speed with respect to the earth would be
increased about 1%, and downwind it would be decreased the same
amount. This seems like a very small change but it is enough to affect
refraction materially. Figure 12-5 illustrates the effect of wind on the
downward refraction case of Fig. 12-4A. A downwind shadow is created
and upwind sound is refracted downward.
Sound
shadow
Sound
shadow
Warm
Warm
Cool
Cool S
S
A
B
F I G U R E 1 2 - 4
Comprehensive illustration of refraction of sound from source S; (A) cool air near the
ground and warmer air above, (B) warm air near the ground and cooler air above. In
(B) note that sound shadow areas result from the upward refraction.
REFRACTION OF SOUND 263
Wind speed near the surface of the earth is usually less than that at
greater heights. A wind gradient exists in such a case that has its effect
on propagation of sound. This is not a true refraction but the effect is
the same. Plane waves from a distant source traveling with the wind
would bend the sound down toward the earth. Plane waves traveling
against the wind will be bent upward.
It is possible, under unusual circumstances, that sound traveling
upwind may actually be favored. For instance, upwind sound is kept
above the surface of the ground, minimizing losses at the ground surface.
After all, does not the sportsman approach his prey upwind?
Doing so keeps footstep noises from being heard by the prey until the
sportsman is quite close.
Refraction of Sound in the Ocean
In 1960 some oceanographers devised an ambitious plan to see how far
underwater sound could be detected.1,2 Charges of 600 lb were discharged
at various depths in the ocean off Perth, Australia. Sounds
from these discharges were detected near Bermuda. The great circle
path the sound presumably followed is shown in Fig. 12-6. Even
though sound in sea water travels 4.3 times faster than in air, it took
13,364 seconds (3.71 hr) for the sound to make the trip. This distance
is over 12,000 miles, close to half the circumference of the earth. Interesting,
but what has this to do with refraction? Everything!
S
Wind
direction
Sound
shadow
F I G U R E 1 2 - 5
Wind gradients refract (not a true refraction) sound. A shadow sound is created upwind
and good listening conditions downwind.
264 CHAPTER TWELVE
An explanation is found in Fig. 12-7. The depth of the oceanic abyss is
5,000 or more fathoms (30,000 ft). At about 700 fathoms (4,200 ft) a very interesting
effect takes place. The sound speed profile shown in Fig. 12-7A is very
approximate to illustrate a principle. In the upper reaches of the ocean the
speed of sound decreases with depth because temperature decreases. At
greater depths the pressure effect prevails causing sound speed to increase
with depth because of the increase in density. The “V” change-over from one
effect to the other occurs near the 700 fathom (4,200 ft) depth.
A sound channel is created by this V-shaped sound-speed profile.
A sound emitted in this channel tends to spread out in all directions.
Any ray traveling upward will be refracted downward, any ray traveling
downward will be refracted upward. Sound energy in this channel
is propagated great distances with modest losses.
Refraction in the vertical plane is very prominent because of the vertical
temperature/pressure gradient of Fig. 12-7A. There is relatively little
N 40°
N 20°
0
520°
540°
560°
Latitude
W 60° W 40° 0 W 20° E 20° E 40° E 60° E 80° E 100° E 120°
Longitude
Perth
Bermuda
F I G U R E 1 2 - 6
Refraction of sound in the ocean. A 600-lb charge was detonated near Perth, Australia,
and the sound was recorded at Bermuda, over 12,000 miles away. The secret
lies in the fact that the sound was confined to a sound channel by refraction that
reduced losses. The sound took 3.71 hours to travel almost half way around the world.
Such long-distance transmission of sound in the sea is being used to study long-range
warming effects of the ocean. (After Heaney et al.2)
REFRACTION OF SOUND 265
horizontal sound speed gradient and therefore very little horizontal
refraction. Sound tends to be spread out in a thin sheet in this sound
channel at about 700 fathom depth. Spherical divergence in three dimensions
is changed to two-dimensional propagation at this special depth.
These long-distance sound channel experiments have suggested
that such measurements can be used to check on the “warming of the
planet” by detecting changes in the average temperature of the oceans.
The speed of sound is a function of the temperature of the ocean.
Accurate measures of time of transit over a given course yield information
on the temperature of that ocean.3
Refraction of Sound in Enclosed Spaces
Refraction is an important effect on a world-sized scale, how about
enclosed spaces? Consider a multi-use gymnasium that serves as an
Sound
speed
Depth
S
A B
F I G U R E 1 2 - 7
How the long range sound experiment of Fig. 12-6 was accomplished; (A) sound speed
decreases with depth in the upper reaches of the ocean (temperature effect) and
increases at greater depths (pressure effect) creating a sound channel at the inversion
depth (about 700 fathoms). (B) A ray of sound is kept in this sound channel by refraction.
Sound travels great distances in this channel because of the lower losses.
266 CHAPTER TWELVE
auditorium at times. With a normal heating and air-conditioning system,
great efforts are made to avoid large horizontal or vertical temperature
gradients. The goals of temperature uniformity and no
troublesome drafts have reduced sound refraction effects to inconsequential
levels.
Consider the same gymnasium used as an auditorium but with less
sophisticated air conditioning. In this case a large ceiling-mounted
heater near the rear acts as a space heater. Working against gravity, the
unit produces copious hot air near the ceiling, relying on slow convection
currents to move some of the heat down to the audience level.
This reservoir of hot air near the ceiling and cooler air below can
have a minor effect on the transmission of sound from the sound system
and on the acoustics of the space. The feedback point of the sound
system might shift. The standing waves of the room might change
slightly as longitudinal and transverse sound paths are increased in
length because of their curvature due to refraction. Flutter echo paths
are also shifted. With a sound radiating system mounted high at one
end of the room, lengthwise sound paths would be curved downward.
Such downward curvature might actually improve audience coverage,
depending somewhat on the directivity of the radiating system.
Endnotes
1Shockley, R.C., J. Northrop, P.G. Hansen, and C. Hartdegen, SOFAR Propagation Paths from
Australia to Bermuda, J. Acous. Soc. Am., 71, 51 (1982).
2Heaney, K.D., W.A. Kuperman, and B.E. McDonald, Perth-Bermuda Sound Propagation
(1960): Adiabatic Mode Interpretation, J. Acous. Soc. Am., 90, 5 (Nov 1991) 2586-2594.
3Spiesberger, John, Kent Metzger, and John A. Ferguson, Listening for Climatic Temperature
Changes in the Northeast Pacific 1983-1989, J. Acous. Soc. Am., 92, 1 (July 1992) 384-396.
Diffusion problems are most troublesome in smaller rooms and at the
lower audio frequencies. The problem with small spaces such as
the average recording studio, control room, or music listening room is
that modal spacings below 300 Hz guarantee a sound field far from diffuse
(Chap. 15).
The Perfectly Diffuse Sound Field
Even though unattainable, it is instructive to consider the characteristics
of a diffuse sound field. Randall and Ward1 have given us a list of
these:
 The frequency and spatial irregularities obtained from steadystate
measurements must be negligible.
 Beats in the decay characteristic must be negligible.
 Decays must be perfectly exponential, i.e., they must be straight
lines on a logarithmic scale.
 Reverberation time will be the same at all positions in the room.
 The character of the decay will be essentially the same for different
frequencies.
Diffusion of Sound
13
267
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
268 CHAPTER THIRTEEN
 The character of the decay will be independent of the directional
characteristics of the measuring microphone.
These six factors are observation oriented. A professional physicist
specializing in acoustics might stress fundamental and basic factors in
his definition of a diffuse sound field such as energy density, energy
flow, superposition of an infinite number of plane progressive waves,
and so on. The six characteristics suggested by Randall and Ward
point us to practical ways of obtaining solid evidence for judging the
diffuseness of the sound field of a given room.
Evaluating Diffusion in a Room
There is nothing quite as upsetting as viewing one’s first attempt at
measuring the “frequency response” of a room. To obtain the frequency
response of an amplifier, a variable-frequency signal is put in
the front end and the output observed to see how flat the response is.
The same general approach can be applied to a room by injecting the
variable frequency signal into “the front end” by means of a loudspeaker
and noting the “output” picked up by a microphone located
elsewhere in the room.
Steady-State Measurements
Figure 13-1 is a graphic-level recorder tracing of the steady-state response
of a studio having a volume of 12,000 cubic feet. In this case, the loudspeaker
was in one lower tricorner of the room, and the microphone was
at the upper diagonal tricorner about one foot from each of the three surfaces.
These positions were chosen because all room modes terminate in
the corners and all modes should be represented in the trace. The fluctuations
in this response cover a range of about 35 dB over the linear 30- to
250-Hz sweep. The nulls are very narrow, and the narrow peaks show evidence
of being single modes because the mode bandwidth of this room is
close to 4 Hz. The wider peaks are the combined effect of several adjacent
modes. The rise from 30 to 50 Hz is due primarily to loudspeaker response
and the 9-dB peak between 50 and 150 Hz (due to radiating into 1⁄4 space)
should not be charged against the room. The rest is primarily room effect.
The response of Fig. 13-1 is typical of even the best studios. Such
variations in response are, of course, evidence of a sound field that is
DIFFUSION OF SOUND 269
not perfectly diffused. A steady-state response such as this taken in an
anechoic room would still show variations, but of lower amplitude. A
very live room, such as a reverberation chamber, would show even
greater variations.
Figure 13-1 illustrates one way to obtain the steady-state response of
a room. Another is to traverse the microphone while holding the loudspeaker
frequency constant. Both methods reveal the same deviations
from a truly homogeneous sound field. Thus, we see that Randall and
Ward’s criteria of negligible frequency and spatial irregularities are not
met in the studio of Fig. 13-1 or, in fact, in any practical recording studio.
Decay Beats
By referring to Chap. 7, Fig. 7-10, we can compare the smoothness of
the reverberation decay for the eight octaves from 63 Hz to 8 kHz. In
Relative sound pressure level - dB
50 100 150 200 250
10 dB
BRUEL & KJAER ¨
Frequency - Hz
F I G U R E 1 3 - 1
Slowly swept sine-wave sound-transmission response of a 12,000-cu ft. video studio.
Fluctuations of this magnitude, which characterize the best of studios, are evidence of
nondiffusive conditions.
270 CHAPTER THIRTEEN
general, the smoothness of the decay increases as frequency is
increased. The reason for this, as explained in Chap. 7, is that the
number of modes within an octave span increases greatly with frequency,
and the greater the mode density, the smoother their average
effect. Beats in the decay are greatest at 63 Hz and 125 Hz. The decays
of Fig. 7-10 indicate that the diffusion of sound in this particular studio
is about as good as can be achieved by traditional means. It is the
beat information on the low-frequency reverberation decay that
makes possible a judgment on the degree of diffusion prevailing.
Reverberation-time measuring devices that yield information only on
the average slope and not the shape of the decay pass over information
that most consultants consider important in evaluating the diffuseness
of a space.
Exponential Decay
A truly exponential decay is a straight line on a level vs. time plot,
and the slope of the line can be described either as a decay rate in
decibels per second or as reverberation time in seconds. The decay
of the 250-Hz octave band of noise pictured in Fig. 13-2 has two
exponential slopes. The initial slope gives a reverberation time of
0.35 second and the final slope a reverberation time of 1.22 seconds.
The slow decay that finally takes over once the level is low enough
is probably a specific mode or group of modes encountering low
absorption either by striking the absorbent at grazing angles or striking
where there is little absorption. This is typical of one type of
nonexponential decay, or stated more precisely, of a dual exponential
decay.
Another type of nonexponential decay is illustrated in Fig. 13-3.
The deviations from the straight line connecting the beginning and
end of the decay are considerable. This is a decay of an octave band of
noise centered on 250 Hz in a 400-seat chapel, poorly isolated from an
adjoining room. Decays taken in the presence of acoustically coupled
spaces are characteristically concave upward, such as in Fig. 13-3, and
often the deviations from the straight line are even greater. When the
decay traces are nonexponential, i.e., they depart from a straight line
in a level vs. time plot, we must conclude that true diffuse conditions
do not prevail.
DIFFUSION OF SOUND 271
Spatial Uniformity of Reverberation Time
When reverberation time for a given frequency is reported, it is usually
the average of multiple observations at each of several positions in the
room. This is the pragmatic way of admitting that reverberatory conditions
differ from place to place in the room. Figure 13-4 shows the
results of actual measurements in a small (22,000 cu ft) video studio.
The multiple uses of the space required variable reverberation time,
which was accomplished by hinged wall panels that can be closed,
revealing absorbent sides, or opened, revealing reflecting sides. Multiple
reverberation decays were recorded at the same three microphone
positions for both “panels-reflective” and “panels-absorptive” conditions.
The circles are the average values, and the light lines represent
average reverberation time at each of the three positions. It is evident
Relative sound pressure level - dB
Time
10 dB
F I G U R E 1 3 - 2
Typical double slope-decay, evidence of a lack of diffuse sound conditions. The slower
decaying final slope is probably due to modes that encounter lower absorption.
272 CHAPTER THIRTEEN
that there is considerable variation, which means that the sound field
of the room is not completely homogeneous during this transient
decay period. Inhomogeneities of the sound field are one reason that
reverberation times vary from point to point in the room, but there are
other factors as well. Uncertainties in fitting a straight line to the decay
also contribute to the spread of the data, but this effect should be relatively
constant from one position to another. It seems reasonable to
conclude that spatial variations in reverberation time are related, at
least partially, to the degree of diffusion in the space.
Standard deviations of the reverberation times give us a measure of
the spread of the data as measured at different positions in a room.
When we calculate an average value, all evidence of the spread of the
data going into the average is lost. The standard deviation is the statistician’s
way of keeping an eye on the data spread. The method of calcu-
Relative sound pressure level - dB
Time
10 dB
F I G U R E 1 3 - 3
The nonexponential form of this decay, taken in a 400-seat chapel, is attributed to
acoustically coupled spaces. The absence of a diffuse sound field is indicated.
DIFFUSION OF SOUND 273
lating the standard deviation is described in the manuals of most scientific
calculators. Plus or minus one standard deviation from the mean
value embraces 68% of the data points if the distribution is normal
(Gaussian), and reverberation data should qualify reasonably well. In
Table 13-1, for 500 Hz, panels reflective, the mean RT60 is 0.56 seconds
with a standard deviation of 0.06 seconds. For a normal distribution,
68% of the data points would fall between 0.50 and 0.62 second. That
0.06 standard deviation is 11% of the 0.56 mean. The percentages listed
in Table 13-1 give us a rough appraisal of the precision of the mean.
In order to view the columns of percentage in Table 13-1 graphically,
they are plotted in Fig. 13-5. Variability of reverberation time values at the
1.0
.08
.06
.04
.02
0
Reverberation time - seconds
63 250 500 1 kHz 2 kHz 4 kHz
Frequency - Hz
Panels absorptive
Panels reflective
F I G U R E 1 3 - 4
Reverberation time characteristics of a 22,000-cu ft studio with acoustics adjustable by
hinged panels, absorbent on one side and reflective on the other. At each frequency, the
variation of the average reverberation time at each of the three positions indicates nondiffuse
conditions especially at low frequencies.
274 CHAPTER THIRTEEN
Table 13-1 Reverberation time of small video studio.
Panels reflective Panels absorptive
Octave band
center Std. dev. Std. dev.
frequency RT60 Std. dev. % of mean RT60 Std. dev. % of mean
63 0.61 0.19 31. 0.69 0.18 26.
125 0.42 0.05 12. 0.32 0.06 19.
250 0.50 0.05 10. 0.24 0.02 8.
500 0.56 0.06 11. 0.24 0.01 4.
1 kHz 0.67 0.03 5. 0.26 0.01 4.
2 kHz 0.75 0.04 5. 0.31 0.02 7.
4 kHz 0.68 0.03 4. 0.33 0.02 6.
8 kHz 0.63 0.02 3. 0.34 0.02 6.
Panels reflective
(RT60 = 0.56sec at 500 Hz)
Panels absorptive
(RT60 = 0.24 sec at 500 Hz)
40
30
20
10
0
63 125 250 500 1 kHZ 2 kHZ 4 kHz 8 kHz
Frequency - Hz
Standard deviation - % of mean
F I G U R E 1 3 - 5
Closer examination of the reverberation time variations of the studio of Fig. 13-4. The standard deviation,
expressed as a percentage of the mean value, shows lack of diffusion, especially below 250 Hz.
DIFFUSION OF SOUND 275
higher frequencies settles down to reasonably constant values in the
neighborhood of 3% to 6%. Because we know that each octave at high frequencies
contains an extremely large number of modes that results in
smooth decays, we can conclude confidently that at the higher audible
frequencies essentially diffuse conditions exist, and that the 3% to 6%
variability is normal experimental measuring variation. At the low frequencies,
however, the high percentages (high variabilities) are the result
of greater mode spacing producing considerable variation in reverberation
time from one position to another. We must also admit that these high
percentages include the uncertainty in fitting a straight line to the wiggly
decay characteristic of low frequencies. However, a glance at Fig. 13-4
shows that there are great differences in reverberation time between the
three measuring positions. For this 22,000 cu ft studio for two different
conditions of absorbance (panels open/closed), diffusion is poor at 63 Hz,
somewhat better at 125 Hz, and reasonably good at 250 Hz and above.
Decay Shapes
If all decays have the same character at all frequencies and that character
is smooth decay, complete diffusion prevails. In the real world,
the decays of Fig. 7-10 with significant changes in character are more
common, especially for the 63-Hz and 125-Hz decays.
Microphone Directivity
One method of appraising room diffusion is to rotate a highly directional
microphone in various planes and record its output to the constant
excitation of the room. This method has been applied with some
success to large spaces, but the method is ill adapted to smaller recording
studios, control rooms, and listening rooms, in which diffusion
problems are greatest. In principle, however, in a totally homogeneous
sound field, a highly directional microphone pointed in any direction
should pick up a constant signal.
Room Shape
How can a room be built to achieve maximum diffusion of sound? This
opens up a field in which there are strong opinions—some of them
276 CHAPTER THIRTEEN
supported by quite convincing experiments—and some just strong
without such support.
There are many possible shapes of rooms. Aside from the general
desirability of a flat floor in this gravity-stricken world, walls can be
splayed, ceilings inclined, cylindrical or polygonal shapes employed.
Some shapes can be eliminated because they focus sound, and focusing
is the opposite of diffusing. For example, parabolic shapes yield beautifully
sharp focal points and cylindrical concavities less sharp but
nonetheless concentrated. Even polygonal concave walls of 4, 5, 6, or 8
sides approach a circle and result in concentrations of sound in some
areas at the expense of others.
The popularity of rectangular rooms is due in part to economy of
construction, but it has its acoustical advantages. The axial, tangential,
and oblique modes can be calculated with reasonable effort and their
distribution studied. For a first approximation,
a good approach is to consider
only the more dominant axial modes,
which is a very simple calculation. Degeneracies
(mode pile-ups) can be spotted and
other room faults revealed.
The relative proportioning of length,
width, and height of a sound sensitive room
is most important. If plans are being made
for constructing such a room, there are usually
ideas on floor-space requirements, but
where should one start in regard to room
proportions? Cubical rooms are anathema.
The literature is full of early quasi-scientific
guesses, and later statistical analyses of
room proportions that give good mode distribution.
None of them come right out and
say, “This is the absolute optimum.” Bolt2
gives a range of room proportions producing
the smoothest room characteristics at low
frequencies in small rectangular rooms (Fig.
13-6). Volkmann’s 2 : 3 : 5 proportion,3 was
in favor 50 years ago. Boner suggested the 1
: 1.26 : 1.59 ratio as optimum.4 Sepmeyer5
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
Length
1.0 1.2 1.4 1.6 1.8 2.0
Height = 1
G
C
F
D
E
H
B
A
WIDTH
F I G U R E 1 3 - 6
A chart of favorable room dimensional ratios to
achieve uniform distribution of modal frequencies of
a room. The broken line encloses the so-called “Bolt-
Area.”2 The letters refer to Table 13-2.
DIFFUSION OF SOUND 277
published a computer statistical study in 1965 that yields several favorable
ratios. An even later paper by Louden6 lists 125 dimension ratios
arranged in descending order of room acoustical quality.
Table 13-2 lists the best proportions suggested by all of these
papers. To compare these with the favorable area suggested by Bolt,
they are plotted in Fig. 13-6. Most of the ratios fall on or very close to
the Bolt area. This gives confidence that any ratio falling in the Bolt
area will yield reasonable low-frequency room quality as far as distribution
of axial modal frequencies is concerned.
One cannot tell by looking at a room’s dimensional ratio whether it
is desirable or not, and it is preferable to make the evaluation, rather
than just take someone’s word for it. Assuming a room height of 10 ft,
and the other two dimensions, an axial mode analysis such as Fig. 13-
7 can be made for each. This has been done and these modes are plotted
in Fig. 13-8. Each is keyed into Table 13-2 for source identification.
All of these are relatively small rooms and therefore suffer the same
fate of having axial-mode spacings in frequency greater than desired.
The more uniform the spacing, the better. Degeneracies, or mode coin-
Table 13-2. Rectangular room dimension ratios for favorable mode
distribution.
Author Height Width Length In Bolt’s range?
1. Sepmeyer5 A 1.00 1.14 1.39 No
B 1.00 1.28 1.54 Yes
C 1.00 1.60 2.33 Yes
2. Louden6 D 1.00 1.4 1.9 Yes
3 best ratios E 1.00 1.3 1.9 No
F 1.00 1.5 2.5 Yes
3. Volkmann3 G 1.00 1.5 2.5 Yes
2 : 3 : 5
4. Boner4 H 1.00 1.26 1.59 Yes
1: 3 2: 3 4
278 CHAPTER THIRTEEN
Length
L = 19 ft 5 in
L = 19.417 ft
f1 = 565/L
Width
W = 14 ft 2 in
W = 14.17 ft
f1 =565/W
Height
H = 8 ft
H = 8.92 ft
f1 = 565/H
Arranged in
ascending
order
Diff
f1
f2
f3
f6
f5
f4
f8
f7
f9
f10
f12
f11
f13
f14
f15
29.1
87.3
116.4
145.5
174.6
203.7
232.8
261.9
320.1
291.0
58.1
39.9
79.7
119.6
159.5
199.4
239.2
279.1
319.0
63.3
126.7
190.0
253.4
316.7
29.1
39.9
58.2
63.3
79.7
87.3
116.4
119.6
126.7
145.5
159.5
174.6
190.0
199.4
203.7
232.8
239.2
253.4
261.9
279.1
291.0
316.7
10.8
18.3
5.1
16.4
7.6
29.1
3.2
7.1
18.8
14.0
15.1
15.4
9.4
4.3
29.1
6.4
14.2
8.5
17.2
11.9
25.7
F I G U R E 1 3 - 7
A convenient data form for studying the effects of room proportions on the distribution of axial modes.
DIFFUSION OF SOUND 279
A
B
C
D
E
F
G
H
2
2 2
2 2 2
3 3
0 50 100 150 200 250 300
Frequency - Hz
F I G U R E 1 3 - 8
Plots of axial mode distribution for the eight “best” room proportions of Table 13-2.
The small numbers indicate the number of modes coincident at those particular frequencies.
A room height of 10 ft is assumed.
280 CHAPTER THIRTEEN
cidences, are a potential problem, and they are identified by the 2 or 3
above them to indicate the number of resonances piled up. Modes very
close together, even though not actually coincident, can also present
problems. With these rules to follow, which of the 8 “best” distributions
of Fig. 13-8 are really the best and which the worst? First, we
reject G with two triple coincidences greatly spaced from neighbors.
Next, F is eliminated because of three double coincidences associated
with some quite wide spacings. We can neglect the effect of the double
coincidences near 280 Hz in C and D because colorations are rarely
experienced above 200 Hz. Aside from the two rejected outright, there
is little to choose between the remainder. All have flaws. All would
probably serve quite well, alerted as we are to potential problems here
and there. This simple approach of studying the axial-mode distribution
has the advantage of paying attention to the dominant axial modes
knowing that the weaker tangential and oblique modes can only help
by filling in between the more widely spaced axial modes.
Figure 13-7 illustrates a data form that makes it easy to study the
axial modes of a room. Analyzing the results requires some experience
and a few rules of thumb are suggested. A primary goal is to avoid coincidences
(pile-ups) of axial modes. For example, if a cubical space were
analyzed, all three columns would be identical; the three fundamentals
and all harmonics would coincide. This produces a triple coincidence
at each modal frequency and great gaps between. Unquestionably,
sound in such a cubical space would be highly colored and acoustically
very poor. The room of Fig. 13-7 has 22 axial modes between 29.1
and 316.7 Hz. If evenly spaced, the spacing would be about 13 Hz, but
spacings vary from 3.2 to 29.1 Hz. However, there are no coincidences—
the closest pair are 3.2 Hz apart. If a new room is to be constructed,
you have the freedom on paper to move a wall this way or that
or to raise or lower the ceiling a bit to improve distribution. The particular
room proportions of Fig. 13-7 represent the end product of many
hours of cut and try. While this cannot be represented as the best proportioning
possible, this room, properly treated, will yield good, uncolored
sound. The proper starting point is proper room proportions.
In adapting an existing space, you lack the freedom to shift walls as
on paper. A study of the axial modes as per Fig. 13-7, however, can still
be very helpful. For example, if such a study reveals problems and
space permits, a new wall might improve the modal situation
DIFFUSION OF SOUND 281
markedly. By splaying this wall, other advantages discussed later may
accrue. If the study points to a coincidence at 158 Hz, well separated
from neighbors, one is alerted to potential future problems with an
understanding of the cause. There is always the possibility of introducing
a Helmholtz resonator tuned to the offending coincidence to
control its effect (see pp. 226–229). All these things are related to
sound diffusion.
Splaying Room Surfaces
Splaying one or two walls of a sound-sensitive room does not eliminate
modal problems, although it might shift them slightly and produce
somewhat better diffusion.7 In new construction, splayed walls
cost no more, but may be quite expensive in adapting an existing
space. Wall splaying is one way to improve general room diffusion,
although its effect is nominal. Flutter echoes definitely can be controlled
by canting one of two opposing walls. The amount of splaying
is usually between 1 foot in 20 feet and 1 foot in 10 feet.
Nonrectangular Rooms
The acoustical benefit to be derived from the use of nonrectangular
shapes in audio rooms is rather controversial. Gilford10 states, “...slanting
the walls to avoid parallel surfaces.... does not remove colorations;
it only makes them more difficult to predict.” Massive trapezoidal
shaped spaces, commonly used as the outer shell of recording studio
control rooms, guarantee asymmetrical low-frequency sound fields
even though it is generally conceded that symmetry with the control
position is desirable.
Computer studies based on the finite element approach have
revealed in minute detail what happens to a low-frequency sound
field in nonrectangular rooms. The results of a study using this
method conducted by van Nieuwland and Weber at Philips Research
Laboratories, The Netherlands, are given in Figs. 13-10 through 13-13.8
Highly contorted sound fields are shown, as expected, for the nonrectangular
case, for modes 1,0, 1,3, 0,4, and 3,0. A shift in frequency of
the standing wave from that of the rectangular room of the same area is
indicated: –8.6%, –5.4%, –2.8%, and +1% in the four cases illustrated.
This would tend to support the common statement that splay282
CHAPTER THIRTEEN
ing of walls helps slightly in breaking up
degeneracies, but shifts of 5% or more are
needed to avoid the effects of degeneracies.
The proportions of a rectangular
room can be selected to eliminate, or at
least greatly reduce, degeneracies, while
in the case of the nonrectangular room,
such a prior examination of degeneracies
is completely impractical. Making the
sound field asymmetrical by splaying
walls only introduces unpredictability in
listening room and studio situations.
If the decision is made to splay walls in
an audio room, say 5%, a reasonable
approximation would be to analyze the
equivalent rectangular room having the
same volume.
Geometrical Irregularities
Many studies have been made on what
type of wall protuberances provide the
best diffusing effect. Somerville and Ward9
reported years ago that geometrical diffusing
elements reduced fluctuations in a
swept-sine steady-state transmission test.
The depth of such geometrical diffusors must be at least 1⁄7 of a wavelength
before their effect is felt. They studied cylindrical, triangular,
and rectangular elements and found that the straight sides of the rectangular-
shaped diffusor provided the greatest effect for both steadystate
and transient phenomena. BBC experience indicates superior
subjective acoustical properties in studios and concert halls in which
rectangular ornamentation in the form of coffering is used extensively.
Absorbent in Patches
Applying all the absorbent in a room on one or two surfaces does not
result in a diffuse condition, nor is the absorbent used most effectively.
Let us consider the results of an experiment that shows the effect of
F I G U R E 1 3 - 9
The use of distributed sound-absorbing modules is an
economical way to achieve maximum absorption as
well as to enhance sound diffusion in the room.
(World Vision International)
DIFFUSION OF SOUND 283
f = 34.3 Hz
100
0
0
0
40 100
 100
f = 31.6 Hz


70
40
F I G U R E 1 3 - 1 0
Comparison of the modal pattern for a 5 x 7 meter two-dimensional room and a nonrectangular
room of the same area. This sound field of the 1,0 mode is distorted in the
nonrectangular room and the frequency of the standing wave is shifted slightly.8
f = 81.1 Hz
100


100
100
50
20
100
100
100
100 100
100
 



 
90
90 90
90
0
0
0
0
0
f = 85.5 Hz

0
0
0
0
0
0
0
80 40
10
10
10
60 10
10
20
20
40
40
20
20
50
F I G U R E 1 3 - 1 1
The 1,3 mode for the 5 x 7 meter room of Fig. 13-10 compared to a nonrectangular room
of the same area. The sound field is distorted and the frequency is shifted.8
284 CHAPTER THIRTEEN
distributing the absorbent.1 The experimental room is approximately a
10-ft cube and it was tiled (not an ideal recording or listening room,
but acceptable for this experiment). For test 1, reverberation time for
the bare room was measured and found to be 1.65 seconds at 2 kHz.
For test 2, a common commercial absorber was applied to 65% of one
wall (65 sq ft), and the reverberation time at the same frequency was
found to be about 1.02 seconds. For test 3, the same area of absorber
was divided into four sections, one piece mounted on each of four of
the room’s six surfaces. This brought the reverberation time down to
about 0.55 seconds.
The startling revelation here is that the area of the absorber was
identical between tests 2 and 3; the only difference was that in test 3 it
was in four pieces, one on each of 3 walls and one piece on the floor.
By the simple expedient of dividing the absorbent and distributing it,
the reverberation time was cut almost in half. Inserting the values of
reverberation time of 1.02 and 0.55 seconds and the volume and area
of the room into the Sabine equation, we find that the average absorpf
= 98.0 Hz
100
100
100






0
0
0
0
0
0
0
0
0
0
100
100 100
f = 95.3 Hz
20
 30
60
70
80
30
10
F I G U R E 1 3 - 1 2
The 0,4 mode of the 5 x 7 meter two-dimensional room of Figs. 13-10 and 13-11. The
high distortion in the rectangular room is accompanied by a shift in standing wave frequency.
8
DIFFUSION OF SOUND 285
tion coefficient of the room increased from 0.08 to 0.15 and the number
of absorption units from 48 to 89 sabins. Where did all this extra
absorption come from? Laboratory testing personnel measuring
absorption coefficients in reverberation chambers have agonized over
the problem for years. Their conclusion is that there is an edge effect
related to diffraction of sound that makes a given sample appear to be
much larger acoustically. Stated another way, the sound-absorbing
efficiency of 65 sq ft of absorbing material is only about half that of
four 16-sq ft pieces distributed about the room, and the edges of the
four pieces total about twice that of the single 65-sq ft piece. So, one
advantage of distributing the absorbent in a room is that its soundabsorbing
efficiency is greatly increased, at least at certain frequencies.
But be warned: The above statements are true for 2 kHz, but at 700 Hz
and 8 kHz, the difference between one large piece and four distributed
pieces is small.
Another significant result of distributing the absorbent is that it
contributes to diffusion of sound. Patches of absorbent with reflective
f = 102.9 Hz
f = 102.9 Hz
100
100



 

100 0 100 0 0 100
30
0
0
0 0
10
0
0
0 60
60
20 10
20
20
80

40
70
20

60
60

F I G U R E 1 3 - 1 3
The 3,0 mode of the 5 x 7 meter two-dimensional room of Figs. 13-10 and 13-11, and
13-12 and resulting distortion of the modal pattern when changed to a nonrectangular
room of the same area.8
286 CHAPTER THIRTEEN
walls showing between the patches have the effect of altering wavefronts,
which improves diffusion. Sound-absorbing modules in a
recording studio such as in Fig. 13-9 distribute the absorbing material
and simultaneously contribute to the diffusion of sound.
Concave Surfaces
A concave surface such as that in Fig. 13-14A tends to focus sound
energy and consequently should be avoided because focusing is just
the opposite of the diffusion we are seeking. The radius of curvature
determines the focal distance; the flatter the concave surface, the
greater the distance at which sound is concentrated. Such surfaces
often cause problems in microphone placement. Concave surfaces
might produce some awe-inspiring effects in a whispering gallery
where you can hear a pin drop 100 ft away, but they are to be avoided
in listening rooms and small studios.
Convex Surfaces: The Poly
One of the most effective diffusing elements, and one relatively easy to
construct, is the polycylindrical diffusor (poly), which presents a convex
section of a cylinder. Three things can happen to sound falling on
B
A
F I G U R E 1 3 - 1 4
Concave surfaces (A) tend to focus sound, convex surfaces (B) tend to diffuse it. Concave
surfaces should be avoided if the goal is to achieve well-diffused sound.
DIFFUSION OF SOUND 287
such a cylindrical surface made of plywood
or hardboard: The sound can be
reflected and thereby dispersed as in Fig.
13-14B; the sound can be absorbed; or the
sound can be reradiated. Such cylindrical
elements lend themselves to serving as
absorbers in the low frequency range
where absorption and diffusion are so
badly needed in small rooms. The reradiated
portion, because of the diaphragm
action, is radiated almost equally throughout
an angle of roughly 120° as shown in
Fig. 13-15A. A similar flat element reradiates
sound in a much narrower angle,
about 20°. Therefore, favorable reflection,
absorption, and reradiation characteristics
favor the use of the cylindrical surface.
Some very practical polys and their
absorption characteristics are presented in
Chap. 9. The dimensions of such diffusors are not critical, although to
be effective their size must be comparable to the wavelength of the
sound being considered. The wavelength of sound at 1,000 Hz is a bit
over 1 ft, at 100 Hz about 11 ft. A poly element 3 or 4 ft across would be
effective at 1000 Hz, much less so at 100 Hz. In general, poly base or
chord length of 2 to 6 ft with depths of 6 to 18 inches meet most needs.
It is important that diffusing elements be characterized by randomness.
A wall full of polys, all of 2-ft chord and of the same depth, might
be beautiful to behold, like some giant washboard, but not very effective
as diffusors. The regularity of the structure would cause it to act as
a diffraction grating, affecting one particular frequency in a much different
way than other frequencies, which is opposite to what the ideal
diffusor should do.
Axes of symmetry of the polys on different room surfaces should be
mutually perpendicular.
Plane Surfaces
Geometrical sound diffusing elements made up of two flat surfaces to
give a triangular cross section or of three or four flat surfaces to give a
120° 20°
A
B
F I G U R E 1 3 - 1 5
(A) A polycylindrical diffusor reradiates sound
energy not absorbed through an angle of about 120°.
(B) A similar flat element reradiates sound in a
much smaller angle.
288 CHAPTER THIRTEEN
polygonal cross section may also be used. In general, their diffusing qualities
are inferior to the cylindrical section.
Endnotes
1Randall, K.E. and F.L. Ward, Diffusion of Sound in Small Rooms, Proc. Inst. Elect. Engs., Vol
107B (Sept. 1960), p. 439-450.
2Bolt, R.H., Note on Normal Frequency Statistics for Rectangular Rooms, J. Acous. Soc. Am.,
18, 1 (July 1946) p. 130-133.
3Volkmann, J.E., Polycylindrical Diffusers in Room Acoustical Design, J. Acous. Soc. Am., 13
(1942), p. 234-243.
4Boner, C.P., Performance of Broadcast Studios Designed with Convex Surfaces of Plywood,
J. Acous. Soc. Am., 13 (1942) p. 244-247.
5Sepmeyer, L.W., Computed Frequency and Angular Distribution of the Normal Modes of
Vibration in Rectangular Rooms, J. Acous Soc. Am., 37, 3 (March 1965), p. 413-423.
6Louden, M.M., Dimension-Ratios of Rectangular Rooms with Good Distribution of
Eigentones, Acustica, 24 (1971), p. 101-103.
7Nimura, Tadamoto and Kimio Shibayama, Effect of Splayed Walls of a Room on Steady-
State Sound Transmission Characteristics, J. Acous. Soc. Am., 29, 1 (January 1957), p. 85-
93.
8van Nieuwland, J.M. and C. Weber, Eigenmodes in Non-Rectangular Reverberation Rooms,
Noise Control Eng., 13, 3 (Nov/Dec 1979), 112-121.
9Somerville, T. and F.L. Ward, Investigation of Sound Diffusion in Rooms by Means of a
Model, Acustica, 1, 1 (1951), p. 40-48.
10Gilford, Christopher, Acoustics for Radio and Television Studios, (1972), London, Peter
Peregrinus, Ltd.
In a strange turn of events, diffraction gratings have become important
in acoustics. The subject of diffraction gratings brings to mind
several memories, widely separated in time and space.
There are three large Morpho butterflies mounted in a frame in our
home, a sweet reminder of first seeing their gorgeous flashes of color as
they flitted about in a jungle clearing in Panama. Their color changes in
a most dazzling fashion. They were even more fascinating to me when
I learned that this color is not pigmentation—it is structural color.
The butterfly’s wing is basically a tawdry tan—the vivid coloring is the
result of breaking down the light that falls upon it into rainbow colors
by diffraction. Viewed through a microscope, the wing surface is made
up of a myriad of very small bumps and grooves.
At Mt. Wilson Observatory is the diffraction grating used by Edwin
Hubble to measure the light of distant stars. Using this diffraction grating,
he formulated his theory of the expanding universe based on the
“red shifts” of starlight he observed. It is a glass plate with precise, parallel
lines cut on it, many thousands to the inch. Sunlight falling on
this grating is broken down into its component colors, just as the light
from the stars.
In early days, the diffraction of X-rays by crystals was studied in
the Physics Department at Stanford University in a safety cage made
of chicken wire screen in a remote corner of the basement. Crystals
The Schroeder Diffusor
14
289
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
290 CHAPTER FOURTEEN
are basically naturally occurring three-dimensional diffraction gratings.
The X-rays, having a wavelength of the same order as the spacings
of the atoms in the crystal lattice, were scattered by the crystal
in an orderly, predictable fashion.
Applying the principle of the diffraction grating to acoustics is further
evidence of the ever-growing cutting edge of science.
Schroeder’s First Acoustic Diffusor
In a remarkable outpouring of fresh, new ideas, Manfred R. Schroeder
of the University of Göttingen, Germany, and AT&T Bell Laboratories
at Murray Hill, New Jersey, has opened new vistas on the diffusion of
sound. Schroeder has lifted from computer and number theory the
idea that a wall with grooves arranged in a certain way will diffuse
sound to a degree unattainable in the past. Maximum-length codes can
be used to create pseudo-random noise by application of certain
sequences of +1 and –1. The power spectrum (from the Fourier transform)
of such noise is essentially flat. A wide and flat power spectrum
is related to reflection coefficients and angles, and this gave rise to the
idea that by applying the +1 and –1 in a maximum-length sequence
something acoustically interesting might result. The –1 suggested a
reflection from the bottom of a groove in a wall with a depth of a quarter
wavelength. The +1 reflection is a reflection from the wall itself
without any groove.
Professor Schroeder’s next move was to test the idea.1 He ordered
a piece of sheet metal to be bent into the shape of Fig. 14-1, planning
to test the idea with 3-cm microwaves. This shape followed the
binary maximum-length sequence with period length 15:
– + + – + – + + + + – – – + –
The resulting reflection pattern (Fig. 14-2(B)) indicated that this piece
of sheet metal was much like any other piece of sheet metal; it gave a
strong specular reflection, but little diffusion of energy. He asked how
deep the steps were and learned that they were half-wavelength, not
the quarter wavelength ordered. No wonder it reflected the
microwaves like a flat sheet of metal! The grooves one-half wavelength
deep gave a reflection coefficient of +1, just like a flat sheet. Back to the
shop for a new metal sheet with grooves a quarter wavelength deep.
THE SCHROEDER DIFFUSOR 291
This time the reflection pattern of Fig.
14-2A resulted, a very encouraging
development.
Professor Schroeder’s associate, who
shaped the metal grating, was one who
always liked to check Schroeder’s theories.
He quietly ran another test with a narrow
strip of metal covering just one of the
grooves, the one marked x in Fig. 14-1. The
reflection pattern of Fig. 14-2B resulted,
which shows essentially specular reflection
of most of the energy back toward the
source. In other words, covering only one of
the slots almost completely destroyed the
favorable reflection pattern of Fig.14-2A.
This success encouraged further development
of the basic idea for acoustical
applications.
One period
/2
/4
x
              
F I G U R E 1 4 - 1
A metal sheet folded to conform to a maximum-length sequence used by Schroeder to
check the diffusion of 3-cm radio waves. The wells are all 1/4 wavelength in depth.
Incident sound
A B
F I G U R E 1 4 - 2
When the wells of Fig. 14-1 were accidentally made
1/2 wavelength deep, the almost pure specular reflection
of B resulted, the same as for a flat sheet of
metal. When another diffuser was made with the correct
1/4 wavelength depths, the very favorable diffusion
pattern of A was obtained. With permission of M.R.
Schroeder and Applied Science Publishers, Ltd., London.
292 CHAPTER FOURTEEN
Maximum-Length Sequences
This experiment was a turning point in acoustics. The reflection pattern
of Fig.14-2A is far superior to the best sound diffusor previously
available. For over a half century, diffusion was sought through adjustment
of room proportions, splaying of walls, the use of semispherical,
polycylindrical, triangular, cubical, and rectangular geometrical protrusions,
and the distribution of absorbing material as shown in Chap.
13. The degree of diffusion achieved by such means was far from sufficient
to meet the need. Considered in this light, the importance of
Professor Schroeder’s simple experiment with the piece of bent sheet
metal becomes apparent.
Because of the one-quarter wavelength groove depth requirement
of the binary maximum-length diffusor, the sound-diffusing properties
of the surface depend on the wavelength of the incident sound. Experience
has indicated that reasonable diffusion results over a band of
plus or minus one-half octave of the frequency around which the diffusor
is designed. For example, consider a maximum-length sequence
diffusor with a sequence length of 15.2 A design frequency of 1,000 Hz
gives a one-half wavelength of 7.8 in, and a one-quarter wavelength
groove depth of 3.9 in. A single period of this diffusor would be about
5 ft in width and effective from about 700 to about 1,400 Hz. Many
such units would be required to provide diffusion over a reasonable
portion of the audible band.
Reflection Phase-Grating Diffusors
The diffraction grating, so long the sole province of optics, is now being
applied to acoustics. Commercial development of the idea is being
aggressively pursued by RPG Diffusor Systems, Inc., of Upper Marlboro,
Maryland.8 In the past, the acoustical designer basically had only
absorption and reflection to use. Diffusion was something earnestly
sought, but only partially achieved. With the large assortment of grating-
type diffusor units commercially available, this situation is rapidly
changing.
The acoustical treatment of large spaces is well served by such diffusors.
A large space is defined as one whose normal mode frequencies
THE SCHROEDER DIFFUSOR 293
are so closely spaced as to avoid low-frequency resonance problems.
This includes music halls, auditoriums, and churches. The sound quality
of a music hall is influenced greatly by the reflections from side
walls. No longer do we completely depend on side walls for necessary
lateral reflections. A row of diffraction-grating diffusors down the center
of the hall at ceiling level can diffuse the sound from the stage laterally
to the people in the seats. Any troublesome specular reflection
can be controlled by strategic placement of diffusors.
In churches, there is always conflict between the intelligibility of
the spoken word and conditions for full enjoyment of the music. The
rear wall is often the source of reflections that create disturbing
echoes. To make this wall absorbent is often detrimental to music conditions.
Making the rear wall diffusive, however, minimizes the echo
problem while at the same time conserving precious music and speech
energy. Music directors are often faced with the problem of singers or
instrument players not hearing each other well. How can they play or
sing together properly if they cannot hear each other? Surrounding the
music group with an array of reflection phase grating diffusors both
conserves music energy and spreads it around to achieve ensemble
between musicians.
Difficult small-room acoustics are helped especially by diffractiongrating-
type diffusing elements. The need to cant walls and distribute
absorption material to achieve some semblance of sound diffusion is
relaxed by the proper use of efficient grating diffusors. For the first
time it is possible, by proper design, to get acceptable voice recordings
from small “announce booths” because diffusing elements create a
larger room sound. It is safe to say that future audio-room design will
never be the same because of these diffusors.
Quadratic-Residue Diffusors
The principle of the reflection phase grating is now applied to
sound, again through Schroeder’s intuitive understanding of the
limitations of the maximum-length diffusor. He reasoned that an
incident sound wave falling on what physicists call a reflection
phase grating would diffuse sound almost uniformly in all directions.
The phase shifts (or time shifts) can be obtained by an array of
294 CHAPTER FOURTEEN
wells of depths determined by a quadratic residue sequence. The
maximum well depth is determined by the longest wavelength to be
diffused. The well width is about a half wavelength at the shortest
wavelength to be scattered.3–5 The depths of the sequence of wells
are determined by the statement:
Well Depth Proportionality Factor = n2 modulo p (14-1)
in which,
p = a prime number.
n = a whole number between zero and infinity.
Webster’s Ninth New Collegiate Dictionary defines a prime number
as Any number (other than 0 or 1) that is not divisible without
remainder by any other integer (except 1 and  the integer itself).
Examples of prime numbers are 5, 7, 11, 13, etc. The modulo refers
simply to residue. For example, inserting p = 11 and n = 5 into Eq.
14-1 gives 25 modulo 11. The modulo 11 means that 11 is subtracted
from 25 until the significant residue is left. In other words, 11 is subtracted
from 25 twice and the residue 3 is our answer. (For a review
of number theory, see Endnotes.6, 7)
In Fig. 14-3 quadratic residue sequences are listed for the prime
numbers 5, 7, 11, 13, 17, 19, and 23, a separate column for each. To
check the above example for n = 5 and p = 11, enter the column
marked 11, run down to n = 5 and find 3, which checks the previous
computation. Figure 14-4 is a model for a quadratic residue reflection
phase grating diffusor for p = 17. The numbers in each column of Fig.
14-3 are proportional to well depths of different quadratic residue diffusors.
The design of Fig. 14-4 is based on the prime 17 sequence of
numbers. At the bottom of each column of Fig. 14-3 is a sketch of a
quadratic residue diffusor profile with well depths proportional to
the numbers in the sequence. The broken lines indicate thin dividers
between the wells.
The Hewlett-Packard HP-41C hand-held calculator has the modulo
function in memory. As an example, for n = 12, p = 19. Equation 14-1
becomes 144 modulo 19, which becomes a bit messy to do by hand. The
HP-41C solves this easily. In an appendix of Schroeder’s excellent book5
a modulo program for calculators is included.
THE SCHROEDER DIFFUSOR 295
n
p
5 7 11 13 17 19 23
0123456789
10
11
12
13
14
15
16
17
18
19
20
21
22
23
0
14410
01422410
014953359410
01493
12
10
10
12
39410
0149
16
82
15
13
13
15
28
16
9410
0149
16
6
17
11
7557
11
17
6
16
9410
0149
16
2
13
3
18
12
8668
12
18
3
13
2
16
9410
Quadratic-residue sequences
Well depth or proportionality = n2 modulo p
n = integer
p = prime number
F I G U R E 1 4 - 3
Quadratic-residue sequences for prime numbers from 5–23. In the diffusor profile at the foot of each column,
the depths of the wells are proportional to the sequence of numbers above.
296 CHAPTER FOURTEEN
Primitive-Root Diffusors
Primitive-root diffusors use a different number theory sequence, as
follows:
Well Depth Proportionality Factor = gn modulo p (14-2)
in which:
p is a prime number, and
g is the least primitive root of p.
Figure 14-5 includes primitive-root sequences for six different combinations
of p and g. The sketches at the bottom of each column are not symmetrical
like those of the quadratic residue diffusors. In most cases this is
a disadvantage but in some cases it is an advantage. There is an acoustical
problem with the primitive root diffusors in that the specular mode is not
suppressed as well as it is in the quadratic-residue diffusor. Commercial
development has largely utilized quadratic residue sequences.
Separators between the wells, usually metallic, are commonly used
to maintain the acoustical integrity of each well. Without separators
the effectiveness of the diffusor is decreased. The stepped phase shifts
for sound arriving at angles other than the perpendicular tend to be
confused in the absence of dividers.
0 1 4 9 16 8 2 15 13 13 15 2 8 16 9 4 1
One period
F I G U R E 1 4 - 4
A typical quadratic-residue diffusor based upon the prime number 17 column of Fig 14-3. The depths of the
wells are proportional to the sequence of numbers in the prime-17 column. Two periods are shown illustrating
how adjacent periods are fitted together.
THE SCHROEDER DIFFUSOR 297
n p = 5
g = 2
p = 7
g = 3
p = 11
g = 2
p = 13
g = 2
p = 17
g = 3
p = 19
g = 2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
2
4
3
1
3
2
6
4
5
1
2
4
8
5
10
9
7
3
6
1
2
4
3
3
6
12
10
9
5
10
7
1
3
9
10
13
5
15
11
16
14
8
7
4
12
2
6
1
2
4
8
16
13
7
14
9
18
17
15
11
3
6
12
5
10
1
Primitive-root sequences
Well depth or proportionality = gn modulo p
p = prime number
g = least primitive root of p
F I G U R E 1 4 - 5
Primitive-root sequences for six combinations of prime number and least-primitive roots. Sound diffusor profiles
at the foot of each column have depths proportional to the sequence of numbers above. Note that these
diffusors are not symmetrical like the quadratic-residue diffusors.
298 CHAPTER FOURTEEN
Quadratic-Residue Applications
Numerous applications of the quadratic residue theory have been
made by RPG Diffusor Systems, Inc.8 For example, Fig. 14-6 shows two
of their model QRD-4311™ below and a QRD Model 1911™ above.
The 19 indicates that it is built on prime 19 and the 11 specifies well
widths of 1.1 inch. The sequence of numbers in the prime 19 column
of Fig. 14-3 specify the proportionality factors for well depths of the
Model QRD 1911™ diffusor of Fig. 14-6.
Following the same pattern, the QRD™ Model 4311 in the lower
portion of Fig. 14-6 is based on prime 43 with well widths also of 1.1
inch. For practical reasons, the columns of Fig. 14-3 stopped at prime
23; primes between 23 and 43 are 29, 31, 37, and 41. A good practical
exercise would be to solve n2 modulo p for n = 0 to 43 and p = 43 and
compare the well depths of the Model 4311 in Fig. 14-6 knowing that
deep wells are dark and dense, shallow wells are light.
This particular cluster of QRD diffusors offers excellent diffusion
in the horizontal hemidisc of Fig. 14-7A. In Fig. 14-6 the specular
reflection from the face of the diffusor is shown in Fig. 14-7B. The vertical
wells of the QRD 4311™ scatter horizontally and the horizontal
wells of the QRD 1911™ scatter vertically. Together they produce a virtual
hemisphere of diffusion.
Performance of Diffraction-Grating Diffusors
In designing an audio space of any kind, the acoustician has three
building blocks at his disposal: absorption, reflection, and diffusion.
In the past, it was a common experience to find too much reflection
but too little diffusion in the room to be treated. Only absorption and
a limited number of geometrical tricks were available for the designer
to carry out his assignment. Absorption, reflection, and diffusion are
now in better balance, at least potentially.
The effects on the incident sound of the three physical principles of
absorption, reflection, and diffusion are compared in Fig. 14-8. Sound
impinging on the surface of a sound absorber is largely absorbed, but a
tiny fraction is reflected. The temporal response shows a greatly attenuated
reflection from the surface of the absorber.
THE SCHROEDER DIFFUSOR 299
F I G U R E 1 4 - 6
A cluster of commercial quadratic-residue diffusors. Below are two QRD-4311™ diffusing
modules with a single QRD-1911™ mounted above. The hemidisc of diffusion for the
lower unit is horizontal, that of the upper unit is vertical. Peter D’Antonio, RPG Diffusor System,
Inc. and the Audio Engineering Society, Inc.
300 CHAPTER FOURTEEN
The same sound wave falling on a hard, reflective surface yields a
reflection of almost the same intensity as the sound falling on the surface
itself, just reduced slightly by losses at the reflective surface. The
polar plot shows the energy concentrated about the angle of reflection.
The width of the polar plot is a function of wavelength and the size of
the reflecting surface.
A sound wave falling upon a diffusor, such as a quadratic residue
type, is diffracted throughout the hemidisc of Fig. 14-7A. The diffused
energy falls off exponentially. The polar diagram shows energy spread
more or less equally throughout 180 degrees, but somewhat reduced at
grazing angles.
The uniformity of the angular distribution of scattered energy
through a wide range of frequency is shown dramatically in Fig. 14-9.
These experimental polar plots, obtained for a commercial quadraticresidue
diffusor, are smoothed by averaging over octave bands. Compa-
A B
F I G U R E 1 4 - 7
(A) A one-dimensional QRD™ diffusor scatters around in a hemidisc. (B) This hemidisc
may be one specularly directed by orientation of the source with respect to the diffusor.
THE SCHROEDER DIFFUSOR 301
rable computer-generated polar plots, for single frequencies based on a
far-field diffraction theory, show a host of tightly packed lobes that have
little practical significance. Near-field Kirchhoff diffraction theory
shows less lobing.
The left column of Fig. 14-9 shows polar distribution of octave
bands of energy centered from 250 Hz to 8,000 Hz, a span of five
octaves. The right column shows the effect for sound incident at
45 degrees for the same frequencies. The results for all angles of
incident sound for the lowest frequency are dependent on the well
depth. The upper frequency is directly proportional to the number
of wells per period and inversely proportional to the well width. In
Fig. 14-9, the diffusion of a flat panel is shown by a light line for
comparison.
Acoustical
treatment
Temporal
response
Spatial
response
Absorption Direct sound
Attenuated
reflection
90
60
90
60
30
0
30
Reflection
Diffusion
Specular
reflection
10 msec
10 dB
Time (msec)
6 dB
Diffusion
F I G U R E 1 4 - 8
A comparision of the three physical principles of absorption, reflection, and diffusion.
Peter D’Antonio, RPG Diffusor Systems, Inc. and the Audio Engineering Society.
250 250 90
60
30
60
90
30
0
500 500
1,000 1,000
2,000 2,000
4,000 4,000
8,000 8,000
F I G U R E 1 4 - 9
Experimental polar plots of a commercial quadratic-residue diffusor, smoothed by averaging
over octave bands. The angular distribution of the energy is excellent over a wide
frequency range and angles of incidence. Peter D’Antonio, RPG Diffusor Systems, Inc.
THE SCHROEDER DIFFUSOR 303
The uniformity of spatial diffusion is determined by the length of the
period. Good broad-bandwidth, wide-angle diffusion, then, requires a
large period with a large number of deep, narrow wells. This is why the
QRD®-4311 has 43 wells of only 1.1 inches width and a maximum well
depth of 16 inches.
A good question about this time would be, “What does a diffusor
do that a hard, flat wall would not do?” The answer is to be found in
Fig. 14-10. This figure compares the return from a flat panel with that
Direct
sound
Specular
reflection
Diffusion
6 dB
10 msec
Time (msec)
F I G U R E 1 4 - 1 0
An energy-time contrast of a specular reflection from a flat panel and the energy diffused
from a quadratic residue diffusor. The peak energy from a diffusing surface is
somewhat lower than that of the flat panel but it is spread out in time. Peter D’Antonio, RPG
Diffusor Systems, Inc. and the Audio Engineering Society.
304 CHAPTER FOURTEEN
from a quadratic-residue diffusor. The large peak to the left is the
direct sound. The second large peak is the specular reflection from a
flat panel. Note carefully that the energy of this sharp specular panel
reflection is only a few dB below that of the incident sound. The diffused
energy from the quadratic residue diffusor is spread out significantly
in time. Most importantly, and revealed in the polar diagrams
of Fig. 14-9, the gratings diffract sound throughout 180 degrees, not
just in the specular direction like the flat panel.
There is a rich and growing literature on the development of the
Schroeder diffraction grating sound diffusor. Keep abreast of the rapid
changes in the field.5,9–25
Expansion of the QRD Line
The appearance of reflection phase grating diffusors Models QRD-
4311™ and QRD-1711™ was followed by numerous other models
designed to meet specific needs. For example, Fig. 14-11 shows Model
QRD-734, which is a 2 ft × 4 ft model suitable for use in suspended
ceiling T-frames as well as other ways. Also shown in Fig. 14-11 is the
Abffusor™, which combines absorption and diffusion in the same
unit. The Triffusor™, also shown in Fig. 14-11, has reflective, absorptive,
and diffusive faces. A group of these, set into a wall, offers wide
options in the acoustics of a space by rotating the individual units.
Solving Flutter Problems
If two opposing reflective surfaces of a room are parallel there is
always the possibility of flutter echoes. This applies to either horizontal
or vertical modes. Successive, repetitive reflections, equally spaced
in time, can even produce a perception of a pitch or timbre coloration
of music and a degradation of intelligibility of speech. The lack of
ornamentation in modern architecture results in a greater flutter possibility.
Flutter can be reduced by careful placement of sound absorbing
material, or by splaying walls. Splaying is impractical in most cases and
increasing absorption often degrades the acoustical quality of a space.
What is needed is a wall treatment that reduces reflections by scattering
rather than absorption. The RPG Flutterfree™ does exactly this. It is an
architectural hardwood molding 4 in wide by 4 or 8 ft length, shown in
THE SCHROEDER DIFFUSOR 305
F I G U R E 1 4 - 1 1
Three proprietary sound diffusing systems: (Upper Left) a broad bandwidth and wideangle
QRD 734 Diffusor™, (Below) the Abffusor™ broad bandwidth diffusor/absorber,
and (Upper Right) the Triffusor™ having one side absorptive, one diffusive, and one
reflective for acoustical variability. Peter D’Antonio, RPG Diffusor Systems, Inc. and the Audio Engineering
Society.
306 CHAPTER FOURTEEN
Fig. 14-12, which reduces specular reflection as well as contributing to
the diffusion of sound. The molding works as a one-dimensional reflection
phase grating diffusor because of the wells routed into its surface.
The depths of the wells follow a prime-7 quadratic residue sequence.
These moldings may be affixed to a wall butted together or spaced, horizontally
or vertically. If they are vertical, spectral reflections are controlled
in the horizontal plane and vice versa.
The Flutterfree™ moldings can be employed as slats for a
Helmholtz slat-type low-frequency absorber (Chap. 9). All the while
low-frequency sound is being absorbed by the Helmholtz absorber, the
surface of each slat performs as a mid-high-frequency sound diffusor.
Application of Fractals
Certain production limitations have been encountered by RPG Diffusor
Systems, Inc. in the development of reflection phase grating diffusors.
For example, the low-frequency limit is determined principally
Isometric view
0'–1 1/16"
Plan view
0'–3 15/16"
F I G U R E 1 4 - 1 2
The Flutterfree™ is a non-absorptive flutter echo control molding. It is a quadraticresidue
diffusor based on the prime-7. It can also serve as slats on a slat-type
Helmholtz low-frequency absorber. Peter D’Antonio, RPG Diffusor Systems, Inc. and the Audio Engineering
Society.
THE SCHROEDER DIFFUSOR 307
by well depth, and the high-frequency limit is determined principally
by well width. Manufacturing constraints place a limit of 1 in on well
width and 16 in on well depth, beyond which the units become
diaphragmatic.
To increase the effective bandwidth,
the self-similarity principle has been
applied in the form of fractals producing
units called Diffractals™.26 These are
really diffusors within diffusors within
diffusors as shown in the progressive
illustration of Fig. 14-13. Three sizes of
quadratic residue diffusors are required
to make up the complete Diffractal™. The
operation is much like that of the multiunit
loudspeaker. The various diffusors
making up a Diffractal™ operate independently
even as the woofer, mid-range,
and tweeter loudspeaker units operate
independently to create the wide-band
system.
Figure 14-14 shows the DFR-82LM Diffractal
™ that is 7 ft 10 in high and 11 ft
wide with a depth of 3 ft. This is a second
generation, low-mid-frequency unit that
covers the range 100 Hz to 5 kHz. The lowfrequency
portion is based on a prime-7
quadratic residue sequence. A mid-range
Diffractal™ is embedded at the bottom of
each well of the larger unit. The frequency
range of each section as well as the crossover
points of these composite units are
completely calculable.
Figure 14-15 shows the larger DFR-
83LMH™ unit that is 16 ft wide, 6 ft 8 in
high and 3 ft deep. This is a three-way
unit covering a frequency range of 100 Hz
to 17 kHz. The depths of the wells of the
F I G U R E 1 4 - 1 3
The Diffractal™ wide-band diffusor is a two- or threeway
crossed over system composed of high-frequency
fractal sound diffusors nested within a low-frequency
diffusor. The three-way system illustrated is a diffusor
within a diffusor within a diffusor, analogous to
a three-way loudspeaker system composed of woofer,
mid-range unit, and tweeter. Peter D’Antonio, RPG Diffusor
Systems, Inc. and the Audio Engineering Society.
308 CHAPTER FOURTEEN
low-frequency unit follow a prime-7 quadratic residue sequence.
Fractals are set in the wells of fractals that are set in the low-frequency
wells.
Diffusion in Three Dimensions
All of the reflection phase grating diffusors discussed previously have
rows of parallel wells. These can be called one dimensional units because
the sound is scattered in a hemidisc as shown in Fig. 14-16-A. There are
occasions in which hemispherical coverage is desired, as shown in Fig.
14-16B. RPG Diffusor Systems™ have met this need by offering the
Omniffusor™. The Omniffusor™ consists of a symmetrical array of
64 square cells as shown in Fig. 14-17. The depth of these cells is based
on the phase-shifted prime-7 quadratic residue number theory sequence.
An off-shoot of the Omniffusor™ (which is made of wood) is the
FRG Omniffusor™, which consists of an array of 49 square cells based
on the two-dimensional phase-shifted quadratic residue number theory.
This unit is made of fiberglass-reinforced gypsum. It is lighter in
weight and one-third the cost of the Omniffusor™, is therefore better
adapted for application to large surface areas.
Isometric view
11'–0"
3'–0"
F I G U R E 1 4 - 1 4
The large sized Model DFR-83LMH Diffractal™, a two-way system composed of a low-freqency
unit (prime-7) with a mid-range Diffractal embedded at the bottom of each well.
Peter D’Antonio, RPG Difusser Systems, Inc.
THE SCHROEDER DIFFUSOR 309
Acoustic Concrete Blocks
The Cinderblox and its many derivatives have been in use since
1917. In 1965 The Proudfoot Company came out with their Sound-
Blox®, which supplies not only load-bearing ability and the mass
required for sound isolation, but also enhanced low-frequency
absorption through Helmholtz resonators formed by slots and cavities
in the blocks. RPG Diffusor System introduced their Diffusor-
Blox® in 1990 that goes one step further: load-bearing ability,
Isometric view
16'–0"
6'–8"
Plan view
Enlarged view
mid/high diffusor
F I G U R E 1 4 - 1 5
The large-sized Model DFR-83LMH Diffractal™, a three-way unit covering a wide frequency
range. Fractals are set in the wells of diffractals in this model. Peter D’Antonio, RPG
Diffusor Systems, Inc. and the Audio Engineering Society.
310 CHAPTER FOURTEEN
transmission loss, low-frequency absorption, and sound diffusion.
The DiffusorBlox® system is made up of three distinct blocks, all of
which are nominally 8 × 16 × 12 inches. A typical block is shown in
Fig. 14-18. These concrete blocks are characterized by a surface containing
a partial sequence of varying well depths, separated by
A B
F I G U R E 1 4 - 1 6
Comparison of the diffraction patterns of (A) the hemicylindrical form of the onedimensional
quadratic-residue diffusor and (B) the hemispherical form of the twodimensional
diffusor. Peter D’Antonio, RPG Diffusor Systems, Inc.
Isometric view
Front elevation
1'–115/8"
1'–115/8"
F I G U R E 1 4 - 1 7
The Omniffusor™ a two-dimensional unit which diffuses sound in both the horizontal
and vertical planes for all angles of incidence. Peter D’Antonio, RPG Diffusor Systems, Inc.
THE SCHROEDER DIFFUSOR 311
dividers; an internal 5-sided cavity that can accept a fiberglass
insert; an optional rear half-flange for reinforced construction; and
an optional low-frequency absorbing slot. Diffusorblox® are fabricated
on standard automatic block machines using molds licensed
from RPG Diffusor Systems, Inc. Typical walls constructed of DiffusorBlox
® are illustrated in Fig. 14-19.
Measuring Diffusion Efficiency
A measure of the effectiveness of a diffusor can be obtained by comparing
the intensity in the specular direction with the intensity at
45° of that direction. This can be expressed as:27
Diffusion Coefficient =
The diffusion coefficient is 1.0 for the perfect diffusor. This coefficient
varies with frequency and is commonly expressed in graphical form.
The variation of diffusion coefficient with frequency for several typical
units is shown in Fig. 14-20. For comparison, diffusion from a flat
panel is included as a broken line. These measurements were all made
under reflection-free condition on sample areas of 64 sq ft and using
the time-delay-spectrometry technique.
The number of wells and well widths affect the performance of the
units. The QRD Model 4311® (Fig. 14-6) having the deepest well
I (45°)
 I (specular)
Isometric view
2' – 8"
0' – 115/8"
F I G U R E 1 4 - 1 8
The DiffusorBlox™ concrete block, which offers the good transmission loss of a heavy
wall, absorption via Helmholtz resonator action, and diffusion through quadratic
residue action. The blocks are formed on standard block machines using licensed molds.
Peter D’Antonio, RPG Diffusor Systems, Inc.
312 CHAPTER FOURTEEN
depths and the narrowest well widths (feasible from a manufacturing
standpoint) gives the highest diffusion coefficient over the greatest frequency
range. For comparison, two other units are shown in Fig. 14-
20, the 1925 and the 734, built with the primes 19 and 7 and well
widths of 2.5 and 3.4 inches. The performance of these, while good, is
somewhat inferior to the QRD Model 4311®.
Comparison of Gratings
with Conventional Approaches
Figure 14-21 compares the diffusing properties of the flat panel (a and
b), flat panel with distributed absorption on it (c and d), the monocylinder
(e and f), the bicylinder (g and h), and quadratic residue diffusors
(i and j). The left column is for sound at 0° incidence and the
right column is for sound at 45° incidence. The “fore-and-aft” scale is
diffraction from 90° through 0° to –90°. The horizontal frequency scale
is basically from 1 through 10 kHz. These three-dimensional read-outs
F I G U R E 1 4 - 1 9
Typical wall configurations using Diffusorblox™. Peter D’Antonio, RPG Diffusor Systems, Inc.
THE SCHROEDER DIFFUSOR 313
cover all pertinent variables and provide a wealth of information,
which requires some skill in interpretation. Specific comments Dr.
D’Antonio has made include the following:
 The first six energy-frequency-curves contain artifacts of the
measurement process which should be disregarded because
they are not in the anechoic condition.
 For 0° incidence, the specular properties of the flat panel with
distributed absorption are quite evident by the pronounced
peak at 0° the specular angle.
0
0.2
0.4
0.6
0.8
1.0
Diffusion coefficient
100 300 500 1 kHz 3 kHz 5 kHz 10 kHz
Frequency - Hz
4,311
1,925
734
43
19
7
1.1"
2.5"
3.4"
Rdg
model
Prime Well
width
4,311
1,925
734
Flat panel
F I G U R E 1 4 - 2 0
Comparison of the variation of diffusion coefficient with frequency of three RPG™ diffusors
and a flat panel. Peter D’Antonio, RPG Diffusor Systems, Inc.
314 CHAPTER FOURTEEN
Energy-frequency-directivity plots comparing diffusion from (a,b) a flat panel, (c,d) a flat panel with distributed
absorption, (e,f) monocylinder, (g,h) bicylinder, and (i,j) quadratic-residue diffusor. These plots
compare many of the early attempts at diffusion with modern quadratic-residue diffusors. Peter D'Antonio, RPG
Diffusor Systems, Inc. and Acoustical Society of America, Inc.
0° Incidence 45° Incidence
Energy (dB)
Directivity
12 dB
90°
0°
90°
a b
1 5 10
c d
frequency (kHz)
e f
g h
i j
Flat
panel
Flat panel with
distributed absorption
Monocylinder
Bicylinder
Quadratic
residue
F I G U R E 1 4 - 2 0
THE SCHROEDER DIFFUSOR 315
 The good spatial diffusion of the monocylinder (e) is illustrated
by the relatively constant energy response from 90° to –90°. The
bicylinder (g) shows two closely spaced peaks in the time
response. Although the spatial diffusion looks good, there is
appreciable equal-spaced comb filtering and broadband highfrequency
attenuation. This accounts, in part, for the poor performance
of cylindrical diffusors.
 The quadratic residue diffusors (i and j) maintain good spatial
diffusion even at 45° incidence. The dense notching is uniformly
distributed across the frequency spectrum and the
energy is relatively constant with scattering angle.
Endnotes
1Mackenzie, Robin, Auditorium Acoustics (1975) London, Applied Science Publishers, Ltd.,
see Chapter 17 especially, “New Results and Ideas for Architectural Acoustics’’ by M. R.
Schroeder, p. 206-209.
2Schroeder, M.R., Diffuse Sound Reflection by Maximum-Length Sequences, J. Acous. Soc.
Am., 57, 1 (Jan 1975) 149-150.
3Schroeder, M.R., and R.E. Gerlach, Diffuse Sound Reflection Surfaces, Proc. Ninth
International Congress on Acoustics, Madrid, 1977, paper D-8.
4Schroeder, M.R., Binaural Dissimilarity and Optimum Ceilings for Concert Halls: More
Lateral Sound Diffusion, J. Acous. Soc. Am., 65, 4 (Apr 1979) 958-963.
5Schroeder, M.R., Number Theory in Science and Communication, 2nd enlarged ed., Berlin,
Springer (1986).
6Beiler, Albert H., Recreations in the Theory of Numbers, 2nd. Ed., (1966) New York, Dover
Publications, Inc.
7Davenport, H., The Higher Arithmetic, (1983) New York, Dover Publications, Inc.
8RPG Diffuser Systems, Inc., 651-C Commerce Drive, Upper Marlboro, MD 20772.
Telephone: (301) 249-0044, FAX: (301) 249-3912.
9Strube, Hans Werner, Scattering of a Plane Wave by a Schroeder Diffusor: A Mode-Matching
Approach, J. Acous. Soc, Am., 67, 2 (Feb 1980) 453-459.
10deJong, B.A. and P.M. van den Berg, Theoretical Design of Optimum Planar Sound
Diffusors, J. Acous. Soc. Am., 68, 4 (Oct 1980) 1,154-1,159.
11Berkout, D.W., van Wulfften Palthe, and D. deVries, Theory of Optimal Plane Diffusors, J.
Acous. Soc. Am. 65, 5 (May 1979) 1334-1336.
12D’Antonio, Peter and John H. Konnert, The Reflection Phase Grating: Design Theory and
Application, J. Audio Eng. Soc., 32, 4 (Apr 1984) 228-238.
13D’Antonio, Peter and John H. Konnert, The Schroeder Quadratic-Residue Diffusor: Design
Theory and Application, 74th Audio Eng. Soc. Convention, New York, October 1983,
preprint #1999.
14D’Antonio, Peter and John H. Konnert, The RFZ/RPG Approach to Control Room
Monitoring, 76th Audio Eng. Soc. Convention, New York, October, 1984, preprint #2157.
15D’Antonio, Peter and John H. Konnert, The RPG Reflection-Phase-Grating Acoustical
Diffusor: Applications, 76th Audio Eng. Soc. Convention, New York, October, 1984,
preprint #2156.
16D’Antonio, Peter and John H. Konnert, The RPG Reflection-Phase-Grating Acoustical
Diffusor: Experimental Measurements. 76th Audio Eng. Soc. Convention, New York,
October, 1984, preprint 2158.
17D’Antonio, Peter and John H. Konnert, The Role of Reflection-Phase-Grating Diffusors in
Critical Listening and Performing Environments, 78th Audio Eng. Soc. Convention,
Anaheim, CA, May 1985, preprint #2255.
18D’Antonio, Peter and John H. Konnert, The Acoustical Properties of Sound Diffusing
Surfaces: The Time, Frequency, and Directivity Energy Response, 79th Audio Eng. Soc.
Convention, New York, October, 1985, preprint #2295.
19D’Antonio, Peter and John H. Konnert, Incorporating Reflection-Phase-Grating Diffusors in
Worship Spaces, 81st Audio Eng. Soc. Convention, Los Angeles, November, 1986, preprint
#2364.
20D’Antonio, Peter and John H. Konnert, New Acoustical Materials and Designs Improve
Room Acoustics, 81st Audio Eng. Soc. Convention, Los Angeles, November 1986, preprint
#2365.
21D’Antonio and John H. Konnert, The Reflection-Phase-Grating Acoustical Diffusor:
Application in Critical Listening and Performing Environments, Proc. 12th International
Congress on Acoustics.
22D’Antonio, Peter, The Reflection-Phase-Grating Acoustical Diffusor: Diffuse It or Lose It, dB
The Sound Engineering Magazine, 19, 5 (Sept/Oct 1985) 46-49.
23D’Antonio, Peter and John H. Konnert, Advanced Acoustic Design of Stereo Broadcast and
Recording Facilities, 1986 NAB Eng. Conference Proc., p. 215-223.
24D’Antonio, Peter and John H. Konnert, New Acoustical Materials Improve Broadcast
Facility Design, 1987 NAB Eng. Conference Proc., p. 399-406.
25D’Antonio, Peter, Control-Room Design Incorporating RFZ, LFD, and RPG Diffusors, dB
The Sound Engineering Magazine, 20, 5 (Sept/Oct 1986) 47-55. Contains bibliography with
40 entries.
26D’Antonio, Peter and John Konnert, The QRD Diffractal: A New One- or Two-Dimensional
Fractal Sound Diffusor, J. Audio Eng. Soc., 40, 3 (Mar 1992), 117-129.
27D’Antonio, Peter and John Konnert, The Directional Scattering Coefficient: Experimental
Determination, J. Audio Eng. Soc., 40, 12 (Dec 1992), 997-1017.
Hermann Von Helmholtz (1821-1894) performed some interesting
acoustical experiments with resonators. His resonators were a
series of metal spheres of graded sizes, each fitted with a neck, appearing
somewhat like the round-bottom flask found in the chemistry laboratory.
In addition to the neck there was another small opening to
which he applied his ear. The resonators of different sizes resonated at
different frequencies, and by pointing the neck toward the sound
under investigation he could estimate the energy at each frequency by
the loudness of the sound of the different resonators.
There were numerous applications of this principle long before the
time of Helmholtz. There is evidence that bronze jars were used by the
Greeks in their open-air theaters, possibly to provide some artificial
reverberation. A thousand years ago Helmholtz-type resonators were
embedded in church walls in Sweden and Denmark with the mouths
flush with the wall surface, apparently for sound absorption.1 The
walls of the modern sanctuary of Tapiola Church in Helsinki, Finland,
are dotted with slits in the concrete blocks2 (Fig. 15-1). These are resonator
“necks” that open into cavities behind, together forming resonating
structures. Energy absorbed from sound in the room causes
each resonator to vibrate at its own characteristic frequency. Part of the
energy is absorbed, part reradiated. The energy reradiated is sent in
every direction, contributing to the diffusion of sound in the room.
Modal Resonances in
Enclosed Spaces
15
317
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
318 CHAPTER FIFTEEN
The resonator principle, old as it is, continually appears in modern,
up-to-the-minute applications.
Resonance in a Pipe
The two ends of the pipe of Fig. 15-2 can be likened to two opposing
walls of a listening room or recording studio. The pipe gives us a simple,
one-dimensional example to work with. That is, what happens
between opposite walls of a rectangular room can be examined without
being bothered by the reflections from the other four surfaces. The
pipe, closed at both ends and filled with air, is a resonator capable of
vibrating at its characteristic frequencies when excited in some way.
Air inside an organ pipe can be set to vibrating by blowing a stream of
air across a lip at the edge of the pipe. It is simpler to place a small
loudspeaker inside the pipe. A sine wave signal is fed to the loudspeaker
and varied in frequency. A small hole drilled in the pipe in the
F I G U R E 1 5 - 1
Helmholtz-type resonators built into the wall of the Tapiola Church in Helsinki, Finland.
The slots and cavities behind them act as both absorbers and diffusors of sound.
MODAL RESONANCES IN ENCLOSED SPACES 319
end opposite the loudspeaker makes possible hearing the low-level
tones radiated by the loudspeaker. As the frequency is increased, nothing
unusual is noted until the frequency radiated from the loudspeaker
coincides with the natural frequency of the pipe. At this frequency, f1,
modest energy from the loudspeaker is strongly reinforced and a relatively
loud sound is heard at the ear hole. As the frequency is
increased, the loudness is again low until a frequency of 2f1 is reached,
at which point another strong reinforcement is noted. Such resonant
peaks can also be detected at 3f1, 4f1 . . . etc.
Now let us assume that means of measuring and recording the
sound pressure all along the pipe are available. In Fig. 15-2 the
graphs below the sketch of the pipe show how the sound pressure
varies along the length of the pipe for different excitation frequencies.
A sound wave traveling to the right is reflected from the right
plug and a sound wave traveling to the left from the left plug. The
left-going waves react with the right-going waves to create, by superposition,
a standing wave at the natural
frequency of the pipe or one of its multiples.
Measuring probes inserted through
tiny holes along the pipe could actually
measure the high pressure near the
closed ends and zero at the center, etc.
Similar nodes (zero points) and antinodes
(maxima) can be observed at 2f1, 3f1,
4f1 . . . etc., as shown in Fig. 15-2. The
dimensions of a studio or listening room
determine its characteristic frequencies
much as though there were a north-south
pipe, an east-west pipe, and a vertical
pipe, the pipes corresponding to the
length, width, and height of the room,
respectively.
Bathroom Acoustics
Why is it that singing in the shower or tub is
such a satisfying experience (to the singer,
at least)? Because here one’s voice sounds
L (ft)
Closed Closed
Lowest frequency
Second multiple
Third multiple
0
0
0
f1 = 1,130
2L
f2 = 2f1
f3 = 3f1
Sound pressure
F I G U R E 1 5 - 2
A pipe closed at both ends helps us to understand
how resonance occurs between two opposing walls of
a listening room or studio. The distance between the
walls determines the characteristic frequency of resonance.
320 CHAPTER FIFTEEN
richer, fuller, and more powerful than anywhere else! The case of the
bathroom baritone clearly illustrates the effect of resonance in a small
room and the resulting reinforcement of sound at certain frequencies
related to the dimensions of the room. Exciting the air in the bathroom at
frequencies far removed from these characteristic frequencies results in
weaker sounds, except at multiples of these frequencies, where the effect
can be very much like that at the lowest natural frequencies.
The person singing in the bathroom is, in a sense, inside a Helmholtz
resonator or an immense organ pipe, but with one important difference;
it is now a three rather than an essentially one-dimensional system like
the pipe. The hard walls of the bathroom are highly reflective. There is
a characteristic modal frequency of resonance associated with the
length, another with the width, and still another with the height of the
bathroom. In the case of the cubical bathroom, all three modal frequencies
coincide to give a mighty reinforcement to the baritone’s voice at
the basic characteristic modal frequency and multiples of it.
Reflections Indoors
Anyone can appreciate the difference between sound conditions
indoors and sound outdoors. Outdoors the only reflecting plane may
be the earth’s surface. If that surface happens to be covered with a foot
of snow, which is an excellent absorber of sound, it may be difficult to
carry on a conversation with someone 20 feet away. Indoors the sound
energy is contained, resulting in a louder sound with a given effort. A
speaker can be heard and understood by hundreds of people with no
reinforcement but that of reflecting surfaces.
Consider sound reflections from a single wall. In Fig. 15-3, a point
source of sound, S, is a given distance from a massive wall. The spherical
wave fronts (solid lines, traveling to the right) are reflected from
this surface (broken lines). Physicists working in various forms of radiation
(light, radio waves, sound) resort to the concept of images
because it makes their mathematical studies much easier. In Fig. 15-3,
the reflections from the surface traveling to the left act exactly as
though they were radiated from another identical point source, S1, an
equal distance from the reflecting surface but on the opposite side.
This is the simplest image case of one source, one image, and a reflecting
surface, all in free space.
MODAL RESONANCES IN ENCLOSED SPACES 321
The isolated reflecting surface of Fig. 15-3 is now brought down to
earth and made into the north wall of a rectangular room as in Fig. 15-4.
Source S still has its image S1 in what is now the north wall of the room.
The source also has other images. S2 is the image in the east wall reflecting
surface, S3 is the image of S in the west wall, and S4 is the image of
S in the south wall. Use your imagination to visualize S5, the image in
the floor, and S6, the image in the ceiling. All of these six images are
assumed to be pulsating just like S and sending sound energy back into
the room. The farther the images are from the reflecting plane, the
weaker will be their contribution at a given point, P, in the room, but
they all make their contribution.
There are images of the images as well. The S1 image has its image in
the south wall at S1’, the image of the S2 image in the west wall at S2’ and
similarly, images S3, S4, S5, and S6 appear at S3’, S4’, S5’, and S6’ (some off
the page). And, then, there are the images of the images of the images,
and so on ad infinitum. The more remote images are so weak that they
can be neglected for the sake of simplicity. Going further in the discussion
of the image is beyond the scope of this book. We have discussed
S S1
Source Image
F I G U R E 1 5 - 3
Sound radiated by a point S is reflected by the rigid wall. The reflected wave can be considered
as coming from S1, an image of S.
322 CHAPTER FIFTEEN
them to this extent only to show how to
visualize how the sound field at some point
P in a room is built up from the direct
sound from the source S plus the vector
sum of the contributions of all the images of
S. This is just another way of saying that the
sound at P is built up from the direct sound
from S plus single or multiple reflections
from all six surfaces.
Two-Wall Resonance
Figure 15-5 shows two parallel, reflective
walls of infinite extent. When a loudspeaker
radiating pink noise excites the space
between the walls, the wall-air-wall system
exhibits a resonance at a frequency of fo =
1,130/2L or 565/L, when L = the distance in
feet between the two walls and 1, 130 the
speed of sound in feet per second. A similar resonance occurs at 2fo, 3fo,
4fo . . . , etc. up through the spectrum. The fundamental frequency fo is
considered a natural frequency of the space between the reflective walls,
and it is accompanied by a train of modes each of which also exhibits
resonance. Other names that have been applied to such resonances are
eigentones (obsolete), room resonances, permissible frequencies, natural
frequencies, or just plain modes, which is preferred to make a studio or
listening room. In adding two more mutually perpendicular pairs of
walls, to form a rectangular enclosure, we also add two more resonance
systems, each with its own fundamental and modal series.
Actually, the situation is far more complicated. So far only axial
modes have been discussed, of which each rectangular room has three,
plus modal trains for each. Each axial mode involves only two opposite
and parallel surfaces. Tangential modes, on the other hand, involve
four surfaces. Oblique modes involve all six surfaces of the room.
Waves vs. Rays
What the diagrams in Fig. 15-6 offer in terms of clarity, they lack in
rigor. In these diagrams, rays of sound are pictured as obeying the law:
North
S´2
S3
S´1 S4
S1 S
S2
P
F I G U R E 1 5 - 4
The surface of Fig. 15-3 is now made into the north
wall of an enclosure. The source S now has six primary
images, one in each of the six surfaces of the enclosure.
Images of the images result in a theoretically
infinite number of images of S. Sound intensity at
point P is made up of the direct sound from S plus
the contributions from all images.
MODAL RESONANCES IN ENCLOSED SPACES 323
The angle of incidence equals the angle of
reflection. For higher audio frequencies,
the ray concept is quite fruitful. When the
size of the enclosure becomes comparable
to the wavelength of the sound in it, however,
special problems arise and the ray
approach collapses. For example, a studio
30 ft long is only 1.3 wavelengths long at
50 Hz. Rays lose all meaning in such a
case. Physicists employ the wave
approach to study the behavior of sound of
longer wavelengths.
Frequency Regions
The audible spectrum is very wide when
viewed in terms of wavelength. At 16 Hz,
considered the low-frequency limit of the
average human ear, the wavelength is
1130/16 = 70.6 ft. At the upper extreme of
hearing, say 20,000 Hz, the wavelength is
only 1130/20,000 = 0.0565 ft or about 11⁄16
of an inch. The behavior of sound is
affected greatly by the wavelength of the
sound in comparison to the size of objects
encountered. In a room, sound of 11⁄16-in
wavelength is scattered (diffused) significantly
by a wall irregularity of a few
inches. The effect of the same irregularity
on sound of 70 ft wavelength would be
insignificantly small. The heart of the acoustical problem in the usual
audio room is that no single analytical approach can cover sound of
such a wide range of wavelengths.
In considering the acoustics of small rooms, the audible spectrum
is divided arbitrarily into four regions, A, B, C, and D (Fig. 15-7).
Region A is the very low frequency region below a frequency of
1130/2L or 565/L where L is the longest dimension of the room. Below
the frequency of this lowest axial mode there is no resonant support
Length L-ft
Second mode
First
mode
Third mode
3f0
Sound pressure
2f0
f0 = 1,130
2L
F I G U R E 1 5 - 5
The air between two parallel, reflective walls can be
considered a resonant system with a frequency of
resonance of f0 = 1,130/2L. This system is also resonant
at integral multiples of f0.
324 CHAPTER FIFTEEN
for sound in the room. This does not mean that such very low frequency
sound cannot exist in the room, only that it is not boosted by
room resonances because there are none in that region.
Region B is that region we have studied in detail in which the
dimensions of the room are comparable to the wavelength of sound
being considered. It is bounded on the low-frequency end by the lowest
axial mode, 565/L. The upper boundary is not definite but an
approximation is given by what has been called the cutoff or crossover
frequency given by the equation:
F2  11,250  (15-1) (RT60)V
Axial Tangential Oblique
F I G U R E 1 5 - 6
Visualization of axial, tangential, and oblique room modes using the ray concept.
Range of audible sound
16 Hz 20 kHz
A B C D
F1 = 565
L
F2 11,250 4F2
(RT60)
V ~ F3 ~ 
F I G U R E 1 5 - 7
When dealing with the acoustics of enclosed spaces, it is convenient to consider the audible frequency range
as composed of four regions: A, B, C, and D demarked by the frequencies F1, F2, and F3. In region D, specular
reflections and ray acoustics prevail. In region B, room modes dominate. Region C is a transition zone in which
diffraction and diffusion dominate. There is no modal boost for sound in region A.
MODAL RESONANCES IN ENCLOSED SPACES 325
where
F2 = cutoff or crossover frequency, Hz.
RT60 = reverberation time of the room, seconds.
V = volume of the room, cu ft.
A 10 × 16 × 23.3 ft room has a volume of 3,728 cu ft. For a reverberation
time of 0.5 second the F2 crossover frequency is 130 Hz. In Fig. 15-7,
F1 = 565/23.3 = 24.2 Hz, and F3  4F2(4)(130)520 Hz.
Region D covers the higher audible frequencies for which the wave
lengths are short enough for geometric acoustics to apply. Specular
reflections (angle of incidence equals angle of reflection) and the
sound ray approach to acoustics prevail. In this region statistical
approaches are generally possible.
Retracing our steps, region C is a transition region between
region B, in which wave acoustics must be used, and region D in
which ray acoustics are valid. It is bounded on the low-frequency
end approximately by the cutoff frequency F2 of Eq. 15-1 and on the
high end approximately by 4F2. It is a difficult region dominated by
wavelengths often too long for ray acoustics and too short for wave
acoustics.
For the 10 × 16 × 23.3 ft room, below 565/23.3 = 24.2 Hz is region A
in which there is no resonant boost for sound. Between 24.2 Hz and
130 Hz (Eq. 15-1) the wave acoustical approach of modal resonances is
essential. Between 130 Hz and (4) (130) = 520 Hz is the transition
region C. Above about 520 Hz the modal density is very high, statistical
conditions generally prevail, and the simpler geometrical acoustics
can be used. Room size determines how the audible spectrum must be
divided for acoustical analysis. Very small rooms, with too few modal
resonances spaced too far apart, are characterized by domination of a
great stretch of the audible spectrum by modal resonances. This is the
“small studio problem” in a nutshell.
Dividing the Audio Spectrum
The very wide frequency range of the ear forces us to apply different
approaches for different frequency ranges in studying the small-room
sound field. An appreciation of this fact is necessary before these different
approaches are presented. The audible spectrum in Fig. 15-7 is
326 CHAPTER FIFTEEN
divided into four parts. At the high-frequency extreme is the D region, in
which the use of sound rays is applicable. In this region, rays of sound
bounce around like billiard balls, following the general rule that the angle
of incidence equals the angle of reflection. In this region, the wavelength
of the sound is very small compared to the dimensions of the room.
Next we slide down the audible spectrum to the range of frequencies
labeled B. In this region, the wavelength is comparable to the
room dimensions. Here the ray concept is without meaning, and the
concept of wave acoustics must be applied.
In between frequency regions D and B of Fig. 15-7 is region C, which
is transitional in nature. The wavelength of sound in this range of frequencies
does not fit either the wave or ray approach; it is too short for
wave acoustics, too long for the application of the ray approach. It is a
region in which diffusion and diffraction of sound dominate.
Region A of Fig. 15-7 is a sort of acoustical never-never land. We
are still indoors, there are reflecting surfaces all about us, yet sound
does not behave like it does outdoors. Nor does it behave like sound
indoors at somewhat higher frequencies. When the wavelength of
the sound is of the same order of magnitude as the dimensions of the
room, as in region B, the modal resonances increase the loudness of
the sound. The upper frequency of region A, however, is well
marked by the axial mode of lowest frequency as determined by the
largest dimension of the room. Below that frequency there is no resonant
boost of modes. The sound of the very low frequencies of
region A is not of outdoor character because it is contained by reflections
from the walls, ceiling, and floor, but without resonance boost.
The low-frequency response of this room is low in region A, is
boosted by room resonances in region B, and is subject to the
vagaries of diffraction and absorption in regions C and D.
Wave Acoustics
At times like this, it is very difficult to keep the treatments in this
book nonmathematical. If we could dispense with such restraints, the
first order of business would be to write down the wave equation. As
a mere glance at this partial differential equation in three dimensions
might strike terror into the hearts of many, such a temptation is
MODAL RESONANCES IN ENCLOSED SPACES 327
resisted. Instead, attention is drawn to one of its solutions for sound
in rectangular enclosures. The geometry used is that of Fig. 15-8,
which judiciously fits the familiar, mutually perpendicular x, y, z
coordinates of three-dimensional space to our studio or listening
room. To satisfy the need for order, the longest dimension L is placed
on the x axis, the next longest dimension W (for width) is placed on
the y axis and the smallest dimension H (for height) on the z axis. The
goal is to be able to calculate the permissible frequencies corresponding
to the modes of a rectangular enclosure. Skipping the mathematics,
we come directly to the answer from the equation stated by
Rayleigh in 1869:
Frequency = + +  (15-2)
where
c = speed of sound, 1,130 ft/sec
L,W,H = room length, width, and height, ft.
p,q,r = integers 0, 1,2,3 . . ., etc.
You might flinch a little even at Eq. 15-2. Although it is not difficult,
it can become a bit messy and tedious. The importance of this
equation is that it gives the frequency of every axial, tangential, and
oblique mode of a rectangular room.
The integers p, q, and r are the only
variables once L, W, and H are set for a
given room. These integers not only provide
the key to the frequency of a given
mode, but also serve to identify the mode
as axial, tangential, or oblique. If p = 1,
q = 0, and r = 0 (shorthand, the 1,0,0
mode), the width and height terms drop
out and Eq. 15-1 becomes:
Frequency = = = =
This is the axial mode corresponding to
the length of the room. The width axial
mode (0,1,0) and the height axial mode
565
L
1,130
2
L
c2
L
p2
L
2
c2
r 2
H
2
q2
W
2
p2
L
2
c2
y
z
x
W
L
H
F I G U R E 1 5 - 8
Orientation of rectangular room of length L, width W,
and height H with respect to the x, y, and z coordinates
for calculating room modal frequencies.
328 CHAPTER FIFTEEN
(0,0,1) are calculated similarly by substituting the appropriate dimension.
If two of the integers are zero, an axial mode frequency is identified
because only one pair of surfaces is involved. In a similar way one
zero identifies tangential modes, and no zero at all specifies oblique
modes.
Mode Calculations—An Example
The utility of Eq. 15-2 is best demonstrated by an example. The room
is small, but convenient for experimental verification. The dimensions
of this room are: length L = 12.46 ft, width W = 11.42 ft, and average
height H = 7.90 ft (the ceiling actually slopes along the length of the
room with a height of 7.13 ft on one end and 8.67 ft on the other).
These values of L, W, and H have been inserted in Eq. 15-2 along with
an assortment of combinations of integers p, q, and r.
Room modes are possible only when p, q, and r are whole numbers
(or zero) because this is the condition that creates a standing
wave. There are many combinations of integers when fundamentals
(associated with 1), second modes (associated with 2), and third
modes (associated with 3), etc., are introduced. Table 15-1 lists some
of the various combinations of p, q, and r and the resulting permissible
modal frequency for each combination. Furthermore, each frequency
is identified as axial, tangential, or oblique by the number of
zeros in that particular p, q, r combination. The lowest natural room
frequency is 45.3 Hz, which is the axial mode, associated with the
longest dimension, the length L, of the room below which there is no
modal boost of the sound. In this case, p = 1, q = 0, and r = 0. Mode
7, the 2,0,0 mode, yields a frequency of 90.7 Hz, which is the second
axial mode associated with the length L. In the same manner, mode
18, with integers 3,0,0 is the third mode of the length axial mode,
and mode 34 the fourth mode. However, there are many tangential
and oblique modes between these, and Table 15-1 provides the
means for carefully identifying all modes and raises the possibility
of studying their relationships.
Axial modes have been emphasized in studio design, and the reason
for this is given later. The message of Eq. 15-2 and Table 15-1 is
that there is much more to room acoustics than axial modes and their
spacing. In between the axial modal frequencies are many other modal
frequencies that have an effect, even though weaker.
MODAL RESONANCES IN ENCLOSED SPACES 329
Table 15-1. Mode calculations. Room dimensions: 12.46 × 11.42 × 7.90 ft.
Mode Integers Mode
number p q r frequency, Hz Axial Tangential Oblique
1 1 0 0 45.3 x
2 0 1 0 49.5 x
3 1 1 0 67.1 x
4 0 0 1 71.5 x
5 1 0 1 84.7 x
6 0 1 1 87.0 x
7 2 0 0 90.7 x
8 2 0 1 90.7 x
9 1 1 1 98.1 x
10 0 2 0 98.9 x
11 2 1 0 103.3 x
12 1 2 0 108.8 x
13 0 2 1 122.1 x
14 0 1 2 122.1 x
15 2 1 1 125.6 x
16 1 2 1 130.2 x
17 2 2 0 134.2 x
18 3 0 0 136.0 x
19 0 0 2 143.0 x
20 3 1 0 144.8 x
21 0 3 0 148.4 x
22 2 2 1 152.1 x
23 3 0 1 153.7 x
24 1 1 2 158.0 x
25 3 1 1 161.5 x
26 0 3 1 164.8 x
27 3 2 0 168.2 x
28 2 0 2 169.4 x
29 1 3 1 170.9 x
30 0 2 2 173.9 x
31 2 3 0 173.9 x
32 2 1 2 176.4 x
33 1 2 2 179.7 x
34 4 0 0 181.4 x
35 3 2 1 182.8 x
330 CHAPTER FIFTEEN
Table 15-1. Continued.
Mode Integers Mode
number p q r frequency, Hz Axial Tangential Oblique
36 2 3 1 188.1 x
37 2 2 2 196.2 x
38 0 4 0 197.9 x
39 3 0 2 197.9 x
40 3 3 0 201.3 x
41 3 1 2 203.5 x
42 0 3 2 206.1 x
43 1 3 2 211.1 x
44 0 0 3 214.6 x
45 1 0 3 219.3 x
46 0 1 3 220.2 x
47 3 2 2 220.8 x
48 1 1 3 224.8 x
49 2 3 2 225.2 x
50 2 0 3 232.9 x
51 4 3 0 234.4 x
52 0 2 3 236.3 x
53 2 1 3 238.1 x
54 3 4 0 240.2 x
55 1 2 3 240.6 x
56 3 3 2 247.0 x
57 2 2 3 253.1 x
58 3 0 3 254.0 x
59 0 3 3 260.9 x
60 3 2 3 272.6 x
61 2 3 3 276.2 x
62 4 0 3 281.0 x
63 0 0 4 286.1 x
64 0 4 3 291.1 x
65 3 0 4 316.8 x
66 0 3 4 322.3 x
MODAL RESONANCES IN ENCLOSED SPACES 331
Experimental Verification
All of the modal frequencies listed in Table 15-1 make up the acoustics of
this particular room for the frequency range specified. To evaluate their
relative effects, a swept-sine-wave transmission experiment was set up. In
effect, this measures the frequency response of the room. Knowing that all
room modes terminate in the corners of a room, a loudspeaker was placed
in one low tricorner and a measuring microphone in the diagonal high tricorner
of the room. The loudspeaker was then energized by a slowly
swept sine-wave signal. The room response to this signal, picked up by
the microphone, was recorded on a graphic level recorder having a paper
speed of 3 mm/second. This resulted in a linear sweep from 50 to 250 Hz
in 38 seconds. The resulting trace is shown in Fig. 15-9.
In the past, attempts have been made to identify the effects of each
mode in reverberation rooms with all six surfaces hard and reflective.
In such cases, the prominent modes stand out as sharp spikes on the
recording. The test room in which the recording of Fig. 15-9 was made
is a spare bedroom, not a reverberation chamber. Instead of concrete,
the walls are of frame construction covered with gypsum board (drywall);
carpet over plywood makes up the floor, closet doors almost
cover one wall. There is a large window, pictures on the wall, and
some furniture, including a couch. It is evident that this is a fairly
absorbent room. The reverberation time at 125 Hz (as we will consider
more fully later) was found to be 0.33 second. This room is much
closer acoustically to studios and control rooms than to reverberation
chambers, and this is why it was chosen.
Mode Identification
A careful study of Fig. 15-9 in an attempt to tie the peaks and valleys
of the transmission run to specific axial, tangential, and oblique modes
is rather disappointing. For one thing, the loudspeaker (JBL 2135)
response is included, although it is quite smooth over this frequency
region. The signal generator and power amplifier are very flat. Most of
the ups and downs must be attributed to modes and the interaction of
modes. Modes close together would be expected to boost room
response if in phase, but cancel if out of phase. There are 11 axial
modes, 26 tangential modes, and 21 oblique modes in this 45.3- to
332 CHAPTER FIFTEEN
254.0-Hz record, and the best we can say is that the transmission trace
shown in Fig. 15-9 is the composite effect of all 58 modes.
The three major dips are so narrow that they would take little energy
from a distributed speech or music spectrum. If they are neglected, the
remaining fluctuations are more modest. Fluctuations of this magnitude
in such steady-state swept-sine transmission tests are characteristic of
even the most carefully designed and most pleasing of studios, control
rooms, and listening rooms. The ear commonly accepts such deviations
from flatness of response. The modal structure of a space always gives
rise to these fluctuations. They are normally neglected because attention
is focused on the response of microphone, amplifier, loudspeaker, and
50 100 150 200 250
10 dB
BRUEL & KJAER ¨
Frequency - Hz
Oblique
Tangential
Axial
Relative sound pressure level - dB
F I G U R E 1 5 - 9
Swept sine-wave transmission run in the test room of Table 15-1. The location of each
axial, tangential, and oblique modal frequency is indicated.
MODAL RESONANCES IN ENCLOSED SPACES 333
other reproducing equipment. The response of the ear and of the room
have yet to receive the attention they deserve.
Mode Decay
The steady-state response of Fig. 15-9 tells only part of the story. The
ear is very sensitive to transient effects, and speech and music are
made up almost entirely of transients. Reverberation decay is one transient
phenomenon that is easily measured. When broadband sound
such as speech or music excites the modes of a room, our interest naturally
centers on the decay of the modes. The 58 modes in the 45.3- to
254.0-Hz band of Fig. 15-9 are the microstructure of the reverberation
of the room. Reverberation is commonly measured in octave bands.
Octave bandwidths of interest are shown in Table 15-2.
Each reverberatory decay by octave bands thus involves an average
of the decay of many modes, but it takes an understanding of the decay
of the individual modes to explain completely the decay of the octave
band of sound. The higher the octave center frequency, the greater the
number of modes included.
All modes do not decay at the same rate. Mode decay depends,
among other things, on the way absorbing material is distributed
in the room. Carpet on the floor of the test room has no effect on
the 1,0,0 or 0,1,0 axial modes involving only walls. Tangential and
oblique modes, which involve more surfaces, would be expected
to die away faster than axial modes that involve only two surfaces.
On the other hand, absorption is greater for the axial modes in
which the sound impinges on the surfaces at right angles than for
Table 15-2. Modes in octave bands.
Limits Modes
(–3 dB
Octave points) Axial Tangential Oblique
63 Hz 45– 89 Hz 3 3 0
125 Hz 89–177 Hz 5 13 7
250 Hz 177–254 Hz 3* 10* 14*
*Partial octave.
334 CHAPTER FIFTEEN
the low angle of incidence common for tangential and oblique
modes.
The actual reverberation decays in the test room using sine-wave
excitation are shown in Figs. 15-10 and 15-11. The measured reverberation
times vary as shown in Table 15-3. The measured reverberation
time varies almost two to one for the selected frequencies in the 180 to
240-Hz range. The decay of the 125-Hz octave band of noise (0.33 sec)
and the 250-Hz octave band (0.37 sec) of Fig. 15-11 averages many
modes and should be considered more or less the “true” values for this
frequency region, although normally many decays for each band
would be taken to provide statistical significance.
The dual slope decay at 240 Hz in Fig. 15-10 is especially interesting
because the low value of the early slope (0.31 sec) is probably
dominated by a single mode involving much absorption, later giving
way to other modes that encounter much less absorption. Actually
identifying the modes from Table 15-1 is difficult, although you
might expect mode 44 to die away slowly and the group of three
near 220 Hz (45, 46, 47) to be highly damped. It is common to force
nearby modes into oscillation, which then decay at their natural frequency.
Mode Bandwidth
Normal modes are part and parcel of room resonances. Taken separately,
each normal mode exhibits a resonance curve such as shown in
Table 15-3. Measured reverberation time of test room.
Average reverberation
Frequency Hz time, seconds
180 0.38
200 0.48
210 0.53
220 0.55
240 0.31 and 0.53 (double slope)
125 Hz octave noise 0.33
250 Hz octave noise 0.37
MODAL RESONANCES IN ENCLOSED SPACES 335
10 dB
180 Hz
RT60 = 0.38 sec
220 Hz
RT60 = 0.55 sec
200 Hz
RT60 = 0.48 sec
210 Hz
RT60 = 0.53 sec
RT60 = 0.31 sec
RT60 = 0.53 sec
240 Hz
F I G U R E 1 5 - 1 0
Pure-tone reverbation-decay recordings made in a test room. Single modes decaying
alone yield smooth, logarithmic traces. Beats between neighboring modes cause the
irregular decay. The two-slope pattern bottom right reveals the smooth decay of a single
prominent mode for the first 20 dB, after which one or more lightly absorbed modes
dominate.
336 CHAPTER FIFTEEN
10 dB
80 Hz
RT 60 = 0.32 sec
125 Hz
Octave
RT 60 = 0.33 sec
250 Hz
Octave
RT 60 = 0.37 sec
90 Hz
RT 60 = 0.29 sec
100 Hz
RT 60 = 0.31 sec
F I G U R E 1 5 - 1 1
The top three pure-tone reverbation decays are dominated by single prominent modes
as indicated by the smooth adherence to logarithmic form. The two lower records are of
octave bands of pink noise, which give the average decay of all modes in those octaves.
MODAL RESONANCES IN ENCLOSED SPACES 337
Fig. 15-12. Each mode, therefore, has a definite bandwidth determined
by the simple expression:
Bandwidth = f2–f1 = (15-3)
RT60 = reverberation time, seconds
Bandwidth is inversely proportional to the reverberation time. In
electrical circuits, the sharpness of the tuning curve depends on the
amount of resistance in the circuit; the greater the resistance, the
broader the tuning curve. In room acoustics, the reverberation time
depends on absorption (resistance). The analogy is fitting (the more
absorption, the shorter the reverberation time, and the wider the mode
resonance). For convenient reference, Table 15-4 lists a few values of
bandwidth in relation to reverberation time.
The mode bandwidth for a typical studio is in the general region of
5 Hz. This means that adjacent modes tend
to overlap for rooms with short reverberation
time, which is desirable. As the skirts
of the resonance curves of adjacent modes
A and B overlap, energizing mode A by driving
the room at frequency A will also tend
to force mode B into excitation. When the
tone of frequency A is removed, the energy
stored in B decays at its own frequency B.
The two will beat with each other during
the decay. The very uniform decays, such
as the top three in Fig. 15-11, are probably
single modes sufficiently removed from
neighbors so as to act independently without
irregularities caused by beats.
Figure 15-13 shows an expanded version
of the 40- to 100-Hz portion of Fig.
15-9. The axial modes from Table 15-1
falling within this range are plotted with
bandwidth of 6 Hz at the –3dB points.
The axial modes (Fig. 15-13A) are peaked
at zero reference level. The tangential
modes have only one-half the energy of
2.2

RT60
3 dB
Frequency
Sound pressure, dB
f1 f2
F I G U R E 1 5 - 1 2
Each mode has a finite bandwidth. The more
absorbent the room, the greater the bandwidth. As
measured at the customary –3-dB points, the modal
bandwidth of a recording studio is near 5 Hz.
338 CHAPTER FIFTEEN
the axial modes,3 which justifies plotting their
peaks 3 dB (10 log 0.5) below the axial modes in
Fig. 15-13B. The oblique modes have only onefourth
the energy of the axial modes, hence the
lone oblique mode at 98.1 Hz that falls within this
range is plotted 6 dB (10 log 0.25) below the axial
mode peak in Fig. 15-13C.
The response of the test room, Fig. 15-13D is
most certainly made up of the collective contributions
of the various modes tabulated in Table 15-1.
Can Fig. 15-13D be accounted for by the collective
contributions of axial modes A, tangential modes B,
and the single oblique mode C? It seems reasonable to account for the
12-dB peak in the room response between 80 and 100 Hz by the combined
effect of the two axial, three tangential, and one oblique mode in
that frequency range. The fall off below 50 Hz is undoubtedly due to
Table 15-4. Mode bandwidth.
Reverberation Mode bandwidth
time, seconds Hz
0.2 11
0.3 7
0.4 5.5
0.5 4.4
0.8 2.7
1.0 2.2
Relative sound pressure level, dB
10
5
0
0
5
10
5
10
15
40 50 60 70 80 90 100
D
A
B
C
Frequency - hertz
F I G U R E 1 5 - 1 3
Attempted correlation of calculated modes and the measured swept-sine response of
the test room over the frequency range 40 Hz to 100 Hz. (A) Axial modes. (B) Tangential
modes. (C) Oblique mode. (D) Reproduction of the 40- to 100-Hz portion of room
response of Fig. 15-9.
MODAL RESONANCES IN ENCLOSED SPACES 339
loudspeaker response. This leaves the 12-dB dip at 74 Hz yet to be
accounted for.
Examining the axial mode at 71.5 Hz, it can be seen that this dip is
the vertical mode of the test room working against a sloping ceiling. The
frequency corresponding to the average height is 71.5 Hz, but the one
corresponding to the height at the low end of the ceiling is 79.3 Hz, and
for the high end it is 65.2 Hz. The uncertainty of the frequency of this
mode is indicated by arrows in Fig. 15-13. If this uncertain axial mode
were shifted to a slightly lower frequency, the 12-dB dip in response
could be better explained. It would seem that a dip in response should
appear near 60 Hz, but none was found.
Rather than being an absolute experimental verification of theory,
this test-room experiment was conducted only to explain the basic features
of the theory. Conditions and techniques lacked the necessary
rigor to produce exact results. The test room is not a rectangular parallelepiped,
which is the basis of Eq. 15-1. Neither is the loudspeaker
response known accurately. In addition, there is the overriding fact
that in combining the effects of the modes of Fig. 15-13 A, B, and C to
get Fig. 15-13 D, phase must be taken into account. These components
must be combined vectorially with both magnitude and phase fully
considered. The main purpose of Fig. 15-13 is to emphasize that room
response is made up of combined modal responses.
Mode Pressure Plots
It is easy to say that the modal pattern of a given room creates a very
complex sound field, but to really drive this point home several
sketches of sound pressure distributions are included. The one dimension
organ pipe of Fig. 15-2 is a starting point that can be compared to
the 1,0,0 mode of Fig. 15-14 for a three-dimensional room. The pressure
is higher near the ends (1.0) and zero along the center of the room.
Figure 15-15 shows sound pressure distribution when only the 3,0,0
axial mode is energized. The sound pressure nodes and anti-nodes in
this case are straight lines as shown in Fig. 15-16.
Three-dimensional sketches of sound pressure distribution
throughout a room become difficult, but Fig. 15-17 is an attempt for
the 2,1,0 tangential mode. We see sound pressure “piled up” in each
corner of the room with two more “piles” at the center edges. This is
340 CHAPTER FIFTEEN
1, 0, 0 Mode
0
1.0
1.0
F I G U R E 1 5 - 1 4
A graphical representation of the sound pressure distribution of the 1,0,0 axial mode of
a room. The sound pressure is zero in the vertical plane at the center of the room and maximum
at the ends of the room. This is comparable to f1 of the organ pipe of Fig. 15-2.
1.0
1.0 1.0
1.0
0 0 0
3, 0, 0 Mode
F I G U R E 1 5 - 1 5
Representation of the sound pressure distribution of the 3,0,0 axial mode of a room.
MODAL RESONANCES IN ENCLOSED SPACES 341
more graphically portrayed in Fig. 15-18 in which the pressure contour
lines are drawn. The broken lines crisscrossing the room between
the “piles” of sound mark the zero pressure regions.
Imagine how complicated the sound pressure pattern would be if all
the modes were concurrently or sequentially excited by voice or music
energy chasing up and down the spectrum while constantly shifting in
intensity. The plot of Fig. 15-18 shows pressure maxima in the corners of
the room. These maxima always appear in room corners for all modes. To
excite all modes, place the sound source in a corner. Conversely, if you
wish to measure all modes, a corner is the place to locate the microphone.
Modal Density
Modal density increases with frequency. In Table 15-1 the one-octave
spread between 45 and 90 Hz, only 8 modal frequencies are counted. In
the next highest octave there are 25 modes. Even in this very limited
0 1.0 1.0 0 1.0 0 1.0
3, 0, 0 Mode
F I G U R E 1 5 - 1 6
Sound pressure contours on a section through a rectangular room for the 3,0,0 axial mode.
342 CHAPTER FIFTEEN
low-frequency range below 200 Hz, you can see modal density increasing
with frequency. Figure 15-19 shows that at somewhat higher frequencies
the rate of increase dramatically rises. Above about 300 Hz or
so, the mode spacing is so small that the room response smooths
markedly with frequency.
Mode Spacing and Coloration
Colorations largely determine the quality of sound for a small studio or
listening room. The big task then is to determine which, if any, of the
hundreds of modal frequencies in a room are likely to create colorations.
The spacing of the modal frequencies is the critically important
factor. In the D-region of Fig. 15-7, the modal frequencies of a small
Z
Y
X
2, 1, 0 Mode
F I G U R E 1 5 - 1 7
Three-dimensional representation of the sound pressure distribution in a rectangular
room for the tangential mode 2,1,0. Brue¨l and Kjaer Instruments, Inc.
MODAL RESONANCES IN ENCLOSED SPACES 343
room are so close together that they tend to merge helpfully and
harmlessly. In the B- and C-regions, below about 300 Hz, their separation
is greater and it is in this region that problems can arise.
How close together must adjacent modal frequencies be to avoid
problems of coloration? Gilford4 states his opinion that an axial mode
separated more than 20 Hz from the next axial mode will tend to be
isolated acoustically. It will tend not to be excited through coupling
due to overlapping skirts but will tend to act independently. In this
isolated state it can respond to a component of the signal near its own
frequency and give this component its proportional resonant boost.
Another criterion for mode spacing has been suggested by Bonello5,6
who considers all three types of modes, not axial modes alone. He
states that it is desirable to have all modal frequencies in a critical band
at least 5% of their frequency apart. For example, one modal frequency
0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.8
1.0 1.0 1.0
1.0 1.0 1.0
E
W
0
0
0
0
0
0
N S
2, 1, 0 Mode
F I G U R E 1 5 - 1 8
Sound-pressure contours of the rectangular room of Fig. 15-16 for the tangential
mode 2,1,0.
344 CHAPTER FIFTEEN
at 20 Hz and another at 21 Hz would be barely acceptable. However, a
similar 1-Hz spacing would not be acceptable at 40 Hz (5% of 40 Hz is
2 Hz). Thus we see that Gilford’s concern was primarily how far apart
axial modes must be spaced to avoid problems resulting from independent
and uncoupled modal action. Bonello’s concern has to do with
separations to avoid degeneracy (coincident) effects.
Zero spacings between modal frequencies are a common source of
coloration. Zero spacing means that two modal frequencies are coincident
(called a degeneracy by acousticians), and such degeneracies
tend to overemphasize signal components at that frequency.
Experiments with Colorations
Any ear can be offended by colorations caused by isolated modes, but
even a critical and trained ear needs some instrumental assistance in
5,000
3,000
2,000
4,000
1,000
0
20 50 100 500 1 kHz 2 kHz
Number of modes
Frequency - Hz
F I G U R E 1 5 - 1 9
The number of modal frequencies increases dramatically with frequency.
MODAL RESONANCES IN ENCLOSED SPACES 345
identifying and evaluating such colorations. The BBC Research
Department made an interesting study.4 Observers listened to persons
speaking at a microphone in the studio under investigation, the voices
being reproduced in another room over a high-quality system.
Observers’ judgments were aided by a selective amplifier that amplified
a narrow frequency band (10 Hz) to a level about 25 dB above the
rest of the spectrum. The output was mixed in small proportions with
the original signal to the loudspeaker, the proportions being adjusted
until it is barely perceptible as a contribution to the whole output. Any
colorations were then made clearly audible when the selective amplifier
was tuned to the appropriate frequency.
In most studios tested this way, and we can assume that they were
well designed, only one or two obvious colorations were found in
each. Figure 15-20 is a plot of 61 male voice colorations observed over
a period of 2 years. Most fall between 100 and 175 Hz. Female voice
colorations occur most frequently between 200 and 300 Hz.
16
12
8
4
0 50 100 150 200 250 300
Frequency - Hz
Number of colorations
F I G U R E 1 5 - 2 0
A plot of 61 male voice colorations observed over a period of 2 years in BBC studios. Most
fall in the 100-175 Hz region. Female voice colorations occurr between 200 and 300 Hz.
After Gilford.4
346 CHAPTER FIFTEEN
Simplified Axial Mode Analysis
Apply what you learned about axial modes to a specific rectangular listening
room or studio. The dimensions of our specimen room are 28 ×
16 × 10 ft. The 28-ft length resonates at 565/28 = 20.2 Hz, the two side
walls 16 ft apart resonate at 565/16 = 35.3 Hz, and the floor-ceiling combination
resonates at 565/10 = 56.5 Hz. These three axial resonances
and the train of multiples for each are plotted in Fig. 15-21. There are 27
axial resonance frequencies below 300 Hz, and for this exercise, the
horde of weaker tangential and oblique modes are neglected.
Because most signal colorations are traceable to axial modes, their
spacings will be examined in detail. Table 15-5 illustrates a convenient
form for this simplified analysis of axial modes. The resonance
frequencies from the L, W, and H columns of Table 15-5 are arranged
in ascending order in the fourth column. This makes it easy to examine
that critical factor, axial mode spacing.
The L-f7 resonance at 141.3 Hz coincides with the W-f4 resonance.
This means that these two potent axial modes act together to create
a potential coloration of sound at that frequency. This coincidence
is also separated 20 Hz from its neighbors. It is also noted from Fig.
15-20 that 141.3 Hz is in a frequency range that is especially troublesome.
Here, then, is a warning of a potential problem.
At 282.5 Hz we see a “pile-up” of L-f14, W-f8, and H-f5 modes which,
together, would seem to be an especially troublesome source of colf
1
f1
f1
20 30 50 70 100 200 300 400 500
Length
Width
Height
Frequency - Hz
F I G U R E 1 5 - 2 1
The axial modal frequencies and multiples of the 16 X 28 X 10 ft room.
MODAL RESONANCES IN ENCLOSED SPACES 347
Table 15-5. Axial mode analysis form.
Room dimensions: Length 28.0 ft.
Width 16.0 ft.
Height 10.0 ft.
Arranged in Axial mode
Axial mode resonances ascending spacing
Hz order Hz
L W H 20.2 15.1
f1 20.2 35.3 56.5 35.3 5.1
f2 40.4 70.6 113.0 40.4 16.1
f3 60.5 105.9 169.5 56.5 4.0
f4 80.7 141.3 226.0 60.5 10.1
f5 100.9 176.6 282.5 70.6 10.1
f6 121.1 211.9 339.0 80.7 20.2
f7 141.3 247.2 100.9 5.0
f8 161.4 282.5 105.9 7.1
f9 181.6 317.8 113.0 8.1
f10 201.8 121.1 20.2
f11 222.0 141.3 0
f12 242.1 141.3 20.1
f13 262.3 161.4 8.1
f14 282.5 169.5 7.1
f15 302.7 176.6 5.0
181.6 20.2
201.8 10.1
211.9 10.1
222.0 4.0
226.0 16.1
242.1 5.1
247.2 15.1
262.3 20.2
282.5 0
282.5 0
282.5 20.2
302.7 20.2


348 CHAPTER FIFTEEN
orations. They are also separated from neighbors by 20 Hz. However,
looking at the experimentally derived plot of Fig. 15-20, practically no
problem with voice colorations was found at 282 Hz, at least for male
voices. The reason for this is probably the salutary presence of tangential
and oblique modes neglected in this study.
With the threat of coloration at 141 Hz, adjustment of dimensions
of a proposed room would be a logical attack. If it is an existing room,
a Helmholtz resonator, sharply tuned and properly located, is a difficult
but possible solution.
The Bonello Criterion
Proof that modal resonances in rooms are an international problem,
this method of evaluating their effect comes from Buenos Aires.
Bonello5,6 suggests a novel method of determining, by computer, the
acoustical desirability of the proportions of rectangular rooms. He
divides the low end of the audible spectrum into bands 1⁄3 octave wide
and considers the number of modes in each band below 200 Hz. The
1⁄3-octave bands are chosen because they approximate the critical
bands of the human ear.
To meet Bonello’s criterion, each 1⁄3 octave should have more modes
than the preceding one, or at least the same number. Modal coincidences
are not tolerated unless at least 5 modes are in that band.
How does a 15.4 × 12.8 × 10-ft room qualify by this criterion? Figure
15-22 shows that it passes this test with flying colors. The graph
climbs steadily upward with no downward anomalies. The horizontal
section at 40 Hz is allowed. The advantage of Bonello’s plan is that it
is well adapted to computer calculation and print-out.
Controlling Problem Modes
The general construction of Helmholtz resonators for normal room
treatment is detailed in chapter 9, but building one with very sharp
tuning (high Q) is more demanding.7 The flexing of wooden boxes
introduces losses that lower the Q. To attain a truly high-Q resonator
with sharp tuning, the cavity must be made of concrete, ceramic, or
other hard, nonyielding material, but fitted with some means of varyMODAL
RESONANCES IN ENCLOSED SPACES 349
ing the resonance frequency. The resonance frequency of a Coca Cola
bottle was measured8 at 185 Hz and was found to have a bandwidth
(–3 dB points) of 0.67 Hz. This yields a Q = 185/0.67 = 276, a very high
value. So, if you are fortunate enough to have your mode problem at
185 Hz ...!
It is also important where the Coke bottle (or other Helmholtz
absorber) is placed, if the goal is to bring a mode or closely spaced
group of modes under control. Let us say that the 2,1,0 mode of Fig.
15-18 is causing a voice coloration and that it is necessary to introduce
a narrow sliver of absorption at the 2,1,0 frequency. If the
Helmholtz absorber were placed at a pressure node (zero pressure) it
would have, obviously, no effect. Placed at one of the antinodes (pressure
peaks) it would have tight interaction with the 2,1,0 mode.
Therefore, any corner would be acceptable, as would the pressure
peaks on the E or W walls.
Frequency - Hz
10 13 16 20 25 32 40 50 63 80 100 125 160 200
Modal density - modes/1/3 octave
18
16
14
12
10
8
6
4
2
0
F I G U R E 1 5 - 2 2
A plot showing the number of modes in 1/3 octave bands for the 15,4 X 12.8 X 10 ft room.
The graph climbs steadily upward with no downward anomalies; hence the room meets the
Bonello criterion.
350 CHAPTER FIFTEEN
Mode Summary
 There are three types of acoustical resonances (natural frequencies,
standing waves, normal modes) in a rectangular enclosure.
These are the axial, tangential, and oblique modes.
 The axial modes are made up of two waves going in opposite
directions, traveling parallel to one axis, and striking only two
walls. Axial modes make the most prominent contribution to
the acoustical characteristics of a space. Because there are three
axes to a rectangular room, there are three fundamental axial
frequencies, each with its own train of modes.
 The tangential modes are formed by four traveling waves that
reflect from four walls and move parallel to two walls. Tangential
modes have only half the energy of axial modes, yet their
effect on room acoustics can be significant. Each tangential
mode has its train of modes.
 The oblique modes involve eight traveling waves reflecting
from all six walls of an enclosure. Oblique modes, having only
one-fourth the energy of the axial modes, are less prominent
than the other two.
 The number of normal modes increases with frequency. Small
rooms whose dimensions are comparable to the wavelength of
audible sound have the problem of excessive separation
between modes, which can contribute to poor characteristics for
recording or other critical work. A musical note falling “in the
cracks” between widely separated modes will be abnormally
weak and will die away faster than other notes. It is almost as
though that particular note were sounded outdoors while the
other notes were simultaneously sounded indoors.
 Axial, tangential, and oblique modes decay at different rates.
Absorbing material must be located on surfaces near which a
given modal pressure is high if it is to be effective in absorbing
that mode. For example, carpet on the floor has no effect on the
horizontal axial modes. Tangential modes are associated with
more surface reflections than axial modes and oblique modes
with even more than tangential ones.
MODAL RESONANCES IN ENCLOSED SPACES 351
 As frequency is increased, the number of modes greatly
increases. Above 300 Hz, average mode spacing becomes so
small that room response tends to become smoother.
 Colorations caused by acoustical anomalies of studios, monitoring
rooms, listening rooms, and other small rooms are particularly
devastating to speech quality. Gilford states4 that axial
modes spaced approximately 20 Hz or more, or a pair of modes
coincident or very close, are frequent sources of colorations. He
also states that colorations are likely to be audible when an axial
mode coincides with a fundamental or first format of at least
one vowel sound of speech, and are in the region of high-speech
energy. Speech colorations below 80 Hz are rare because there is
so little energy in speech in that part of the spectrum. There are
essentially no speech colorations above 300 Hz for either male
or female voices. Modal colorations are more noticeable in
speech than in music.
 The bandwidth of room modes, measured at the –3 dB points,
increases as reverberation time is shortened. Modes in ordinary
studios have bandwidths on the order of 5 Hz. Harmonics of
modes have the same bandwidths as their fundamentals.
The serious student wishing to pursue this subject further is urged
to consult references 9 through 17. These papers, spanning a half century,
trace the growth in our understanding of room modes.
Endnotes
1Brüel, Per V., Sound Insulation and Room Acoustics, London, Chapman and Hall (1951),
2Ruusuvuori, Aarno, Architect; Mauri Parjo, Acoustical Consultant.
3Morse, Philip M. and Richard H. Bolt, Sound Waves in Rooms, Reviews of Modern Physics,
16, 2 (Apr 1944) 69-150.
4Gilford, C.L.S., The Acoustic Design of Talks Studios and Listening Rooms, Proc. Inst. Elect.
Engs., 106, Part B, 27 (May 1959) 245-258. Reprinted in J. Audio Eng. Soc., 27, 1/2 (1979) 17-
31.
5Bonello, O.J., A New Computer-Aided Method for the Complete Acoustical Design of
Broadcasting and Recording Studios, IEEE International Conference on Acoustics and
Signal Processing, ICASSP 79, Washington 1979, p. 326-329.
6Bonello, O.J., A New Criterion for the Distribution of Normal Room Modes, J. Audio Eng.
Soc., 29, 9 (Sept 1981) 597-606. Correction in 29, 12 (1981) 905.
352 CHAPTER FIFTEEN
7van Leeuwen, F.J., The Damping of Eigentones in Small Rooms by Helmholtz Resonators,
European Broadcast Union Review, A, 62 (1960) 155-161.
8Siekman, William, Private communication. Measurements were made at Riverbank
Acoustical Laboratories when Siekman was manager. He presented the results of these
measurements at the April 1969 convention of the Acoustical Society of America.
9Hunt, Frederick V., Investigation of Room Acoustics by Steady-State Transmission
Measurements, J. Acous. Soc. Am., 10 (Jan 1939) 216-227.
10Hunt, F.V., L.L. Beranek, and D.Y. Maa, Analysis of Sound Decay in Rectangular Rooms, J.
Acous. Soc. Am., 11 (July 1939) 80-94.
11Bolt, R. H., Perturbation of Sound Waves in Irregular Rooms, J. Acous. Soc. Am., 13 (July
1942) 65-73.
12Bolt, R.H., Note on Normal Frequency Statistics in Rectangular Rooms, 18, 1 (July 1946)
130-133.
13Knudsen, Vern O., Resonances in Small Rooms, J. Acous. Soc. Am. (July 1932) 20-37.
14Mayo, C. G., Standing-Wave Patterns in Studio Acoustics, Acustica, 2, 2 (1952) 49-64.
15Meyer, Erwin, Physical Measurements in Rooms and Their Meaning in Terms of Hearing
Conditions, Proc. 2nd International Congress on Acoustics, (1956) 59-68.
16Louden, M.M., Dimension Ratios of Rectangular Rooms with Good Distribution of
Eigentones, Acustica, 24 (1971) 101-104.
17Sepmeyer, L.W., Computed Frequency and Angular Distribution of the Normal Modes of
Vibration in Rectangular Rooms, J. Acous. Soc. Am., 37, 3 (March 1965) 413-423.
The perceptual effect of sound reflections depends on the size of the
room. The situation in a recording studio, control room, or listening
room is quite different from that in a music hall or large auditorium.
The case of sound reflections in small rooms is considered first in this
discussion.
Law of the First Wavefront
Imagine two people in a small room as illustrated in Fig. 16-1. The
first sound of the person speaking to reach the listener is that traveling
a direct path because it travels the shortest distance. This direct
sound, which arrives first at the ears of the listener, establishes the
perception of the direction from which the sound came. Even
though it is immediately inundated by a stream of reflections from
the various surfaces of the room, this directional perception persists,
tending to lock out the effects of all later reflections insofar as
direction is concerned. Cremer has called this the law of the first
wavefront. This fixation of the direction to the source of sound is
accomplished within a small fraction of a millisecond and, as
already mentioned, is unaffected by the avalanche of reflections following
the arrival of the direct sound.
Sound Reflections in
Enclosed Spaces
16
353
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
354 CHAPTER SIXTEEN
Mean Free Path
The average (mean) distance sound travels
between successive reflections is called
the mean free path. This average distance
is given by the expression 4V/S, in which
V is the volume of the space and S is the
surface area of the space. In a room 25 × 20
× 10 ft, the sound travels a distance of 10.5
ft between reflections on the average.
Sound travels 1.13 ft per millisecond. At
that speed it takes only 9.3 ms to traverse
the mean-free distance of 10.5 ft. Viewed
another way, about 107 reflections take
place in the space of a second.
Figure 16-2 shows cathode-ray oscillograms
of the reflections occurring during
the first 0.18 second in a recording studio having a volume of 16,000
cu ft. The microphone was placed, successively, in four different locations
in the room. The impulsive sound source was in a fixed position.
F I G U R E 1 6 - 1
Of the many reflections of sound in a small room, the
direct wave arrives first and establishes the receiver’s
sense of the direction of arrival of the sound. Later
reflections do not affect this directional perception.
M-1 M-3
M-2 M-4
F I G U R E 1 6 - 2
Individual reflections are resolved in these echograms taken at four different positions
in a studio of 16,000 cu ft volume and having a reverberation time of 0.51 second. The
horizontal time scale is 20 msec/div.
SOUND REFLECTIONS IN ENCLOSED SPACES 355
The sound source was a pistol that punctured a paper with a blast of
air, giving an intense pulse of sound of less than a millisecond duration.
The reflection patterns at the four positions show differences but,
in each, scores of individual reflections are clearly resolved. These
“echograms” define the transient sound field of the room during the
first 0.18 second as contrasted to the steady-state condition. The question
now reduces to the perceptual effects of all these reflections on
the human auditory system.
The Effect of Single Reflections
Numerous research studies on the audibility of simulated reflections
have been conducted over a period of many years. Most of these have
used an arrangement of loudspeakers very much like the traditional
high-fidelity, stereophonic arrangement as shown in Fig. 16-3. The
observer is seated at the apex so that lines drawn to the two loudspeakers
are approximately 60 degrees apart (this angle varies with the investigator).
The mono signal (commonly
speech) is fed to one of the loudspeakers,
which represents the direct signal. The signal
to the other loudspeaker can be delayed
any amount: This represents a lateral
reflection. The two variables under study
are: the level of the reflection compared to
that of the direct, and the delay of the
reflection with respect to the direct signal.
Perception of Sound Reflections
The earlier researchers in this field were
interested primarily in the effect of reflections
on the perception of music in music
halls. Recent work by Olive and Toole1,2 was
aimed more specifically at listening conditions
in small rooms, such as recording studio
control rooms and home listening
rooms. The work of Olive and Toole is summarized
in Fig. 16-4. This graph plots reflection
level against reflection delay, the two
variables mentioned above. A reflection
In
Direct
Delay
Reflection
level
control
Simulated
lateral reflection
F I G U R E 1 6 - 3
Typical equipment arrangement used by many investigators
of the audible effect of a simulated lateral
reflection on the direct signal. Reflection level (with
respect to the direct) and reflection delay are the
variables under control.
356 CHAPTER SIXTEEN
level of 0 dB means that the reflection is the same level as the direct signal.
A reflection level of –10 dB means that the reflection level is 10 dB
below the direct. In all cases reflection delay is in milliseconds later than
the direct signal.
Figure 16-4, curve A is the absolute threshold of audibility of the
echo. This means that at any particular delay, the reflection is not
heard for reflection levels below this line. Note that for the first 20 milliseconds,
this threshold is essentially constant. At greater delays, progressively
lower reflection levels are required for a just-audible
reflection. It will soon become apparent that for a home listening room
Echo region
Reflection
inaudible
Image effects(shifting, spreading)  spaciousness
Spaciousness
C
B
A
20
10
0
10
20
30
40
0 20 40 60 80
Reflection delay - ms
Reflection level - dB
F I G U R E 1 6 - 4
Results of investigations into the effect of a simulated lateral reflection in anechoic
environment and with speech as the test signal. Curve A is the absolute threshold of
detection of the reflection (Olive and Toole1). Curve B is the image-shift threshold (Olive
and Toole1). Curve C indicates the points at which the reflection is heard as a discreet
echo. This is a composite of the results of Refs. 3 and 5. Journal of the Audio Engineering Society.
SOUND REFLECTIONS IN ENCLOSED SPACES 357
or other small room, delays in the 0–20 ms range are of greatest significance.
In this range the reflection audibility threshold varies little
with delay.
Perception of Spaciousness
Assume a reflection delay of 10 ms with the reflection coming from the
side. As the level of the reflection is increased from a very low level,
the reflection is completely inaudible. As the level of the reflection is
increased, it finally becomes audible as its level reaches about 15 dB
below the direct signal. As the reflection level is increased beyond this
point, the room takes on a sense of spaciousness; the anechoic room in
which the tests were made sounded more like a normal room. The listener
is not aware of the reflection as a discrete event, nor of any directional
effect, only this sense of spaciousness.
Image Changes
As the level of the reflection is further increased other effects become
audible. At about 10 dB above the threshold of audibility of the reflection,
changes in the apparent size and location of the front auditory
image become apparent. At greater delays, the image tends to become
smeared toward the reflection.
Reviewing what happens in the 10–20 ms delay range, as the reflection
level is increased above the threshold of audibility, spatial effects
dominate. As the reflection level is increased roughly 10 dB above the
audibility threshold, image effects begin to enter, including image size
and shifting of position of the image.
Discrete Echoes
Reflections having a level another 10 dB above the image shift threshold
introduce another perceptual threshold. The reflections now are
discrete echoes superimposed on the central image.3 Such discrete
echoes are very damaging to sound quality in any practical situation.
For this reason, reflection level/delay combinations that result in such
echoes are to be shunned in practical installations.
Effect of Angle of Incidence on Audibility of Reflection
German researchers have shown that the direction from which the reflection
arrives has practically no effect on the perception of the reflection
with one important exception. When the reflection arrives from the same
358 CHAPTER SIXTEEN
direction as the direct signal, it can be up to 5 to 10 dB louder than the
direct before it is detected. This is due to masking of the reflection by the
direct signal. If the reflection is recorded along with the direct signal and
reproduced over a loudspeaker, it will be masked by this 5 to 10 dB
amount.
Effect of Signal Type on Audibility of Reflection
The type of signal has a major effect on the audibility of reflections.
Consider the difference between continuous and noncontinuous
sounds. Impulses, in the form of 2 clicks per second, are of the noncontinuous
type. Pink noise illustrates the continuous type. Speech
and music lie in between the two types. In Fig. 16-5 the differences in
audibility thresholds of continuous vs. noncontinuous sounds is illustrated.
Anechoic speech comes closer to being a noncontinuous sound
than either music or pink noise. At delays less than 10 ms, the level of
impulses for threshold detection must be much higher than continuous
sounds. The threshold curves for Mozart music and pink noise are
very close together. This confirms the
belief that pink noise is a reasonable surrogate
for music for measurements.
Effect of Spectrum on Audibility of
Reflection
Most of the researchers used sounds having
the same spectrum for both the direct and
reflected simulations. In real life, reflections
depart from the original spectrum because
sound-absorbing materials invariably absorb
high frequencies more than low frequencies.
In addition to this, off-axis loudspeaker
response lowers the high-frequency content
even more. Threshold audibility experiments
have shown that rather radical
low-pass filtering of the reflection signal
produced only minor differences in thresholds.
The conclusion is that alteration of
reflection spectrum does not change audibility
thresholds appreciably.
Clicks
Speech
Music
Pink noise
0
0 10 20 30
10
20
30
Reflection level - dB
Reflection delay - ms
F I G U R E 1 6 - 5
Absolute thresholds of perception of reflections of
different types of signals, ranging from 2/second
clicks (noncontinuous), to pink noise (continuous).
The closeness of pink noise to Mozart music gives
assurance that pink noise is a reasonable surrogate
for music in measurements. After Olive and Toole.1
Journal of the Audio Engineering Society.
SOUND REFLECTIONS IN ENCLOSED SPACES 359
Using Reflection Data
Figure 16-4 is a remarkable concentration of the perceptual effects of
lateral reflections. A single lateral reflection affects the size and position
of the auditory image and controls spaciousness. How about the
two-loudspeaker stereo installation? Olive and Toole investigated this
and found that the effects obtained from a single loudspeaker are
directly applicable to the stereo case. This means that the information
presented earlier is immediately available for application to stereo listening
situations such as the home music-reproduction room.
Those interested in the reproduction of high-fidelity sound will see
the practicality of the results of these reflection studies. The possibility
of adjusting the spaciousness of the listening room as well as the stereo
image sharpness is intriguing. All this is possible by careful and knowledgeable
manipulation of lateral reflections. Of course, lateral reflections
can come into their own only after interfering early reflections are
reduced. These possibilities are explored further in later chapters.
Large Spaces
Echoes
Larger enclosed spaces, such as performance theaters, music halls,
churches, and auditoriums, are potential producers of discrete echoes.
Architects must be continually alert to surfaces that might produce
reflections of sufficient level and delay to be perceived as discrete
echoes. This is a gross defect for which there is little tolerance, audible
to everyone with normal hearing.
Figure 16-6 is plotted on the same general coordinates as Fig. 16-4.4,5
The main difference is that the delay scale is carried out to 600 ms, in
line with larger spaces.
Reverberation time affects the audibility of echoes. The reverberation
time of the graph of Fig. 16-6 is 1.1 seconds. The heavy broken line
shows the decay rate representing this reverberation time (60 dB in 1.1 seconds
or 30 dB in 550 ms). The shaded area represents combinations of echo
level and echo delay, experimentally determined, which result in echoes
disturbing to people. The upper edge is for 50%, the lower edge is for 20%
of the people disturbed by the echo. Concert hall reverberation time usually
hovers around 1.5 seconds; many churches are closer to 1 second to favor
360 CHAPTER SIXTEEN
speech. Measurements of this type, made in spaces having other reverberation
times, show that the shaded area indicating the level/delay region
causing troublesome echoes is close to being tangent to the reverberation
decay line. This presents the possibility of estimating the echo threat of any
large room by drawing the reverberation line. For example, the lighter broken
line in Fig. 16-6 is drawn for a reverberation time of 0.5 second (60 dB
decay in 500 ms or 30 dB decay in 250 ms). An echo-interference area just
above this decay line can be very roughly inferred in this way.
Spaciousness
Acoustical consultants and architects routinely design music halls to
give lateral reflections of appropriate levels and delays to add a sense of
50% disturbed
Echo region
RT60 = 0.5 sec
RT60 = 1.1 sec
20% disturbed
10
0
10
20
30
40
Echo region - dB
Echo delay - ms
0 100 200 300 400 500 600
F I G U R E 1 6 - 6
Acceptable echo levels for speech under reverberant conditions (Reverberation Time 1.1
seconds). After Nickson, Muncey, and Dubout.4
SOUND REFLECTIONS IN ENCLOSED SPACES 361
spaciousness to the music for those in the audience. This is a highly specialized
area of limited interest to readers of this book. This application
does emphasize the importance of informed manipulation of reflections
in large spaces to achieve desirable results, even as in smaller rooms.
Endnotes
1Olive, Sean E. and Floyd E. Toole, The Detection of Reflections in Typical Rooms, J. Audio
Eng. Soc., 37, 7/8, (July/Aug 1989) 539-553.
2Toole, Floyd E., Loudspeakers and Rooms for Stereophonic Sound Reproduction, Proc.
Audio Eng. Soc. 8th International Conference, Washington, D.C., (1990) 71-91.
3Meyer, E. and G.R. Schodder, On The Influence of Reflected Sound on Directional
Localization and Loudness of Speech, Nachr. Akad. Wiss., Göttingen; Math. Phys. Klasse IIa,
6, (1952) 31-42.
4Nickson, A.F.B., R.W. Muncey, and P. Dubout, The Acceptability of Artificial Echoes with
Reverberant Speech and Music, Acustica, 4 (1954) 515-518.
5Lochner, J.P.A. and J.F. Burger, The Subjective Masking of Short Time Delayed Echoes by
Their Primary Sounds and Their Contribution to the Intelligibility of Speech, Acustica, 8
(1958) 1-10.
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The term comb filter has been widely used in the popular audio press
as an explanation of delayed reflection effects. Comb filtering is a
steady-state phenomenon. It has limited application to music and
speech, which are highly transient phenomena. With transient
sounds, the audibility of a delayed replica is more the result of successive
sound events. A case might be made for proper application of
combing to brief snatches of speech and music that approach steady
state, but already there is an etymological impasse. The study of the
audible effects of delayed reflections is better handled with the generalized
threshold approach of Chap. 16.
What Is a Comb Filter?
A filter changes the shape of the frequency response or transfer function
of a system. An electronic circuit used to shape the frequency response
of a system to achieve a certain desired end could be a filter. A filter
could also be a system of pipe and cavities used to change an acoustical
system, such as is used in some microphones to adjust the pattern.
In the early days of multitrack recording, experimenters were constantly
developing new, different, and distinctive sounds. Phasing and
flanging were popular words among these experimenters.1 At first
multiple-head tape recorders were used to provide delayed replicas of
Comb-Filter Effects
17
363
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
364 CHAPTER SEVENTEEN
sounds that were then mixed with the original sound to produce some
unusual and eerie effects. Currently special electronic circuits are
used to generate these delays. Whatever the means, these audible colorations
of sound are the result of comb filters.2
Superposition of Sound
A sound contractor was concerned about the aiming of his horns in a
certain auditorium. The simplest mechanical mounting would cause
the beam of one horn to cut across the beam of the other horn. What
happens in that bit of space where the two beams intersect? Would the
beams tend to spread out? Would sound energy be lost from the beams
as one beam interacts with the other? Relax—nothing happens.
Imagine a physics lab with a large, but shallow, ripple tank of water
on the lecture table. The instructor positions three students around the
tank, directing them to drop stones in the tank simultaneously. Each
stone causes circular ripples to flow out from the splash point. Each
set of ripples expands as though the other two ripple patterns were not
there.
The principle of superposition states that every infinitesimal volume
of the medium is capable of transmitting many discrete disturbances
in many different directions, all simultaneously and with no
detrimental effect of any one on the others. If you were able to observe
and analyze the motion of a single air particle at a given instant under
the influence of several disturbances, you would find that its motion is
the vector sum of the various particle motions required by all the disturbances
passing by. At that instant, the air particle moves with
amplitude and direction of vibration to satisfy the requirements of
each disturbance just as a water particle responds to several disturbances
in the ripple tank.
At a given point in space, assume an air particle responds to a passing
disturbance with amplitude A and 0° direction. At the same
instant another disturbance requires the same amplitude A, but with a
180° direction. This air particle satisfies both disturbances at that
instant by not moving at all.
A microphone is a rather passive instrument. Its diaphragm
responds to whatever fluctuations in air pressure occur at its surface.
If the rate of such fluctuations (frequency) falls within its operating
COMB-FILTER EFFECTS 365
range, it obliges with an output voltage proportional to the magnitude
of the pressure involved. In Fig. 17-1, a 100-Hz tone from loudspeaker
A actuates the diaphragm of a microphone in free space, and a 100-Hz
voltage appears at the microphone terminals. If a second loudspeaker
B lays down a second 100-Hz signal at the diaphragm of the microphone
identical in pressure but 180 degrees out of phase with the first
signal, one acoustically cancels the other, and the microphone voltage
falls to zero. If an adjustment is made so that the two 100-Hz acoustical
signals of identical amplitude are in phase, the microphone delivers
twice the output voltage, an increase of 6 dB. The microphone
slavishly responds to the pressures acting on its diaphragm. In short,
the microphone responds to the vector sum of air pressure fluctuations
impinging upon it. This characteristic of the microphone is intimately
involved in acoustical comb-filter effects.
Tonal Signals and Comb Filters
A 500-Hz tone is shown as a line on a frequency scale in Fig. 17-2A.
All of the energy concentrated in this pure tone is located at this frequency.
Figure 17-2B shows an identical signal except it is delayed by
0.5 ms in respect to the signal of A. The signal has the same frequency
and amplitude, but the timing is different. Consider both A and B as
acoustical signals combining at the diaphragm of a microphone. Signal
A could be a direct signal and B a reflection of A off a nearby sidewall.
What is the nature of the combined signal the microphone puts out?
Because signals A and B are pure tones,
simple sine waves, both vary from a positive
peak to a negative peak 500 times per
second. Because of the 0.5 ms delay, these
two tonal signals will not reach their positive
or negative peaks at the same instant.
Often along the time axis both are positive,
or both are negative, and at times one is
positive while the other is negative. When
the sine wave of sound pressure representing
signal A and the sine wave of sound
pressure representing signal B combine
(with due respect to positive and negative
A
B
F I G U R E 1 7 - 1
The microphone diaphragm responds to the vector
sum of sound pressures from multiple sources.
366 CHAPTER SEVENTEEN
signs) they produce another sine wave of the same frequency, but of
different amplitude.
Figure 17-2 treats the two 500-Hz tones as lines on a frequency scale.
Figure 17-3 treats the same 500-Hz direct tone and the delayed tone in
the more familiar sine-wave form. The delay is accomplished by feeding
the 500-Hz tone into a digital delay device and combining the original
and the delayed tones in a common three-resistor summing circuit.
In Fig. 17-3A the direct 500-Hz tone is shown originating at zero
time. It takes 2 ms for one cycle of a 500-Hz tone (1500  0.002 sec). One
cycle is also equivalent to 360 degrees. The 500-Hz signal, e, is plotted
according to the time and degree scales at the bottom of the figure.
A delay of 0.1 ms is equivalent to 18 degrees; a delay of 0.5 ms is
equivalent to 90 degrees; a delay of 1 ms is equivalent to 180 degrees. The
effect of these three delays on the tonal signals is shown in Fig. 17-3B
A
C
B
Amplitude
Amplitude
Amplitude
Tone
A
Combination
of A & B
Tone
B
0 0.5 1 1.5 2 0 0.5 1 1.5 2
0 0.5 1 1.5 2
Frequency - kHz
F I G U R E 1 7 - 2
Tonal signals and comb filters; (A) a sine wave having a frequency of 500 Hz, (B)
another sine wave of 500 Hz that is delayed 0.5 ms from A, and C is a summation of A
and B. The 500-Hz signal and its delayed counterpart reach their peaks at slightly different
times, but adding them together simply yields another sine wave; there is no
comb filtering (see Figs. 1-11, 1-13).
COMB-FILTER EFFECTS 367
(later the same delays will be compared
with music and speech signals). The combination
of e and e1 reaches a peak of approximately
twice that of e (6 dB). A shift of 18
degrees is a very small shift, and e and e1 are
practically in phase. The curve e  e2, at 90
degree phase difference has a lower amplitude,
but still a sine form. Adding e to e3
(delay 1 ms, shift of 180 degrees) gives zero
as adding two waves of identical amplitude
and frequency but with a phase shift of 180
degrees results in cancellation of one by the
other.
Adding direct and delayed sine waves
of the same frequency results in other sine
waves of the same frequency. Adding
direct and delayed sine waves of different
frequencies gives periodic waves of irregular
wave shape. Conclusion: Adding direct
and delayed periodic waves does not create
comb filters. Comb filters require signals
having distributed energy such as
speech, music, and pink noise.
Combing of Music and Speech
Signals
The spectrum of Fig. 17-4A can be considered
an instantaneous slice of music,
speech, or any other signal having a distributed
spectrum. Figure 17-4B is essentially
the same spectrum but delayed 0.1
ms from Fig. 17-4A. Figure 17-4C is the
acoustical combination of the A and B
sound pressure spectra at the diaphragm
of the microphone. The resulting overall response of Fig. 17-4C
appears like a sine wave, but combining spectra is different from combining
tonal signals. This sine-wave appearance is natural and is actually
a sine-wave shape with the negative loops made positive.
Symbol Delay, ms Angle
e
e e
e1
e2
e3
e e1
e2
0 0
0.1 18
0.5 90
1.0 180
Delay
Amplitude
Amplitude
1
1
2
0
0
1
1
2
e  e1
e  e3
e  e2
Time - ms
Angle - degrees
0 1 2 3
0 180 360 540
A
B
F I G U R E 1 7 - 3
An exercise to demonstrate that combining sine
waves yields not comb filters, but simply other sine
waves. A distributed spectrum is required for the formation
of comb filters. 500 Hz sine waves are displayed
with delays of 0.1, 0.5, and 1.0 ms to conform
to the distributed spectrum cases in Fig. 17-5.
368 CHAPTER SEVENTEEN
Combing of Direct and Reflected Sound
The 0.1 ms delay in Fig. 17-4 could have been from a digital-delay
device, or it could have been a reflection from a wall or other object.
The spectral shape of a signal will be changed somewhat upon reflection,
depending on the angle of incidence, the acoustical characteristics
of the surface, etc.
A reflection delayed 0.1 ms will have traveled (1,130 ft/sec) (0.001
sec)  1.13 ft further than the direct signal. This difference in path
length, only about 111⁄32 inch, could result from a grazing angle with both
source and listener, or microphone, close to the reflecting surface.
Greater delays are expected in more normal situations such as those of
Fig. 17-5. The spectrum of Fig. 17-5A is from a noise generator. This is a
“shhh” sound similar to the between-station noise of an FM radio
receiver. Random noise of this type is used widely in acoustic measurements
because it is a continuous signal, its energy is distributed throughout
the audible frequency range, and it is closer to speech and music
signals than sine or other periodic waves. In Fig. 17-5B, this random
noise signal drives a loudspeaker, which faces a reflective surface; a
A B
C
Amplitude
0 5 10 15 20
Amplitude
0 5 10 15 20
Amplitude
0 5 10 15 20
Frequency - Hz
F I G U R E 1 7 - 4
Combing of signals having distributed spectra; (A) instantaneous spectrum of music
signal. (B) A replica of A, which is delayed 0.1 ms from A. (C) A summation of A and C
showing typical comb filtering.
COMB-FILTER EFFECTS 369
Relative amplitude - dB
Frequency - kHz
0.1 ms
0.5 ms
1.0 ms
0 5 10 15 20
0.7"
3.4"
6.75"
Delay
A
B
C
D
F I G U R E 1 7 - 5
A demonstration of comb filtering in which direct sound from a loudspeaker is acoustically
combined with a reflection from a surface at the diaphragm of a microphone. (A)
No surface, no reflection. (B) Placing the microphone 0.7 in from the surface creates a
delay of 0.1 ms and the combination of the direct and the reflected rays shows cancellations
at 5 and 15 kHz and every 10 kHz. (C) A delay of 0.5 ms creates cancellations
much closer together. (D) A delay of 1 ms results in cancellations even more closely
together. If t is taken as the delay in seconds, the first null is 1/2t and spacing between
nulls or between peaks is 1/t.
370 CHAPTER SEVENTEEN
nondirectional microphone is then placed at varying distances from the
reflective surface.
In Fig. 17-5B, the microphone diaphragm is placed about 0.7 inches
from the reflective surface. Interference takes place between the direct
sound the microphone picks up from the loudspeaker and the sound
reflected from the surface. The output of the microphone shows the
comb-filter pattern characteristic of a 0.1 ms delay.
Placing the microphone diaphragm about 3.4 inches from the
reflective barrier, as in Fig. 17-5C, yields a 0.5 ms delay, which results
in the comb-filter pattern shown. Plotted on a linear frequency scale,
the pattern looks like a comb; hence, the name combfilter. Increasing
the delay from 0.1 to 0.5 ms has increased the number of peaks and the
number of nulls five-fold.
In Fig. 17-5D, the microphone is 6.75 inches from the reflective barrier
giving a delay of 1.0 ms. Doubling the delay has doubled the number
of peaks/nulls once again.
Increasing the delay between the direct and reflected components
increases the number of constructive and destructive interference
events proportionally. Starting with the flat spectrum of Fig. 17-5A,
the far-from-flat spectrum of B is distorted by the presence of a reflection
delayed only 0.1 ms. An audible response change would be
expected. One might suspect that the distorted spectrum of D might be
less noticeable because the multiple, closely spaced peaks and narrow
notches tend to average out the overall response aberrations.
Reflections following closely after the arrival of the direct component
are expected in small rooms because the dimensions of the room
are limited. Conversely, reflections in large spaces would have greater
delays, which generate more closely spaced comb-filter peaks and nulls.
Thus, comb-filter effects resulting from reflections are more commonly
associated with small room acoustics. The size of various music halls
and auditoriums renders them relatively immune to audible comb-filter
distortions, because the peaks and nulls are so numerous and packed so
closely together that they merge into an essentially uniform response.
Figure 17-6 illustrates the effect of straining a music signal through a 2
ms comb filter. The relationship between the nulls and peaks of response
is related to the piano keyboard as indicated. Middle C, (C4), has a frequency
of 261.63 Hz, and is close to the first null of 250 Hz. The next
higher C, (C5), has a frequency close to twice that of C4 and is treated
COMB-FILTER EFFECTS 371
favorably with a 6-dB peak. Other Cs up the keyboard will be either discriminated
against with a null, or especially favored with a peak in
response—or something in between. Whether viewed as fundamental frequencies
or a series of harmonics, the timbre of the sound suffers.
The comb filters illustrated in Figs. 17-4, 17-5, and 17-6 are plotted
to a linear frequency scale. In this form the comb appearance and visualization
of the delayed effects are most graphic. A logarithmic-frequency
scale, however, is more common in the electronics and audio
industry. A comb filter resulting from a delay of 1 ms plotted to a logarithmic
frequency scale is shown in Fig. 17-7.
Comb Filters and Critical Bands
Is the human auditory system capable of resolving the perturbations of
Fig. 17-5D? The resolution of human hearing is circumscribed by the
10
0
10
20
Relative amplitude - dB
C4 C5 C6 C7 C8 C9 C10
Middle C
0 0.5 1 1.5 2 2.5
Frequency - kHz
F I G U R E 1 7 - 6
Passing a music signal through a 2 ms comb filter affects the components of that signal
as indicated. Components spaced one octave can be boosted 6 dB at a peak or essentially
eliminated at a null, or can be given values between these extremes.
372 CHAPTER SEVENTEEN
critical band tuning curves of the inner ear. The critical bandwidths at
representative frequencies are recorded in Table 17-1. For example,
the average critical bandwidth of the human auditory system at 1,000
Hz is about 128 Hz. A peak-to-peak combfilter
frequency of 125 Hz corresponds to a
reflection delay of about 8 ms (1⁄0.008  125
Hz), which corresponds to a difference in
path length between the direct and
reflected components of about 9 ft (1,130
ft/sec 0.008 sec  9.0 ft). This situation
is plotted in Fig. 17-8B. To illustrate what
happens for greater delays, Fig. 17-8C is
sketched for a delay of 40 ms. Shorter
delays are represented by Fig. 17-8A for a
delay of 0.5 ms.
The relative coarseness of the critical
band cannot analyze and delineate the
Relative level - dB
20
10
0
10
20
20 50 100 200 500 1 kHz 2 kHz 5 kHz 10 kHz 20 kHz
Frequency - Hz
F I G U R E 1 7 - 7
Up to this point all comb filters have been plotted to a linear scale to demonstrate the
origin of the term comb. Plotting to the more convenient and familiar logarithmic scale
aids in estimating the effects of a given comb on a given signal.
Table 17-1 Auditory critical bands.
Width of
Center frequency critical band*
(Hz) (Hz)
100 38
200 47
500 77
1,000 128
2,000 240
5,000 650
*Calculated equivalent rectangular band as proposed by Moore
and Glasberg.3
COMB-FILTER EFFECTS 373
Critical
band
Delay
0.5 ms
Delay
8 ms
Delay
40 ms
800 900 1,000 1,100 1,200
800 900 1,000 1,100 1,200
8 900 1,000 1,100 1,200
A
B
C
Frequency - Hz
F I G U R E 1 7 - 8
The audibility of combing is an important but not a well-understood factor. To assist in
estimating the perceptual importance of comb filters, they are compared to the auditory
critical band effective at a frequency of 1,000 Hz. (C) At a delay of 40 ms the width
of the critical band is so coarse, relatively, that no analyzing of the comb filter is possible.
(A) On the other hand the width of the auditory critical band is comparable to the
comb peak at 0.5 ms delay. (B) is an example in between A and C. This would seem to
confirm the observation that in large spaces (long delays) comb filters are inaudible,
while they often are very troublesome in small spaces (short delays).
374 CHAPTER SEVENTEEN
numerous peaks and nulls resulting from a 40-ms delay (Fig. 17-8C).
Therefore, the human ear would not be expected to interpret response
aberrations resulting from 40-ms combing as a coloration of the signal.
On the other hand, the combing resulting from the 0.5 ms delay (Fig.
17-8A) could be delineated by the ear’s critical band at 1,000 Hz resulting
in a perceived coloration of the signal. Figure 17-8B illustrates an
intermediate example in which the ear is marginally able to analyze
the combed signal. The width of the critical bands of the auditory system
increases rapidly with frequency. It is difficult to imagine the
complexity of the interaction between a set of critical bands and a constantly
changing music signal, with diverse combing patterns from a
host of reflections. Only carefully controlled psychoacoustical experiments
can determine whether the resulting colorations are audible
(Chap. 16).
Comb Filters in Stereo Listening
In the standard stereo listening arrangement, the input signals to each
ear come from two loudspeakers. These signals are displaced in time
with respect to each other because of the loudspeaker spacing; the
result is the generation of comb-filters. Blauert indicated that comb-filter
distortion is not generally audible.4 The auditory system has the
ability of disregarding these distortions as the perception of timbre is
formed. This is called binaural suppression of colorations of timbre;
however, no generally accepted theory exists to explain how the auditory
system accomplishes this.4 Distortion can be heard by plugging
one ear; however, this destroys the stereo effect. Comparing the timbre
of signals from two loudspeakers producing comb-filter distortion and
one loudspeaker (that does not) will demonstrate that stereo comb-filter
distortion is barely audible. The timbre of the two is essentially the
same. Furthermore, the timbre of the stereo signal changes little as the
head is turned.
Coloration and Spaciousness
A reflected wave reaching the ear of a listener is always somewhat different
from its direct wave. The characteristics of the reflecting wall
vary with frequency. By traveling through the air, both the direct and
COMB-FILTER EFFECTS 375
reflected components of a sound wave are altered slightly, due to the
air’s absorption of sound, which varies with frequency. The amplitudes
and timing of the direct and reflected components differ. The
human auditory system responds to the frontal, direct component
somewhat differently than to the reflection from the side. The perception
of the reflected component is always different than the direct
component. The amplitudes and timing will be related, but with an
interaural correlation less than maximum.
Weakly correlated input signals to the ears contribute to the
impression of spaciousness. If no reflections occur, such as when listening
outdoors, there is no feeling of spaciousness (see Fig. 16-4). If
the input signals to the ears are correct, the perception of the listener
is that of being completely enveloped and immersed in the sound. The
lack of strong correlation is a prerequisite for the impression of spaciousness.
Combing in Stereo Microphone Pickups
Because two microphones separated in space pick up a sound at
slightly different times, their combined output will be similar to the
single microphone with delayed reflections. Therefore, spaced microphone
stereo-pickup arrangements are susceptible to comb-filter problems.
Under certain conditions the combing is audible, imparting a
phasiness to the overall sound reproduction, interpreted by some as
room ambience. It is not ambience, however, but distortion of the time
and intensity cues presented to the microphones. It is evident that
some people find this distortion pleasing, so spaced microphone pickups
are favored by many producers and listeners.
Audibility of Comb-Filter Effects
Chapter 16 showed that spaciousness is the result of reflections combining
with the direct signal. This chapter demonstrated that combining
a signal with a close replica of itself delayed a small amount
creates comb filters. The audibility of comb filters is thus clearly stated
in the Olive-Toole thresholds of Fig. 16-4. Only through psychoacoustical
measurements of this type can the audibility of comb-filters
be determined.
376 CHAPTER SEVENTEEN
Comb Filters in Practice
Example 1: Figure 17-9 illustrates three microphone placements that
produce comb filters of varying degree. A close source-to-microphone
distance is shown in Fig. 17-9A. The direct component travels 1 ft and
the floor-reflected component travels 10.1 ft (see Table 17-2). The difference
between these (9.1 ft) means that the floor reflection is delayed
8.05 ms (9.1/1130  0.00805 sec.). The first null is therefore at 62 Hz
with subsequent null and peak spacing of 124 Hz. The level of the
1.0'
4.0'
10.3'
A
C
B
F I G U R E 1 7 - 9
Common microphone placements compared with respect to production of comb filters
(see Table 17-1). (A) Reflection 20 dB down, minimum comb-filter problems. (B)
Reflection only 8 dB down, comb-filter problems expected. (C) Reflection almost same
level as direct, comb-filter problems certain. A microphone on the floor of (C) would
reduce the difference in path length between direct and reflected components (and the
combing) to almost zero.
COMB-FILTER EFFECTS 377
reflection is -20 dB referred to the direct component (20 log 1.0/10.1 
20 dB).
Similar calculations for Fig. 17-9B and C are included in Table 17-
2. In A the direct component is 10 times stronger than the floor reflection.
The effect of the comb filter would be negligible. Figure 17-9C
has a reflection almost as strong as the direct, and the comb-filter effect
would be maximum. Figure 17-9B is intermediate between A and C. A
microphone is shown on the floor in Fig. 17-9C. A floor bounce would
occur, but the difference between the direct and reflected path length
would be very small, essentially eliminating
the comb filter.
Example 2: Two microphones on a
podium, Fig. 17-10, are very common. Are
they used as stereo microphones? Stereo
reproduction systems are quite rare in
auditoriums. The chances are very good
that the two microphones are fed into the
same mono system and thus become an
excellent producer of comb-filter effects.
The common excuse for two microphones
is “to give the speaker greater freedom of
movement” or “to provide a spare microphone
in case of failure of one.” Assuming
the microphones are properly polarized
and the talker is dead center, there would
Table 17-2 Comb-filter situations (Refer to Fig. 17-9).
First Pk/null
Fig Path length, null spacing Refl.
8-13 ft. Difference 1/2t 1/t level
Direct Refl. Ft. (t) ms. Hz Hz dB
A 1.0 10.1 9.1 8.05 62 124 20
B 4.0 10.0 6.0 5.31 94 189 8
C 10.3 11.5 1.2 1.06 471 942 1
F I G U R E 1 7 - 1 0
An infamous example of comb-filter production, two
microphones feeding into the same mono amplifier
with a sound source that moves about.
378 CHAPTER SEVENTEEN
be a helpful 6-dB boost in level. Assume also that the microphones are
24 in apart and the talker’s lips are 18 in from a line drawn through the
two microphones and on a level with the microphones. If the talker
moves laterally 3 in, a 0.2 ms delay is introduced, reducing important
speech frequencies. If the talker does not move, the speech quality
would probably not be good, but it would be stable. Normal talker
movements shift nulls and peaks up and down the frequency scale
with quite noticeable shifts in quality.
Example 3: A common situation with comb-filter possibilities is
the singing group with each singer holding a microphone (Fig. 17-11),
and each microphone fed to a separate channel but ultimately mixed
together. The voice of A, picked up by both microphones, is mixed,
producing comb filters resulting from the path difference. Each
singer’s voice is picked up by all microphones but only adjacent
singers create noticeable comb filters. Experiments reported by Burroughs5
indicate that if singer A’s mouth is at least three times farther
from singer B’s microphone than from A’s own microphone, the
A B
d1 d2
F I G U R E 1 7 - 1 1
For group singing, if d2 is at least three times as great as d1, the comb-filter effect is
minimized.
COMB-FILTER EFFECTS 379
comb-filter effects are overshadowed by other problems. This 3:1 rule
works because maintaining this distance means that delayed replicas
are at least 9 dB below the main signal. This assures that comb-filter
peaks and nulls are / 1 dB or less in amplitude and thus essentially
imperceptible.
Example 4: Dual mono loudspeakers, one on stage left and the
other on stage right, or variations of this theme, are quite common
(Fig. 17-12). Two sources radiating identical signals create comb filters
over the audience area. On the line of symmetry (often down the
center aisle) both signals arrive at the same time and no comb filters
are produced. Equi-delay contours range out from stage center over
the audience area, the 1-ms delay contour nearest the center line of
symmetry, and greater delays as the sides of the auditorium are
approached.
Example 5: Multi-element loudspeakers can have their own combfilter
sources. In Fig. 17-13 it is apparent that frequency f1 is radiated
by both bass and mid-range units, that both are essentially equal in
magnitude, and that the two radiators are physically displaced. These
are the ingredients for comb-filter production in the audience area.
25' 0
54321
12345
Delay - milliseconds
F I G U R E 1 7 - 1 2
In the common split system in which two loudspeakers radiate identical signals, zones
of constructive and destructive interference result which degrade sound quality in the
audience area.
380 CHAPTER SEVENTEEN
The same process is at work between the mid-range and the tweeter
units. Only a narrow band of frequencies is affected, the width of
which is determined by the relative amplitudes of the two radiations.
The steeper the crossover curves, the narrower the frequency range
affected.
Example 6: Permanently mounted microphones may be flushmounted
with the advantage of an approximate 6-dB gain in sensitivity
due to the pressure rise at the table surface. Another advantage
is minimizing comb-filter distortions. In Fig. 17-14, a direct ray from
the source activates the microphone diaphragm, which is shielded
from reflections.
Estimating Comb-Filter Response
Remembering a few simple relationships enables you to estimate the
effect of comb filters on the response of a system. If the delay is t seconds,
the spacing between peaks and the spacing between nulls is 1/t
Hz. For example, a delay of 0.001 second (1 ms) spaces the peaks
1/0.001  1,000 Hz, and the nulls will also be spaced the same amount
(Table 17-3).
The frequency at which the first null (i.e., the null of lowest frequency)
will occur is 1/(2t) Hz. For the same delay of 0.001 s, or 1 ms,
dB Bass Mid range High frequency
Frequency
f1 f2
F I G U R E 1 7 - 1 3
Comb-filter distortion can occur in the crossover region of a multielement loudspeaker
because the same signal is radiated from two physically separated units.
COMB-FILTER EFFECTS 381
the first null will occur at 1/[2 0.001]  500 Hz. For this 1-ms delay,
you can almost figure out the system response in your head; the first
null is at 500 Hz, nulls are spaced 1,000 Hz, and peaks are spaced
1,000 Hz apart. Of course, there is a peak between each adjacent pair
of nulls at which the two signals are in phase. Adding two sine waves
with the same frequency, the same amplitude, in phase, doubles the
amplitude, yielding a peak 6 dB higher than either component by itself
S
F I G U R E 1 7 - 1 4
The flush-mounted microphone. Sounds from the source S that strike the surface do
not reach the microphone, thus avoiding comb-filter effects. Another advantage of
this mounting is an increase of level due to the pressure buildup near the reflecting
surface.
Table 17-3 Comb-filter peaks and nulls.
Frequency of Spacing between nulls Spacing between peaks
Delay (ms) lowest null (Hz) (Hz)
0.1 5,000 10,000
0.5 1,000 2,000
1. 500 1,000
5. 100 200
10. 50 100
50. 10 20
382 CHAPTER SEVENTEEN
(20 log 2  6.02 dB). The nulls, of course,
will be at a theoretical minimum of minus
infinity as they cancel at phase opposition.
In this way, the entire response curve can
be sketched as the phase of the two waves
alternates between the in-phase and the
phase-opposition condition down through
the spectrum.
An important point to observe is that
the 1/(2t) expression above gives a null at
500 Hz, which robs energy from any distributed
signal subject to that delay. A
music or speech signal passing through a
system having a 1-ms delay will have
important components removed or
reduced. This is nothing short of signal
distortion, hence the common phrase
comb-filter distortion.
If the mathematics of the 1/t and the
1/(2t) functions seem too laborious, Figs.
17-15 and 17-16 are included as graphical
solutions.
Peak height
Null depth
10
8
6
4
2
0
2
4
6
8
10
12
14
16
18
20
Level change - dB
0.2 0.4 0.6 0.8 1.0
Amplitude ratio
F I G U R E 1 7 - 1 5
The effect of amplitude ratios on comb-filter peak
height and null depth.
COMB-FILTER EFFECTS 383
First cancellation
Second cancellation
Third cancellation
Fourth cancellation
10 k
5 k
2 k
1 k
500
200
100
Frequency - Hz
0.1 0.2 0.5 1 2 5 10
Delay - ms
F I G U R E 1 7 - 1 6
The magnitude of the delay determines the frequencies at which destructive interference
(cancellations) and constructive interference (peaks) occur. The broken lines indicate
the peaks between adjacent cancellations.
Endnotes
1Bartlett, Bruce, A Scientific Explanation of Phasing (Flanging), J. Audio Eng. Soc., 18, 6
(1970) 674–675.
2The New Stereo Soundbook by Ron Streicher and F. Alton Everest, Audio Engineering
Associates, 1029 N. Allen Ave., Pasadena, CA 91104.
3Moore, Brian C.J. and Brian Glasberg, Suggested Formulae for Calculating Auditory-Filter
Bandwidths, J. Acous. Soc. Am., 74, 3 (Sept 1983) 750–753.
4Blauert, Jens, Spatial Hearing, (1983), Cambridge, MA, MIT Press, 325–326.
5Burroughs, Lou, Microphones: Design and Application, Plainview, NY, Sagamore
Publishing Co., (1974), Chapters 10 and 11.F I G U R E 1 7 - 1
384 CHAPTER SEVENTEEN
The background noise levels in recording studios, control rooms, and
listening rooms must be kept under control if these rooms are to be
of maximum use in their intended way. Hums, buzzes, rumbles, aircraft
noises, tooting auto horns, dogs barking, or typewriter sounds are
most incongruous if audible during a lull in a program. Such sounds
might not be noticed outside the studio when they are a natural part of
the situation, but during a pause or a quiet musical or speech passage
they stand out like the proverbial sore pollex.
In a studio, interfering sounds can come from control-room monitors
operating at high level or from equipment in adjacent areas. Control
rooms have their own noise problems, some intruding from the
outside, some generated by recorders, equipment cooling fans, etc.
There is one source of noise, however, that is common to all soundsensitive
rooms, and that is the noise coming from the air-conditioning
diffusors or grilles, the subject of this chapter. A certain feeling of
helplessness in approaching air-conditioning noise problems is widespread
and quite understandable.
The control of air-conditioning noise can be expensive. A noise specification
in an air-conditioning contract for a new structure can escalate
the price. Alterations of an existing air-conditioning system to correct
high noise levels can be even more expensive. It is important for studio
designers to have a basic understanding of potential noise problems in
Quiet Air
for the Studio
18
385
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
386 CHAPTER EIGHTEEN
air-conditioning systems so that adequate control and supervision can
be exercised during planning stages and installation. This applies
equally to the most ambitious studio and to the budget job.
Selection of Noise Criterion
The single most important decision having to do with background
noise is the selection of a noise-level goal. The almost universal
approach to this is embodied in the family of Balanced Noise Criterion
(NCB) curves of Fig. 18-1.1 The selection of one of these contours
establishes the goal of maximum allowable noise-pressure level in
each octave band. Putting the noise goal in this form makes it easily
checkable by instruments. The downward slope of these contours
reflects both the lower sensitivity of the human ear at low frequencies
and the fact that most noises with distributed energy drop off with frequency.
To determine whether the noise in a given room meets the
contour goal selected, sound-pressure level readings are made in each
octave and plotted on the graph of Fig. 18-1. The black dots represent
such a set of measurements made with a sound-level meter equipped
with octave filters. A convenient single number NCB-20 applies to this
particular noise. If the NCB-20 contour had been specified as the highest
permissible sound-pressure level in an air-conditioning contract,
the above installation would just barely be acceptable.
Which contour should be selected as the allowable limit for background
noise in a recording studio? This depends on the general studio
quality level to be maintained, on the use of the studio, and other
factors. There is little point in demanding NCB-15 from the air-conditioning
system when intrusion of traffic and other noise is higher
than this. In general, NCB-20 should be the highest contour that
should be considered for a recording studio or listening room, and
NCB-15 is suggested as a practical and attainable design goal for the
average studio. NCB-10 would be excellent and it would probably
take special effort and expense to reduce all noise to this level.
The advent of digital recording has changed our view of which NCB
contour to select as a goal. Figure 5-12 showed that signal-to-noise
ratios of more than 100 dB must be accommodated, for the top-flight
studios at least. This means a lower noise floor. A lower noise floor
means tighter construction practices and HVAC (heating, ventilating,
QUIET AIR FOR THE STUDIO 387
and air-conditioning) contract noise specifications that substantially
increase the cost.
If you want to know just how an NCB-15 or NCB-20 background
noise really sounds, the following procedure is suggested. Beg, borrow,
buy, or rent a sound-level meter with built-in octave filters. Measure
Approximate
threshold of
hearing for
continuous
noise
63 125 250 500 1,000 2,000 4,000 8,000
Octave band center frequencies, Hz
NC 70
NC 60
NC 50
NC 40
NC 30
NC 20
Octave band sound-pressure level, dB re 20 µPa
20
30
40
50
60
70
80
90
10
A
B
F I G U R E 1 8 - 1
Balanced noise-criterion (NCB) curves for occupied rooms. Octave-band sound-pressure
levels of the magnitudes indicated in the regions A and B may induce audible rattles or
feelable vibrations in lightweight partitions and ceiling constructions. After Beranek.1
388 CHAPTER EIGHTEEN
the sound levels in several studios that you suspect have high noise or
studios you consider quite acceptable. By the time you have measured
four or five such rooms with the HVAC turned on and off, you will have
little NCB numbers in your head for ready recall in future discussions
and you, too, will have become an expert.
Fan Noise
The fan is a chief contributor to HVAC noise in the studio, but it is by
no means the only contributor. The sound power output of the fan is
largely fixed by the air volume and pressure required in the installation,
but there are certainly variations between the types of fans. Figure
18-2 gives the specific sound power output of just two types of
fans: the airfoil centrifugal and the pressure blower. Specific sound
power level means that the measurements have been reduced to the
standard conditions of 1 cu ft per minute and a pressure of 1 in of
water. On this basis, noise of various types of fans can be compared
equitably.
The centrifugal fan is one of the quietest fans available. The amazing
thing is that large fans are quieter than small fans. This is also true
of the pressure blower fan as shown in Fig. 18-2.
Aside from the term specific, sound power in general is foreign to
the audio field. All the acoustical power radiated by a piece of machinery
must flow out through a hemisphere. The manufacturer is the
source of noise data on any particular fan. Sound power is usually evaluated
by sound-pressure level readings over the hemisphere. Sound
power is proportional to sound pressure squared. Noise ratings of fans
in terms of sound power can be converted back to sound-pressure levels
applicable to a given room by means of the following formula:
Sound-pressure level = sound-power level – 5 log V – 3 log f – 10 log
r  25 dB
where
V = room volume, cu ft
f = octave-band center frequency, Hz
r = distance source to reference point, ft
There is usually a tone generated by the fan with a frequency given by
(rev/sec) (number of blades). This tone adds to the level of the octave
QUIET AIR FOR THE STUDIO 389
band in which it falls. Three dB of the centrifugal
fans and eight dB for the pressure
fans should be added to the one octave
band level to account for the contribution
of the fan tone.
ASHRAE
It might sound like an Egyptian goddess,
but our new word for today is ASHRAE (as
in ashtray). It stands for the American
Society of Heating, Refrigerating, and Air-
Conditioning Engineers. Although its primary
purpose is to enlighten and
standardize its own engineers, this highly
respected organization is a prolific source
of help for the studio designer who is a
novice in HVAC. In this brief chapter it is
impractical to go deeply into specific
design technologies, but the ASHRAE
Handbooks can be faithful guides to those
faced with such problems.2 Chapter 7 of
this handbook entitled Sound and Vibration
Fundamentals (1985) introduces fundamental
principles, including the
source/path/receiver concept; basic definitions and terminology; and
acoustical design goals.
Chapter 32 (1984) covers Sound and Vibration Control. The objective
of this chapter is to achieve a noise level appropriate for the functions
of the space—not the lowest possible level. Overdesign is as
unforgivable as underdesign. The material in chapter 32 includes:
noise control for outdoor equipment installations; system noise control
for indoor air-conditioning systems; general design considerations
for good noise control; mechanical-equipment-room noise isolation;
vibration isolation and control; and troubleshooting for noise and
vibration problems.
These two chapters are excellent sources for the designer or manager
of any kind of audio room. The Handbook is massive and expensive, but
Pressure blower
Small Large
Large
Small
Airfoil centrifugal
Octave band center frequency - Hz
Sound-power level - dB re 1 pw
F I G U R E 1 8 - 2
Noise sound-power output of different types of fans
commonly used for air-conditioning. (Data are from
Table 22-2.2)
390 CHAPTER EIGHTEEN
it is available in local libraries leaning toward service to industries. For
those with some engineering training and a modicum of determination,
these chapters in the Handbook can provide a background for intelligent
dealing with the highest caliber of air-conditioning contractors. For the
lower caliber type, this background is indispensable for avoiding big
and expensive mistakes.
Machinery Noise
The first step in the reduction of HVAC noise in the studio is wisely
locating the HVAC machinery. If this is left to chance, Murphy’s Law
will result in the equipment room being adjacent to or on the roof
directly above the studio. The wall or roof panel, vibrating like a giant
diaphragm, is remarkably efficient in radiating airborne noise into the
studio. So, step number one is to locate the equipment as far removed
from the sound-sensitive areas as possible.
The next step is to consider some form of isolation against structure-
borne vibration. If the equipment is to rest on a concrete slab
shared with the studio and plans are being drawn up, the machineryroom
slab should be isolated from the main-floor slab. Compressed
and treated glass fiber strips are available that are suitable for separating
slabs during pouring. Other precautions would include proper
vibration isolation mounts, designed accurately or they will be useless
or downright damaging to the situation. Flexible joints in pipes and
ducts where they leave the machinery room might be advisable.
Air Velocity
In air distribution systems, the velocity of air flow is a very important
factor in keeping HVAC noise at a satisfactorily low level. Noise generated
by air flow varies approximately as the 6th power of the velocity.
As air velocity is doubled, the sound level will increase about 16
dB at the room outlet. Some authorities say that air-flow noise varies
as the 8th power of the velocity and give 20 dB as the figure associated
with doubling or halving the air velocity.
A basic design parameter is the quantity of air the system is to deliver.
There is a direct relationship between the quantity of air, air velocity, and
size of duct. The velocity of the air depends on the cross-sectional area of
QUIET AIR FOR THE STUDIO 391
the duct. For example, if a system delivers 500 cu ft/minute and a duct
has 1 sq ft of cross-sectional area, the velocity is 500 ft/min. If the area is
2 sq ft, the velocity is reduced to 250 ft/min; if 0.5 sq ft, velocity is
increased to 1,000 ft/min. An air velocity maximum of 500 ft/min is suggested
for broadcast studios, and this value is also about right for topflight
recording studios and other critical spaces. Specifying a low
velocity eliminates many headaches later.
High-pressure, high-velocity, small-duct systems are generally less
expensive than low-velocity systems. True budget systems commonly
ensure noise problems in studios because of high air velocity and its
resulting high noise. Compromise can be made by flaring out the ducts
just upstream from the grille. The increasing cross-sectional area of the
flare results in air velocity at the grille being considerably lower than
in the duct feeding it.
Effect of Terminal Fittings
Even if fan and machinery noise are sufficiently
attenuated, by the time the air reaches
the sound-sensitive room, air turbulence
associated with nearby 90° bends, dampers,
grilles, and diffusors can be serious noise
producers as suggested by Fig. 18-3.
“Natural” Attenuation
The designer must be careful to avoid
expensive overdesign of an air distribution
system by neglecting certain attenuation
effects built into the system. When a plane
wave sound passes from a small space,
such as a duct, into a larger space, such as a
room, some of the sound is reflected back
toward the source. The effect is greatest for
low-frequency sound. Recent research has
also indicated that the effect is significant
only when a straight section of ductwork,
three to five diameters long, precedes the
A
B
C D
F I G U R E 1 8 - 3
Air turbulence caused by discontinuities in the flow
path can be a serious producer of noise. (A) 90-
degree miter bend. (B) Damper used to control quantity
of air. (C) Sound radiated from duct walls set
into vibration by turbulence or noise inside the duct.
(D) Grilles and diffusors.
392 CHAPTER EIGHTEEN
duct termination. Any terminal device,
such as a diffusor or grille, tends to nullify
this attenuation effect. A 10-in duct dumping
air into a room without a grille can give
rise to a 15-dB reflection loss in the 63-Hz
octave. This is about the same attenuation
that a 50- to 75-ft run of lined ductwork
would give. Figure 18-4 shows several
methods of attenuation.
A similar loss occurs at every branch or
takeoff. There is also an attenuation of
noise in bare, rectangular sheet ducts due to
wall flexure amounting to 0.1 to 0.2 dB per
foot at low frequencies. Round elbows
introduce an attenuation, especially at the
higher frequencies. Right-angle elbows also
introduce attenuation (see Fig. 18-5). All of
these losses are built into the air-handling
system and serve to attenuate fan other
noise coming down the duct. It’s there, it’s
free, so take it into consideration to avoid
overdesign.
Duct Lining
The application of sound-absorbing materials
to the inside surfaces of ducts is a
standard method of reducing noise levels.
Such lining comes in the form of rigid
boards and blankets and in thicknesses of
1⁄2-inch to 2-inches. Such acoustical lining
also serves as thermal insulation when it is
required. The approximate attenuation
offered by 1-in duct lining in typical rectangular
ducts depends on the duct size as
shown in Fig. 18-6. The approximate
attenuation of round ducts is given in Fig.
18-7. Duct attenuation is much lower in
A B
C D
F I G U R E 1 8 - 4
Air turbulence noise can be materially reduced by
(A) use of deflectors, (B) radius bends, (C) airfoils,
and (D) carefully shaped grilles and diffusors.
0
125 250 500 1 kHz 2 kHz 4 kHz 8 kHz 63
10
20
30
40
50
2,000 ft /min
700 ft /min
Without
vanes
Without
vanes
With
vanes
With
vanes
Frequency – Hz
Octave-sound power level - dB
F I G U R E 1 8 - 5
Noise produced by 12” x 12” square cross ssection
90-degree-miter elbow with and without deflection
vanes and at 2,000 and 700 fpm air velocities. These
curves have been calculated following the procedures
in Ref.2.
QUIET AIR FOR THE STUDIO 393
the round ducts than in lined rectangular
ducts comparable cross-sectional areas.
Plenum Silencers
A sound-absorbing plenum is an economical
device for achieving significant attenuation.
Figure 18-8 shows a modest-sized
plenum chamber, which if lined with 2-in
thickness of 3 lb/cu ft density glass fiber,
will yield a maximum of about 21-dB attenuation.
The attenuation characteristics of
this plenum are shown in Fig. 18-9 for two
thicknesses of lining. With a lining of 4-in
of fiberboard of the same density, quite uniform
absorption is obtained across the audible
band. With 2 inches of glass fiberboard,
attenuation falls off below 500 Hz. It is
apparent that the attenuation performance
of a plenum of given size is determined primarily
by the lining.
Figure 18-10 gives actual measurements
on a practical, lined-plenum muffler
approximately the same horizontal dimensions
as that of Fig. 18-8, but only half the
height and with baffles inside. Attenuation
of 20 dB or more above the 250-Hz octave
was realized in this case, and it solved an
otherwise intolerable problem.
Plenum performance can be increased
by increasing the ratio of the cross-sectional
area of the plenum to the cross-sectional
area of the entrance and exit ducts,
and by increasing the amount or thickness
of absorbent lining. A plenum located at
the fan discharge can be an effective and
economical way to decrease noise entering
the duct system.
1" Duct liner
6" 12"
12" 24"
24" 36"
6
5
4
3
2
1
0
Approximate attenuation - dB/Ft.
63 125 250 500 1 k 2 k 4 k 8 k
Frequency - Hz
F I G U R E 1 8 - 6
Attenuation offered by 1” duct lining on all four
sides of rectangular duct. Dimensions shown are the
free area inside the duct with no air flows. This is a
plot of the data from Ref.2.
63 125 250 500 1 k 2 k 4 k 8 k
24"dia
6"dia
12"dia
Frequency - Hz
Approximation attenuation - dB/Ft.
3
2
1
0
F I G U R E 1 8 - 7
Attenuation offered by spiral wound round ducts with
perforated spiral wound steel liner. Dimensions
shown are the free area inside the duct, with no air
flow. This is a plot of the data from Ref.2.
394 CHAPTER EIGHTEEN
Packaged Attenuators
Numerous packaged proprietary noise
attenuators are available.3 Cross sections
of several types are shown in Fig. 18-11
with their performance plotted below. For
comparison, the attenuation of the simple
lined duct is given in curve A. Some of the
other attenuators have no line-of-sight
through them, i.e., the sound must be
reflected from the absorbing material to
traverse the unit and hence will have
somewhat greater attenuation. The
absorbing material is usually protected by
perforated metal sheets in these packaged
silencers. The attenuation of such units is
very high at midband speech frequencies
but not as good at low frequencies.
Reactive Silencers
Several passive, absorptive silencers have
been considered that rely for their effectiveness
on the changing of sound energy into
heat in the interstices of fine glass fibers.4
Another effective principle used in silencers
is that of the expansion chamber as shown
in Fig. 18-12. This type performs by reflecting
sound energy back toward the source,
thereby canceling some of the sound energy.
Because there is both an entrance and exit
discontinuity, sound is reflected from two points. Of course, the destructive
interference (attenuation nulls) alternates with constructive interference
(attenuation peaks) down through the frequency band, the
attenuation peaks becoming lower as frequency is increased. These peaks
are not harmonically related, therefore they would not produce high
attenuation for a noise fundamental and all its harmonics, but rather
attenuate slices of the spectrum. By tuning, however, the major peak can
eliminate the fundamental while most of the harmonics, of much lower
5' 5'
24"
24"
12"
12"
3'
F I G U R E 1 8 - 8
Dimensions of a plenum that will yield about 21 db
attentuation throughout much of the audible spectrum.
63 125 250 500 2 kHz 1 kHz 4 kHz 8 kHz
Frequency - Hz
0
5
10
15
20
25
Attenuation - dB
4"
2" Lining
F I G U R E 1 8 - 9
Calculated attenuation characteristics of the plenum
of Fig. 18-8 lined with 2” and 4” glass of 3 lb/cu ft
density. Calculations follow ASHRAE procedures.
QUIET AIR FOR THE STUDIO 395
amplitude, would receive some attenuation.
By putting two reactive silencers of this type
in series and tuning one to fill in the nulls of
the other, continuous attenuation can be
realized throughout a wide frequency range.
No acoustical material is required in this
type of silencer, which operates like an automobile
muffler.
Resonator Silencer
The resonator silencer illustrated in Fig. 18-
13 is a tuned stub that provides high attenuation
at a narrow band of frequencies. Even a
small unit of this type can produce 40- to 60-
dB attenuation. This type of silencer offers
little constriction to air flow, which can be a
problem with other types of silencers.
Duct Location
Why build an STC-60 dB wall between studio and control room, for
instance, and then serve both rooms with the same supply and exhaust
ducts closely spaced as in Fig. 18-14? This is a tactical error that
results in a short path speaking tube from one room to the other, nullifying
quite effectively the 60-dB wall. With an untrained air-conditioning
contractor doing the work without a supervisor sensitive to the
acoustical problem, such errors can easily happen. To obtain as much
as 60-dB attenuation in the duct system to match the construction of
the wall requires the application of many of the principles discussed
earlier. Figure 18-15 suggests two approaches to the problem; to separate
grilles as far as possible if they are fed by the same duct, or better
yet, to serve the two rooms with separate supply and exhaust ducts.
Some Practical Suggestions
 The most effective way to control air-flow noise is to size the
ducts so as to avoid high velocities. The economy of the smaller
ducts, however, may be more than enough to pay for silencers to
Sound level at
muffler input
Sound level
5' from
supply grille
31.5 63 125 250 500 1 k 2 k 4 k 8 k 16 k
0
20
30
40
50
60
70
Frequency - Hz
Octave sound-pressure level - decibels
A
F I G U R E 1 8 - 1 0
Measured noise sound pressure levels. (A) At plenum
input. (B) In room 5 feet from the supply grilles.
This plenum is approximately the size of that of Fig.
18-8 but only half the height and it contains baffles.
396 CHAPTER EIGHTEEN
63 125 250 500 1 k 2 k 4 k 8 k
0
10
30
20
40
50
Frequency - Hz
Noise reduction - dB
A
B
C
D
12"
12"
12"
14"
24"
24"
24"
34"
48"
46"
A C
B D
F I G U R E 1 8 - 1 1
Attenuation characterisitcs of three packaged silencers compared to that of the lined
duct, curve A. Adapted from Doelling.3
QUIET AIR FOR THE STUDIO 397
bring the higher noise to tolerable
levels.
 Right-angle bends, dampers, etc.,
create noise due to air turbulence.
Locating such fittings 5 to 10 diameters
upstream from the outlet allows
the turbulence to smooth out.
 Noise and turbulence inside a duct
cause the duct walls to vibrate and
radiate noise into surrounding
areas. Rectangular ducts are worse
offenders than round ones. Such
noise increases with air velocity
and duct size, but can be controlled
with external treatment of thermal
material.
 Acoustical ceilings are not good
sound barriers, hence in a soundsensitive
area, the space above a layin
ceiling should not be used for
high-velocity terminal units.
/4
/4 /2  3/2 2
D
0
10
20
30
Attenution, dB
F I G U R E 1 8 - 1 2
Attenuation characteristics of a reactive type of
silencer, the expansion chamber. Sound is attenuated
by virtue of the energy reflected back toward the
source, canceling some of the oncoming sound.
Adapted from Sanders.4
Frequency
Attenuation, dB
0
10
20
30
40
F I G U R E 1 8 - 1 3
Attenuation characteristics of the tuned-stub silencer. Adapted from Sanders.4
398 CHAPTER EIGHTEEN
 The ear can detect sounds far below the
prevailing NCB contour noise (see Fig.
18-1). The goal should be to reduce
noise in the studio to a level at which it
cannot be heard on a playback of a
recording at normal level and without
noise reduction.
 Plenums are effective and straightforward
devices adaptable to studio quieting
programs, and they offer
attenuation throughout the audible
spectrum. They are especially effective
at the fan output.
 Some of the noise energy is concentrated
in the highs, some in the lows.
There must be an overall balance in the
application of silencers so that the
resulting studio noise follows roughly
the proper NCB contour. Otherwise
overdesign can result.
Endnotes
1Beranek, Leo L., Balanced Noise Criterion (NCB)
Curves, J. Acous. Soc. Am., 86,2 (Aug 1989) 650-664.
2Anon., Sound and Vibration Control, ASHRAE
Handbook, 1984 Systems, Chapter 32. Refer also to
ASHRAE annual Product Directories and Systems.
Published by Am Soc. Heating, Ventilating, and Air-
Cond. Eng., 1791 Tullie Circle, N.E., Atlanta, GA 30329.
3Doelling, Norman, How Effective Are Packaged
Attenuators?, ASHRAE Journal, 2, 2 (Feb 1960) 46-50.
4Sanders, Guy J., Silencers: Their Design and
Application, Sound and Vibration, 2, 2 (Feb 1968) 6-13.
Control
room
Studio
60 - dB wall
a/c duct
F I G U R E 1 8 - 1 4
An expensive, high-transmission-loss wall can be
bypassed by sound traveling over a short duct path
from grille to grille.
A
B
F I G U R E 1 8 - 1 5
Two possible solutions to the problems of Fig. 18-14.
(A) Separate grilles as far as possible. (B) Feed each
room with a separate duct. Both approaches increase
duct path length and thus duct attenuation.
In the present context, “Listening Room” is taken as the normal home
high-fidelity music center (or, possibly, the listening area in a professional
work room). The control room or monitoring room of a recording
studio is a very special type of listening room, which is treated in
Chap. 21. This chapter considers that portion of a home set up for
enjoyment of recorded music. Families having the means to dedicate a
certain space exclusively for music are fortunate. The rest of us must
be satisfied with a multipurpose space, typically a living room serving
also as a listening room.
Ideally, all members of the household will be of one mind in such a
project. Realistically, there must be compromises between the technical
and the aesthetic. Just where the compromise point comes to rest
(if ever) may turn out to be a measure of the relative force of personalities
and strength of wills. If the persons involved share a genuine
appreciation of good music, the war is over except for a few skirmishes
along the way.
The Acoustical Link
The acoustics of the space is a vital part of both the recording and reproducing
process. In every acoustical event, there is a sound source and
some sort of receiving device with an acoustical link between the two.
Acoustics of the
Listening Room
19
399
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
400 CHAPTER NINETEEN
Disc or tape recordings have the imprint of the acoustics of the recording
environment recorded on them. If the sound source is a symphony
orchestra and the recording is made in the performing hall, the reverberation
of the hall is very much a part of the orchestral sound. If the reverberation
time of the hall is 2 seconds, a 2-second tail is evident on every
impulsive sound and sudden cessation of music, and it affects the fullness
of all the music. In playing this record in our home listening area,
what room characteristics will best complement this type of music?
Another recording may be of the popular type. This music was probably
recorded in a very dead studio by a multitrack system. Basic rhythm
sections playing in this very dead studio and well separated acoustically
are laid down on separate tracks. During subsequent sessions the vocals
and other instruments are recorded on still other tracks. Finally, all are
combined at appropriate levels in a mix down, with a bit of “sweetening”
added. The position in the stereo field of the sounds on each track is
adjusted by turning a panpot knob. In the mix down, many effects,
including artificial reverberation, are added. What listening room characteristics
are best for playback of this recording?
If the taste of the high-fidelity enthusiast is highly specialized, the
listening-room characteristics can be adjusted for relative optimum
results for one type of music. If the taste is more universal, the acoustical
treatment of the listening room might need trimming for different
types of music.
The dynamic range of reproduced music in a listening room depends
at the loud extreme on amplifier power, the power-handling ability of
the loudspeakers, and the tolerance of family members and neighbors.
The social limitation usually comes into play at a level far lower than
the average electronic and transducer limitation. The low-level end of
the dynamic range scale is limited by noise, environmental or electronic.
Household noise usually determines the lower limit. The usable
range between these two extremes is far, far less than, say, the range of
an orchestra in a concert hall. Expanders, compressors, companders,
etc., are capable of restoring some of the original range (see Fig. 5-12).
Peculiarities of Small-Room Acoustics
The ten-octave spread of the audible spectrum is so great that the acoustical
analysis of small rooms is quite different from that for large rooms.
ACOUSTICS OF THE LISTENING ROOM 401
The reason is apparent when room size is considered in terms of the
wavelength of sound. The 20-Hz to 20-kHz audio band covers sound
wavelengths from 56.5 ft to 0.0565 ft (11⁄16 inch). Below about 300 Hz
(wavelength 3.8 ft), the average studio or listening room must be considered
as a resonant cavity. It is not the studio that resonates, it is the air
confined within the studio. As frequency increases above 300 Hz the
wavelengths become smaller and smaller with the result that sound may
be considered as rays and specular reflections prevail.
In this book, small rooms such as listening rooms, audio workrooms,
control rooms, and studios are emphasized. The design of large
rooms such as concert halls, theaters, and auditoriums is left to the
professional acoustical consultant.
Reflections of sound from the enclosing surfaces dominate both the
low- and high-frequency regions. At the lower frequencies reflections
result in standing waves, and the room becomes a chamber resonating at
many different frequencies. Reflections of sound from the room surfaces
also dominate at the midband and higher audible frequencies, without the
cavity resonances, but with specular reflections as the major feature.
For the professional acoustician and the critical audiophile, the
listening room is as much of a challenge as the design of a professional
recording studio, but it has not received as much attention.
All the major acoustical problems are involved in the design of a listening
room and any other small audio room. The acoustics of the
listening room in this chapter are therefore considered as an introduction
to the acoustics of other types of small audio spaces in following
chapters.
Room Size
Problems are inevitable if sound is recorded or reproduced in spaces that
are too small. Gilford1 states that studio volumes less than approximately
1,500 cu ft are so prone to sound coloration that they are impractical.
Rooms smaller than this produce sparse modal frequencies with exaggerated
spacings, which are the source of audible distortions.
Room Proportions
Chapter 13, especially Table 13-2, lists the room proportions yielding
the most favorable distribution of room modes. With new construction
it is strongly advised to use these proportions as a guide, confirming
402 CHAPTER NINETEEN
any dimensions seriously considered by calculation and close study of
the spacing of axial-mode frequencies.
In the home listening room the room shape and size in most cases are
already fixed. The existing room dimensions should then be used for axial
mode calculations after the pattern of Table 19-1. A study of these modal
Table 19-1 Axial modes.
Room dimensions  21.5' × 16.5' × 10.0'
Length Width Height Axial
L = 21.5 ft W = 16.5 ft H = 10.0 ft Arranged in mode
f1 = 565/L f1 = 565/W f1 = 565/H ascending spacing
Hz Hz Hz order Hz
f1 26.3 34.2 56.5 26.3 7.9
f2 52.6 68.4 113.0 34.2 18.4
f3 78.9 102.6 169.5 52.6 3.9
f4 105.2 136.8 226.0 56.5 11.9
f5 131.5 171.0 282.5 68.4 10.5
f6 157.8 205.2 339.0 78.9 23.7
f7 184.1 239.4 102.6 2.6
f8 210.4 273.6 105.2 7.8
f9 236.7 307.8 113.0 18.5
f10 263.0 131.5 5.3
f11 289.3 136.8 21.0
f12 315.6 157.8 11.7
169.5 1.5
171.0 13.1
184.1 21.1
205.2 5.2
210.4 15.6
226.0 10.7
236.7 2.7
239.4 23.6
263.0 10.6
273.6 8.9
282.5 6.8
289.3 18.5
307.8
Mean axial mode spacing  11.7 Hz
Standard deviation  6.9 Hz
ACOUSTICS OF THE LISTENING ROOM 403
frequencies will then reveal the presence of coincidences (two or more
modes at the same frequency) or isolated modes spaced 25 Hz or more from
neighbors. Such faults pinpoint frequencies at which colorations may
occur.
Reverberation Time
Reverberation time has been demoted from being a primary determinant
of acoustical quality of small rooms to a nonentity. The amount of
overall absorbance in a listening room is still important in establishing
the general listening conditions. If the room is excessively dead or too
live and reverberant, listener fatigue might develop and music quality
may deteriorate.
The old familiar Sabine equation for reverberation time (Eq. 7-1, of
Chap. 7) makes possible an estimate of the amount of absorbing material
required for a reasonable reverberant condition. Even though we
place little importance on the reverberation time figure itself, it is expedient
to assume a reasonable reverberation time, say about 0.3 sec, for
the purposes of these calculations. From this, the total number of
sabins of absorption can be estimated, which would result in reasonable
listening conditions. In most home listening/living rooms, the
structure and the furnishings often supply most of the basic absorbance
required. Careful listening tests must determine the degree of room
ambience most suitable for the favorite type of music.
The Listening Room: Low Frequencies
The bare room of Fig. 19-1 is the starting point. The room is 21.5 ft
long, 16.5 ft wide, with a 10-ft ceiling. These dimensions fix the
axial mode resonances and their multiples (harmonics). Following
discussions in Chap. 15, axial mode effects will be emphasized and
tangential and oblique modes will be neglected. Axial mode frequencies
out to 300 Hz were calculated for the length, width, and
height dimensions and tabulated in Table 19-1. These axial modal
frequencies were then arranged in ascending order of frequency,
irrespective of the dimensional source (length, width, or height).
Spacings between adjacent modes were than entered in the righthand
column. No coincidences are noted; only a single pair are as
close as 1.5 Hz. The dimensional ratios 1 : 1.65 : 2.15 are well within
the “Bolt area” of Fig. 13-6.
404 CHAPTER NINETEEN
The calculated axial modal frequencies listed in Table 19-1 must
now be applied to the listening room space. This is done graphically
in Fig. 19-2, following the “stylized impressions” of Toole.5 The positions
of the nulls locate null-lines, which are drawn through the listening
room. Lines representing the length-mode nulls are drawn
through both the elevation view and the plan view because these nulls
actually form a null “sheet” that extends from floor to ceiling. In other
words, the position of the listener’s seat can be moved to avoid these
particular nulls at 26, 53, and 79 Hz, but remember, there are 8 more
below 300 Hz.
The three lowest axial-mode nulls associated with the height of the
room (56, 113, and 170 Hz), are sketched on the elevation view of Fig.
19-2. These nulls are horizontal “sheets” at various heights. The head
of the listener in the elevation view lies between two nulls and at the
peak of the 79-Hz resonance.
The three axial modes of lowest frequency are sketched on the plan
view. The nulls in this case are vertical “sheets” extending from floor
21' - 6"
16' - 6"
10' - 0"
F I G U R E 1 9 - 1
Dimensions of listening room assumed for analysis.
ACOUSTICS OF THE LISTENING ROOM 405
to ceiling. The listener, situated on dead center of the room, intercepts
the nulls of every odd axial mode.
The resonance nulls have been sketched because their location is
definite, but between any two nulls of a given axial mode, a peak
exists. Although nulls are capable of removing a sizeable chunk of
1, 0, 0
(26 Hz)
3, 0, 0
(79 Hz)
2, 0, 0
(53 Hz)
0, 0, 3
(170 Hz)
0, 0, 2
(113 Hz)
0, 0, 1
(56 Hz)
0, 3, 0
(103 Hz)
0, 2, 0
(68 Hz)
0, 1, 0
(34 Hz)
Elevation
Plan
F I G U R E 1 9 - 2
Plan and elevation views of listening room of Fig. 19-1 showing the low-frequency
modal pressure distributions of the first three axial modes of Table 19-1.
406 CHAPTER NINETEEN
spectrum, the low-frequency acoustics of the room is dominated by
the wide, relatively flat peaks.
The extreme complexity of the modal structure of listening room
acoustics is becoming apparent. Only the first three axial modes of
length, width, and height have been shown. All of the axial modal frequencies
listed in Table 19-1 have an important part to play in the lowfrequency
acoustics of the space. These axial modes exist only when
they are excited by the low-frequency sound of the signal being reproduced
in the room. The spectrum of the music is continually shifting,
therefore, the excitation of the modes is also continually shifting. The
length axial mode at 105.2 Hz (Table 19-1) comes alive only as spectral
energy in the music hits 105.2 Hz. If only our eyes were sensitive to
sound pressure (intensity) and frequency (color), what an awesome
sight the constantly shifting room resonances would be as Beethoven’s
Eighth Symphony is reproduced! And all this represents only the
sound energy below about 300 Hz.
Control of Modal Resonances
The low-frequency sound field at the listener’s ears is made up of the
complex vectorial sum of all axial, tangential, and oblique modes at
that particular spot in the room. The loudspeakers energize the modal
resonances prevailing at their locations. The modes that have nulls at a
loudspeaker location cannot be energized, but those having partial or
full maxima at this location will be energized proportionally. The interaction
of low-frequency resonances in the listening room at the loudspeaker
and listening positions is too complex and transient to grasp
fully, but they can be understood if broken down into the contributions
of individual modes.
Loudspeakers should be located as far away from reflecting surfaces
as practicable. Loudspeaker positions should be considered tentative,
moving them slightly if necessary to improve sound quality. The same
is true for listening position.
With a basic understanding of the complexity of the low-frequency
modal sound fields of a listening room, one must be amazed that
sound quality is as good and natural as it is.
Bass Traps for the Listening Room
It is not practical to acoustically treat each mode separately. A general
low-frequency treatment is usually sufficient to control “room
ACOUSTICS OF THE LISTENING ROOM 407
boom” and other resonance anomalies. In fact, a room of wood-frame
construction could very well have sufficient structural low-frequency
absorption built in to provide all the general modal control
necessary.
In addition to the adjustment of the general ambience of the
room, low-frequency absorption in the two corners of the room near
the loudspeakers can have an important effect on the stereo image.
Figure 19-3 suggests four ways that such absorption can be obtained.
The first, Fig. 19-3A, is a Helmholtz resonator trap, home-built in
the corner. This could employ either a perforated face or spaced
slats. Some design frequency must be assumed, perhaps 100 Hz, and
an average depth of the triangular shape must be estimated. With
these figures, Eq. 9-5 (for slats) or Table 9-4 (for perforations) can be
used for completing the design. In fact, the design of Fig. 9-24 for a
diaphragmatic absorber is still another one that can be used.
The easiest solution of the problem would be the selection of one of
the three proprietary absorbers of Figs. 19-3B, C, or D. Tube Traps™,
offered by Acoustic Sciences Corporation,2 provide the necessary
A B C D
F I G U R E 1 9 - 3
Four possible ways of providing low-frequency absorption for the corners of the listening
room nearest the loudspeakers. (A) Home built Helmholtz resonator, (B) The use of
Tube Trap™ or (C) Snap Traps™ available from Acoustic Sciences Corporation.2 (D) The
use of Korner Killers™ available from RPG Diffusor Systems, Inc.3
408 CHAPTER NINETEEN
absorption by stacking a 9-in diameter unit on top of an 11-in unit,
resulting in a total height of 6 ft. These traps are fibrous cylinders with
wire mesh skeleton, together acting as resonant cavities. Half of the
periphery is a reflective, limp-mass sheet. Low-frequency energy (below
400 Hz) readily penetrates this sheet while high-frequency energy is
reflected from it. Used in a corner such as this, the cylinder would be
rotated so that this reflective side faces the room. The cylinder thus contributes
to diffusion of the room as well as to deep-bass absorption.
The corner treatment of Fig. 19-3C is based on another product of
Acoustic Sciences Corporation3 called the Snap Trap™. Mounting
tracks of 1" × 1⁄2" J-metal are installed in the corner to hold the edges of
the panel. The Snap Trap™ sheet is then bent and snapped in place.
The air space behind the acoustic panel ensures good low-frequency
absorption. A curved membrane reflector strip within the panel provides
wide-angle reflection above 500 Hz.
Still another possible corner treatment offered by RPG Diffusor Systems,
Inc.3 is shown in Fig. 19-3D. The Triffusor™ is especially
adapted to variable acoustic control with an absorbent side, a diffusing
side, and a reflective side. An adaptation of the Triffusor™, called the
Korner Killer™, has two absorptive sides and one quadratic residue
diffusor side. With the absorptive sides into the corner for modal control,
the diffusive side faces the room. This RPG™ diffusor face not
only diffuses the sound energy falling on it, it also reduces the amplitude
of the energy returned to the room. The nominal dimensions of
the Korner Killer™: height, 4 ft, faces 24 in.
With a pair of any of the devices of Fig. 19-3 in the corners of the
listening room near the loudspeakers, the chances are good that sufficient
modal control is introduced to clean up any stereo image problems
resulting from room resonances. There are two more corners of
the room that could be treated similarly in the unusual case that more
modal control is required.
Modal Colorations
The more obvious source of low-frequency colorations of sound results
from the momentary deviations from flatness of the room response,
which results from concentrations of modes or great spacings between
modes (see Fig. 15-21). Transient bursts of music energy result in
unequal, forced excitation of the modes. As the transient excitation is
ACOUSTICS OF THE LISTENING ROOM 409
removed, each mode decays at its natural (and often different) frequency.
Beats can occur between adjacent decaying modes. Energy at new and
different frequencies is injected, which is a coloration of the signal.
The Listening Room: The Mid-High Frequencies
The propagation of sound of shorter wavelengths, above about 300 Hz,
can be considered in the form of rays that undergo specular reflection.
Figure 19-4 shows the same listening room and listener for the consideration
of the mid-high-frequency reflections of the sound from the
loudspeakers. Sound from the right loudspeaker is studied in detail as
characteristic of the symmetrical room.
The first sound to reach the listener’s ears is the direct sound, travelling
the shortest distance. Reflection F from the floor arrives next.
Reflections from the ceiling (C), the near side wall (W) and the far side
wall (W) arrive later. One other early reflection is the one labelled D
resulting from the diffusion of sound from the edges of the loudspeaker
cabinet (Fig. 11-10). The reflection of this sound from the front
wall is shown in the plan view of Fig. 19-4.
These constitute the early reflections, as contrasted to reflections
from the rear surfaces of the room and general reverberation, arriving
much later. The direct ray carries important information concerning
the signal being radiated. If it is accompanied by the early reflections,
the sharp stereo perception of the direct ray tends to be blurred.
The important research of Olive and Toole4,5 is partially summarized
in Fig. 19-5 (a repeat of Fig. 16-4 for convenience). The variables
here are reflection level and reflection delay. It is instructive to compare
the levels and delays of the early reflections of Fig. 19-4 with those
of the Olive/Toole graphs of Fig. 19-5. Table 19-2 lists estimations of the
level and delay of each of the early reflections identified in Fig. 19-4.
These reflections, plotted on Fig. 19-5, all fall within the audible region
between the reflection threshold and the echo-production threshold.
The pure direct signal is immediately followed by a competitive swarm
of early reflections of various levels and delays, producing comb-filter
distortion.
The need is obvious: Reduce the level of the competing reflections so
that the direct signal stands out in all its beauty—with one exception.
Figure 19-5 is a study of a direct signal and a single lateral reflection.
410 CHAPTER NINETEEN
Allowing a single lateral reflection of adjustable level would place control
of the spaciousness and image effects in the hands of the experimenter/
listener! The potential of this concept is staggering. Therefore,
the next step in improving the listening conditions in this listening
room will be the effective elimination of all of the early reflections,
except those lateral reflections off the left and right side wall that will be
adjusted for optimum sound quality.
21' - 6"
10' - 0"
16' - 6"
Plan
Elevation
Direct
C
F
W
G
F
D
W
F I G U R E 1 9 - 4
Plan and elevation views for listening room showing the early reflections from the floor,
the ceiling, the side walls, and diffusion form edges of the loudspeaker cabinet. The
later reflections F and G are the beginning of the reverberant component.
ACOUSTICS OF THE LISTENING ROOM 411
Identification and Treatment of Reflection Points
One method of reducing the levels of the early reflections is to treat the
entire front portion of the room with sound absorbing material. This
would also kill the lateral reflection and would probably make the
room too dead for listening comfort. The principle recommended here
is to add a minimum of absorbing material to treat only the specific
surfaces responsible for the reflections.
Locating these reflection points is easy with a helper with a mirror.
With the listener/experimenter seated at the “sweet spot” the
assistant moves a mirror on the floor until the observer can see
20
10
0
0 20 40 60 80
10
20
30
40
Reflection
inaudible
Echo region
Image effects(shifting, spreading)  spaciousness
Spaciousness
x
x
x
x
xF
DC W
W
Reflection delay - ms
Reflection level - dB
F I G U R E 1 9 - 5
A repeat of Fig. 16-4 for more convenient reference. The small x’s represent the early
room reflections calculated in Table 19-2. After Olive and Toole.5
412 CHAPTER NINETEEN
the tweeter of the right loudspeaker reflected in it. This is the point
on the floor where the floor reflection hits. This point is carefully
marked and the procedure repeated for the tweeter of the left loudspeaker,
and the second floor reflection spot is marked. A small rug
covering these two marks should reduce the floor reflections to
inaudibility.
The same procedure is carried out for locating the reflection points
for the left and right side-wall and the ceiling reflections. Each of these
points should also be covered with enough absorbing material to ensure
ample coverage of the reflection points.
The point of reflection for the sound energy diffracted from the
edges of the loudspeaker cabinet is more difficult to locate. Installing
an absorber on the wall between the loudspeakers should subdue diffraction
reflections.
When all the covered reflection points are in place (Fig. 19-6) try a
preliminary listening test. The stereo image and the music will probably
be much clearer and more precise now that the early reflections
have been reduced.
Table 19-2. Reflections: Their amplitudes and delays.
Path Path Reflection Reflection
Sound length direct level* delay**
paths (ft) (ft) (dB) (ms)
Direct 8.0 0.0 — —
F (Floor) 10.5 2.5 –2.4 2.2
D (Diff) 10.5 2.5 –2.4 2.2
C (Cell) 16.0 8.0 –6.0 7.1
W (near wall) 14.0 6.0 –4.9 5.3
W (far wall) 21.0 13.0 –8.4 11.5
F (rear) 30.6 22.6 –11.7 20.0
G (rear) 44.3 36.3 –14.9 32.1
*Reflection level = 20 log
**Reflection delay =
(Assuming perfect reflection and inverse square propagation.)
(reflected path) – (direct path)
 1,130
direct path
 reflected path
ACOUSTICS OF THE LISTENING ROOM 413
Lateral Reflections: Control of Spaciousness
The lateral reflections from the side walls have been essentially eliminated
by the absorbing material placed on the wall.The next listening test should
be on the same music with the side-wall absorbers temporarily removed,
but with the floor, ceiling, and diffraction absorbers still in place. The
Olive/Toole graphs of Fig. 19-5 can now be tested. Does the fullstrength lateral
reflection give the desired amount of spaciousness and image shifting
or spreading? The lateral reflections can be reduced somewhat by hanging
a cloth instead of the heavy absorber. The adjustment of the magnitude of
F I G U R E 1 9 - 6
The room of Fig. 19-1 with minimum sound absorbing treatment to reduce the level of
the early reflections from room surfaces. The reflectivity of the side-wall absorbers may
be adjusted to control spaciousness and image effects in the listening room. Additional
absorbing material may be needed to adjust the average reverberant character of the
room for best listening.
414 CHAPTER NINETEEN
the lateral reflections can be explored by using sound absorbers of varying
absorbance (velour, heavy cloth, light cloth, oudoor carpet, indoor carpet)
on the side-wall reflection points.
This is a new and highly promising field: The ability to adjust the
lateral reflections to achieve the desired spaciousness and stereo
image effect to suit the individual listener or to optimize conditions
for different types of music.
Endnotes
1Gilford, C.L.S., The Acoustic Design of Talks Studios and Listening Rooms, Proc. Inst. Elect.
Engrs., 106, Part B, 27 (May 1959), 245-258. Reprinted in J. Audio Eng. Soc., 27, 1/2 (1979)
17-31.
2Acoustic Sciences Corporation, P.O. Box 1189, Eugene, OR 97440, telephone 1-800-ASCTUBE,
FAX 503-343-9245.
3RPG Diffusor Systems, Inc., 651-C Commerce Drive, Upper Marlboro, MD 20772, telephone
301-249-0044, FAX 301-249-3912.
4Toole, Floyd E., Loudspeaker and Rooms for Stereophonic Sound Reproduction, Proc.
Audio Eng. Soc. 8th International Conference, Washington, D.C. (1990) 71-91.
5Olive, Sean E., and Floyd E. Toole, The Detection of Reflections in Typical Rooms, J. Audio
Eng. Soc., 37, 7/8 (July/Aug 1989) 539-553.
This is the day of the small recording studio. Musicians are interested
in making demonstration records to develop their style and to sell
their sounds. There are hundreds of small recording studios operated
by not-for-profit organizations that turn out a prodigious quantity of
material for educational, promotional, and religious purposes. Studios
are required for the production of campus and community radio, television,
and cable programs. All of these have limited budgets and limited
technical resources. The operator of these small studios is often
caught between a desire for top quality and the lack of means, and
often the know-how to achieve it. This chap. is aimed primarily to
those in these needy groups, although the principles expounded are
more widely applicable.
What is a good recording studio? There is only one ultimate criterion—
the acceptability of the sound recorded in it by its intended
audience. In a commercial sense, a successful recording studio is one
fully booked and making money. Music recorded in a studio is pressed
on discs or recorded on tape and sold to the public. If the public likes
the music, the studio passes the supreme test. There are many factors
influencing the acceptability of a studio beside sound studio quality,
such as the type of program and the popularity of the performers, but
studio quality is vital, at least for success on a substantial, long-range
basis.
Acoustics of the
Small Recording Studio
20
415
C H A P.
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416 CHAPTER TWENTY
Public taste must be pleased for any studio to be a success. Producing
a successful product, however, involves many individuals along
the way whose decisions may make or break a studio. These decisions
may be influenced by both subjective and technical factors. The
appearance of a studio, convenience, and comfort might outweigh
acoustical quality, sometimes because the more tangible esthetic qualities
are better understood than the intangible acoustical qualities.
This chap. has little to say on the artistic, architectural, and other such
aspects of a studio, but their importance cannot be denied. They just
require a different kind of specialist.
Acoustical Characteristics of a Studio
Sound picked up by a microphone in a studio consists of both direct
and indirect sound. The direct sound is the same as would exist in
the great outdoors or in an anechoic chamber. The indirect sound,
which immediately follows the direct, is the sound that results from
all the various nonfree-field effects characteristic of an enclosed
space. The latter is unique to a particular room and may be called
studio response. Everything that is not direct sound is indirect,
reflected sound.
Before dissecting indirect sound, let us
look at the sound in its all-inclusive form
in a studio, or any other room for that matter.
Figure 20-1 shows how sound level
varies with distance from a source, which
could be the mouth of someone talking, a
musical instrument, or a loudspeaker.
Assume a pressure level of 80 dB measured
1 foot from the source. If all surfaces of the
room were 100 percent reflective, we
would have a reverberation chamber to
end all reverberation chambers, and the
sound pressure level would be 80 dB
everywhere in the room because no sound
energy is being absorbed. There is essentially
no direct sound; it is all indirect.
Graph B represents the fall off in sound
Sound-pressure level - dB
80
70
40
1 2 3 4 5 7 10 15 20 30 40
50
60
(A) All surfaces 100%
reflective
D
Partially
absorptive
C
(B) All surfaces
100% absorptive
(free field)
Distance from source - feet
F I G U R E 2 0 - 1
The sound-pressure level in an enclosed space varies
with distance from the source of sound according to
the absorbency of the space.
ACOUSTICS OF THE SMALL RECORDING STUDIO 417
pressure level with distance from the sound source with all surfaces
100 percent absorptive. In this case all the sound is direct; there is no
indirect component. The best anechoic rooms approach this condition.
It is the true free field illustrated in Chap. 4, and for this condition the
sound pressure level decreases 6 dB for each doubling of the distance.
Between the indirect “all reverberation” case of graph A of Fig. 20-1
and the direct “no reverberation” case of graph B lie a multitude of other
possible “some reverberation” cases, depending on room treatment. In
the area between these two extremes lies the real world of studios in
which we live and move and have our being. The room represented by
graph C is much more dead than that of graph D. In practical studios, the
direct sound is observable a short distance out from the source, but after
that the indirect sound dominates. A sudden sound picked up by a
microphone in a studio would, for the first few milliseconds, be dominated
by the direct component, after which the indirect sound arrives at
the microphone as a torrent of reflections from room surfaces. These are
spread out in time because of the different path lengths traveled.
A second component of indirect sound results from room resonances,
which in turn are the result of reflected sound. The direct sound flowing
out from the source excites these resonances, bringing into play all the
effects listed in Chap. 15. When the source excitation ceases, each mode
dies away at its own natural frequency and at its own rate. Sounds of
very short duration might not last long enough to fully excite room resonances.
Distinguishing between reflections and resonances is an acknowledgment
that neither a reflection concept nor a resonance concept will carry
us through the entire audible spectrum. Resonances dominate the lowfrequency
region in which the wavelengths of the sound are comparable
to room dimensions. The ray concept works for higher frequencies and
their shorter wavelengths (Chap. 16). Around the 300- to 500-Hz region
is a difficult transition zone. But with this reminder of the basic limitations
of our method we can return to analyzing the components of sound
in a small studio.
The third component of indirect sound is involved with the materials of
construction—doors, windows, walls, and floors. These too are set into
vibration by sound from the source, and they too decay at their own particular
rate when excitation is removed. If Helmholtz resonators are
involved in room treatment, sound not absorbed is reradiated.
418 CHAPTER TWENTY
The sound of the studio, embracing these three components of
indirect sound plus the direct sound, has its counterpart in musical
instruments. In fact, it is helpful to consider our studio as an instrument
that the knowledgeable musician, technician, or engineer can
play. It has its own characteristic sound, and a certain skill is required
to extract from it its full potential.
Reverberation
Reverberation is the composite, average effect of all three types of indirect
sound. Measuring reverberation time does not reveal the individual
components of which reverberation is composed. Herein lies the
weakness of reverberation time as an indicator of studio acoustical
quality. The important action of one or more of the indirect components
may be obscured by the averaging process. This is why it is said
that reverberation time is an indicator of studio acoustical conditions,
but not the only one.
There are those who feel it is improper and inaccurate to apply
the concept of reverberation time to relatively small rooms. It is
true that a genuine reverberant field may not exist in small spaces.
Sabine’s reverberation equation is based on the statistical properties
of a random sound field. If such an isotropic, homogeneous
distribution of energy does not prevail in a small room, is it proper
to apply Sabine’s equation to compute the reverberation time of the
room? The answer is a purist “no,” but a practical “yes.” Reverberation
time is a measure of decay rate. A reverberation time of 0.5
seconds means that a decay of 60 dB takes place in 0.5 seconds.
Another way to express this is 60 dB/0.5 second = 120 dB/second
decay rate. Whether the sound field is diffuse or not, sound decays
at some particular rate, even at the low frequencies at which the
sound field is least diffuse. The sound energy stored at the modal
frequencies decays at some measurable rate, even though only a
few modes are contained in the band being measured. It would
seem to be a practical step to utilize Sabine’s equation in small
room design to estimate absorption needs at different frequencies.
At the same time, it is well to remember the limitations of the
process.
ACOUSTICS OF THE SMALL RECORDING STUDIO 419
Studio Design
In a general book of this type, space is too limited to go into anything but
basic principles. Fortunately, there is a rich literature on the subject,
much of it written in easy-to-understand language. In designing a studio,
attention should be given to room volume, room proportions, and sound
decay rate, diffusion, and isolation from interfering noise.
Studio Volume
A small room almost guarantees sound colorations resulting from
excessive spacing of room resonance frequencies. This can be minimized
by picking one of the favorable room ratios suggested by Sepmeyer
(see Fig. 13-6) 1.00 : 1.28 : 1.54, applying it to a small, a medium,
and a large studio and seeing what happens. Table 20-1 shows the
selected dimensions, based on ceiling heights of 8, 12, and 16 feet
resulting in room volumes of 1,000, 3,400, and 8,000 cubic feet. Axial
mode frequencies were then calculated after the manner of Table 15-5
and plotted in Fig. 20-2, all to the same frequency
scale. As previously noted, the
room proportions selected do not yield
perfect distribution of modal frequencies,
but this is of no consequence in our investigation
of the effects of room volume. A
visual inspection of Fig. 20-2 shows the
increase in the number of axial modes as
volume is increased, which of course
results in closer spacing. In Table 20-2 the
number of axial modes below 300 Hz is
shown to vary from 18 for the small studio
to 33 for the large. The low-frequency
response of the large studio, 22.9Hz, is
shown to be far superior to that of the two
smaller studios at 30.6 and 45.9 Hz. This is
an especially important factor in the
recording of music.
We must remember that modes other
than axial are present. The major diagonal
Small studio
Medium studio
Large studio
0 50 100 150 200 250 300
0 50 100 150 200 250 300
0 50 100 150 200 250 300
F I G U R E 2 0 - 2
Comparison of the axial-mode resonances of a small
(1,000 cu ft), a medium (3,400 cu ft), and a large
(8,000 cu ft) studio all having the proportions 1.00:
1.28: 1.54.
420 CHAPTER TWENTY
dimension of a room better represents the lowest frequency supported
by room resonances because of the oblique modes. Thus, the
frequency corresponding to the room diagonal listed in Table 20-2 is
a better measure of the low-frequency capability of a room than the
lowest axial frequency. This approach gives the lowest frequency for
the large room as 15.8 Hz, compared to 22.9 Hz for the lowest axial
mode.
The average spacing of modes, based on the frequency range from
the lowest axial mode to 300 Hz, is also listed in Table 20-2. The average
spacing varies from 8.4 Hz for the large studio to 14.1 Hz for the
small studio.
The reverberation times listed in Table 20-2 are assumed, nominal
values judged fitting for the respective studio sizes. Given these reverberation
times, the mode bandwidth is estimated from the expression
2.2/RT60. Mode bandwidth varies from 3 Hz for the large studio to 7 Hz
Table 20-2. Studio resonances in Hz.
Small Medium Large
studio studio studio
Number of axial modes below 300 Hz 18 26 33
Lowest axial mode 45.9 30.6 22.9
Average mode spacing 14.1 10.4 8.4
Frequency corresp. to room diagonal 31.6 21.0 15.8
Assumed reverb, time of studio, second 0.3 0.5 0.7
Mode bandwidth (2.2/RT60) 7.3 4.4 3.1
Table 20-1. Studio dimensions.
Ratio Small studio Medium studio Large studio
Height 1.00 8.00 ft 12.00 ft 16.00 ft
Width 1.28 10.24 ft 15.36 ft 20.48 ft
Length 1.54 12.32 ft 18.48 ft 24.64 ft
Volume 1,000 cu ft 3,400 cu ft 8,000 cu ft
ACOUSTICS OF THE SMALL RECORDING STUDIO 421
for the small studio. The advantage of closer spacing of axial modes in
the large studio tends to be offset by its narrower mode bandwidth. So,
we see conflicting factors at work as we realize the advantage of the
mode skirts overlapping each other. In general, however, the greater
number of axial modes for the large studio, coupled with the extension
of room response in the low frequencies, produces a response superior
to that of the small studio.
The examples of the three hypothetical studios considered above
emphasize further the appropriateness of musical instrument analogy of
a studio. We can imagine the studio as a stringed instrument, one string
for each modal frequency. These strings respond sympathetically to
sound in the room. If there are enough strings tuned to closely spaced
frequencies, and each string responds to a wide enough band of frequencies
to bridge the gaps between strings, the studio-instrument responds
uniformly to all frequency components of the sound in the studio. In
other words, the response of the studio is the vector sum total of the
responses of the individual modes. If the lines of Fig. 20-2 are imagined
to be strings, it is evident that there will be dips in response between
widely spaced frequencies. The large studio, with many strings, yields
the smoother response.
Conclusion: A studio having a very small volume has fundamental
response problems in regard to room resonances; greater studio volume
yields smoother response. The recommendation based on BBC
experience still holds true, that coloration problems encountered in
studios having volumes less than 1,500 cubic feet are severe enough to
make small rooms impractical. For reasons of simplicity, the axial
modes considered in the previous discussion are not the only modes,
but they are the dominant ones.
Room Proportions
If there are fewer axial modes than are desired in the room under consideration,
sound quality is best served by distributing them as uniformly as
possibly. The cubical room distributes modal frequencies in the worst
possible way—by piling up all three fundamentals, and each trio of multiples
with maximum gap between. Having any two dimensions in multiple
relationship results in this type of problem. For example, a height of
422 CHAPTER TWENTY
8 ft and a width of 16 ft means that the second harmonic of 16 ft coincides
with the fundamental of 8 ft. This emphasizes the importance of proportioning
the room for best distribution of axial modes.
The perfect room proportions have yet to be found. It is easy to
place undue emphasis on a mechanical factor such as this. I urge you
to be well informed on the subject of room resonances and to be aware
of certain consequences, but let us be realistic about it—all of the
recording that has ever taken place has been done in spaces less than
perfect. In our homes and offices, conversations are constantly taking
place with serious voice colorations, and we listen to and enjoy
recorded music in acoustically abominable spaces. The point is that in
striving to upgrade sound quality at every stage of the process, reducing
sound colorations by attention to room modes is just good sense.
Reverberation Time
Technically, the term “reverberation time” should not be associated
with relatively small spaces in which random sound fields do not
exist. However, some first step must be taken to calculate the amount
of absorbent needed to bring the general acoustical character of a room
up to an acceptable level. While reverberation time is useful for this
purpose, it would be unfortunate to convey the impression that the
values of reverberation time so obtained have the same meaning as
that in a large space.
If the reverberation time is too long (sound decays too slowly),
speech syllables and music phrases are slurred and a definite deterioration
of speech intelligibility and music quality results. If rooms are
too dead (reverberation time too short), music and speech lose character
and suffer in quality, with music suffering more. These effects are
not so definite and precise as to encourage thinking that there is a specific
optimum reverberation time, because many other factors are
involved. Is it a male or female voice, slow or fast talker, English or
German language (they differ in the average number of syllables per
minute), a stand-up comic or a string ensemble, vocal or instrumental,
hard rock or a waltz? In spite of so many variables, readers need guidance,
and there is a body of experience from which we can extract
helpful information. Figure 20-3 is an approximation rather than a true
optimum—but following it will result in reasonable, usable conditions
ACOUSTICS OF THE SMALL RECORDING STUDIO 423
for many types of recording. The shaded area of Fig. 20-3 represents a
compromise in rooms used for both speech and music.
Diffusion
Before the advent of the Schroeder (diffraction grating) diffusor, there
was little advice to give regarding diffusion in the small studio. Splaying
walls and the use of geometrical protuberances have only a modest
diffusing effect. Distributing the absorbing material is a useful means
of not only achieving some diffusion, but increasing the absorbing efficiency
as well.
Modular diffusing elements are on the market that really diffuse as
shown in Chap. 14. There are even 2 ft-×-4 ft-modular units that offer
high-quality diffusion and excellent broadband absorption (0.82 coefficient
at 100 Hz, for example), all within a 2-in thickness (the Abffusor
™). The application of this new principle of diffusion, with or
without the absorption feature, contributes a feeling of spaciousness
through the diffusion of room reflections and the control of resonances.
1.2
1.4
1.0
0.8
0.6
0.4
0.2
0
1,000 2,000 5,000 10,000 20,000 30,000
Speech
Music
Reverberation time - seconds
Room volume - cu ft
F I G U R E 2 0 - 3
Suggested reverberation times for recording studios. The shaded area is a compromise
region for studios in which both music and speech are recorded.
424 CHAPTER TWENTY
Noise
Noise is truly something in the ear of the “behearer.” One person’s
beautiful music is another person’s noise, especially at 2 AM. It is a
two-way street, and fortunately, a good wall that protects a studio area
from exterior noise also protects neighbors from what goes on inside.
The psychological aspect of noise is very important—acceptable if
considered a part of a situation—disturbing if considered extraneous.
Chap. 18 has already treated the special case of air-conditioning
noise.
Studio Design Procedure
We have considered reverberation and how to compute it (Chap. 7),
the reality of room resonances (Chap. 15), the need for diffusion
(Chaps. 13 and 14), various types of dissipative and tuned absorbers
(Chap. 9), and as mentioned, one of the most serious studio noise
producers, the air-conditioning equipment (Chap. 18). All of these
are integral parts of studio design. The would-be designer should
also sample the literature to see how others have solved similar
problems.1–3
Some Studio Features
A glance into other people’s studios often makes one aware of
“things I want to do” or “things I definitely don’t like.” Figures 20-4
and 20-5 show the treatment of a budget 2,500 cu ft studio. Built on
the second floor of a concrete building with an extensive printing
operation below, certain minimum precautions were advisable. The
studio floor is 3⁄4-inch plywood on 2-×-2-inch stringers resting on
1⁄2-inch soft fiberboard. Attenuation of noise through the double 5⁄8 
drywall ceiling is augmented by a one inch layer of dry sand, a
cheap way to get amorphous mass. The wall modules, containing
a 4-inch thickness of Owens-Corning Type 703 Fiberglas (3 lb/cu ft
density), help to absorb and diffuse the sound.
The studio of Fig. 20-6 with a volume of 3,400 cu ft has a couple
of interesting features. The wall modules (Fig. 20-5) feature carefully
stained and varnished frames and a neat grille cloth. Those of
Fig. 20-7 are of two kinds, one sporting a very attractive fabric
ACOUSTICS OF THE SMALL RECORDING STUDIO 425
F I G U R E 2 0 - 4
View of a 2,500-cu ft voice studio looking into the control room. World Vision, International.
F I G U R E 2 0 - 5
Rear view of the 2,500-cu ft voice studio of Fig. 20-4. Wall modules containing 4-inch
thicknesses of dense glass fiber contribute to diffusion of sound in the room. World
Vision, International.
426 CHAPTER TWENTY
design, the other more subdued. The studio
of Fig. 20-6 has a rather high ceiling,
hence a virtual, visual ceiling was established
at a height of 8 feet. This consists
of four 5-×-7-foot suspended frames as
shown in Fig. 20-8, which hold fluorescent
lighting fixtures and patches of glass
fiber. The plastic louvers are acoustically
transparent.
The voice studio of Fig. 20-9, with a
volume of 1,600 cu ft, employs wall
absorbing panels manufactured by the
L.E. Carpenter Co. of Wharton, New Jersey.
These panels feature a perforated
vinyl wrapping and a 3⁄8-in rigid composition
board backing. The concrete floor
rests on soft fiberboard with distributed
cork chips under it. The low-frequency
deficiencies of carpet and wall panels
require some Helmholtz correction, and
F I G U R E 2 0 - 6
A 3,400-cu ft studio used for both recording and editing voice tapes. Decorator-type
fabric provides an attractive visual design for the absorber/diffusor wall modules.
Mission Communications Incorporated.
F I G U R E 2 0 - 7
Close view of the wall modules of Fig. 20-6. Mission
Communications Incorporated.
ACOUSTICS OF THE SMALL RECORDING STUDIO 427
thirteen 20 × 40 × 8-inch boxes are mounted in the suspended ceiling
frame out of sight.
Figure 20-10 is a 3,700-cu ft music studio that is also used for voice
work. Low-frequency compensation is accomplished by the same
Helmholtz boxes mentioned above, 14 of them in each of two suspended
ceiling frames.
Elements Common to All Studios
Chap. 4 of Reference 1 treats sound lock treatment, doors and their
sealing, wall constructions, floor/ceiling constructions, wiring pre-
F I G U R E 2 0 - 8
The high structural ceiling of the studio of Fig. 20-6
allows the use of four 5 X 7 ft suspended frames to
bring the visual ceiling down to 8 ft and to support
illumination fixtures and absorbing material. The
plastic louvers are acoustically transparent. Mission
Communications Incorporated.
F I G U R E 2 0 - 9
Voice studio with a volume of 1,600 cu ft. A 7 X 10 ft
suspended ceiling frame hides 13 Helmholtz resonators
for low-frequency absorption. Wall modules
are proprietary units covered with perforated vinyl.
Far-East Broadcasting Company.
428 CHAPTER TWENTY
cautions, illuminating fixtures, observation windows, and other
things common to all studios and which can create serious problems
if not handled properly.
Endnotes
1Ballou, Glen, ed., Handbook for Sound Engineers—The New Audio Cyclopedia,
Indianapolis, IN, Howard W. Sams & Co., 2nd ed. (1991): Chap. 4, Common Factors in All
Audio Rooms, by F. Alton Everest, p 67-101; Chap. 5, Acoustical Design of Audio Rooms, by
F. Alton Everest, p. 103-141; Chap. 6, Contempory Practices in Audio Room Design, by F.
Alton Everest, p. 143-170, Chap. 7, Rooms for Speech and Music, by Rollins Brook, p. 171-
201.
2Allison, Roy F. and Robert Berkowitz, The Sound Field in Home Listening Rooms, J. Audio.
Eng. Soc., 20, 6 (July/Aug 1972), 459-469.
3Kuhl, Walter, Optimal Acoustical Design of Rooms for Performing, Listening, and
Recording, Proc. 2nd. International Congress on Acoustics, (1956) 53-58.
F I G U R E 2 0 - 1 0
Music studio with a volume of 3,700 cu ft employing two 9 x 11 foot suspended ceiling
frames that hold a total of 28 Helmholtz resonators. Far East Broadcast Company
The acoustical design of control rooms has come of age during the
past decade. Rather than discuss the numerous subjective opinions
and approaches to control-room design that have characterized the literature
of the past, we shall go directly to current practice, which
promises even greater refinements in the future.
The Initial Time-Delay Gap
Every recording bears indelibly the marks of the room in which the
sounds were recorded. Beranek1 made an intensive study of concert
halls around the world. He noted that those rated the highest by qualified
musicians had certain technical similarities. Among them was
what he called the initial time-delay gap. This is the time between the
arrival of the direct sound at a given seat and the arrival of the critically
important early reflections. He was impressed by the fact that
halls rating high on the quality scale had a well-defined initial timedelay
gap of about 20 milliseconds. Halls having this time delay gap
confused by uncontrolled reflections were rated inferior by qualified
listeners.
Davis’s attention was directed to the initial time-delay gap of recording
studios and control rooms through work with a newly introduced
measuring technique known as time-delay spectrometry.2 The factors
Acoustics
of the Control Room
21
429
C H A P T E R
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430 CHAPTER TWENTY-ONE
generating the initial time delay of a typical recording studio are illustrated
in Fig. 21-1A. The direct sound travels a short distance from
source to microphone. Later, the sound reflected from the floor, ceiling,
or nearby objects arrives at the microphone. This time gap between the
arrival of the direct and reflected components is determined by the
geometry of the particular setup in a particular studio. Although it
varies with each setup, the delay gap for a studio typically falls within
a fairly narrow range.
The operator in a conventional control room cannot hear the studio
delay gap because it is masked by early control-room reflections. This
Reflected
Reflected
Direct
Direct
Studio
Control room
A
B
F I G U R E 2 1 - 1
The wider time-delay gap of a live-end-dead-end control room avoids masking that of
the studio. After Davis.2
ACOUSTICS OF THE CONTROL ROOM 431
means that the operator is deprived of an important component of the
studio sound in his critical evaluation. Davis figured that the way to
allow the operator to hear the studio delay gap was to eliminate or
reduce the early reflections in the control room. Time-delay spectrometry
dramatically revealed the comb-filter effects associated with early
reflections from surfaces near the loudspeakers and from the console
face. This clarification of the problem led directly to the solution of
placing absorbing material on the surfaces surrounding the loudspeakers
in the front part of the control room (Fig. 21-1B)
Chips Davis (not related to Don Davis), who was a party to this
early experimentation and analysis in 1978 in a Syn-Aud-Con seminar
and a man of action, decided to test the idea.3,4 He mounted absorbing
material on the surfaces of the front part of his control room in Las
Vegas. The results were very encouraging; the sounds from the studio
had improved clarity and the ambience of the control room took on a
new spacious dimension.
The delay gap of the control room was now narrow enough to avoid
masking that of the studio. Giving the control room a precise initial
time-delay gap gave listeners the impression of a much larger room.
In simplified and idealized form, Fig. 21-2 shows the energytime
relationships essential for a properly designed and adjusted
control room. At time = 0, the signal leaves the monitor loudspeaker.
After an elapsed transit time, the direct sound reaches the
ears of the operator. There follows some insignificant low-level
“clutter” (to be neglected if 20 dB down), after which the first
return from the rear of the room arrives. In the early days, emphasis
was placed on the first, second, and third significant reflections.
Suffice it to say that these important, prominent, delayed reflections
constitute the end of the time-delay gap and the first signs of
an exponential decay.
The Live End
The deadening of the front end of the control room near the observation
window seemed like a fairly straightforward procedure—just
cover the surfaces with absorbent. The resulting improvement of
stereo image and sound quality gave no indication that other
approaches to the treatment of the front end might work even better.
432 CHAPTER TWENTY-ONE
Rather, once the front end is made absorbent, attention was naturally
shifted to the rear, live end.
Delays offered by reflections from the rear wall dominated the
thinking in the early days. Attention was correctly directed to the
ability of the human auditory system to integrate these delayed
reflections from the rear wall. The desire to make the rear wall diffusive
was present in early thinking, but the only means at hand was
to introduce the relatively ineffective geometrical irregularities. Had
reflective phase-grating diffusors been available in 1978, they would
have been eagerly utilized.
Specular Reflections vs. Diffusion
For the specular reflection of sound energy from the monitor loudspeakers
from the rear wall (Fig. 21-3A), all of the acoustical energy
from a given point on the reflector surface arrives in a single instant of
time. If the same sound energy is incident on a reflection phase-grating
Diffuse sound
Time
Energy (specular and diffused)
from rear of room
Initial time-delay gap
Direct sound at operator's position
Time of travel
source to operator
Energy density
F I G U R E 2 1 - 2
The definition of the initial time-delay gap for a control room.
ACOUSTICS OF THE CONTROL ROOM 433
diffusor on the rear wall (Fig. 21-3B), the back-scattered energy is
spread out in time.5 Each element of the diffusor returns energy, the
respective reflected wavelets arriving at different times. This temporal
distribution of reflected (diffused) energy results in a rich, dense,
nonuniform mixture of comb filters that the human auditory system
perceives as a pleasant ambience. This is in contrast to the sparse specular
reflections that combine to form unpleasant wideband colorations.
With the reflection phase-grating diffusor, the reflected wavelets
are not only spread out in time, they are also spread out in space.
The one-dimensional diffusor spreads its reflected energy in the
horizontal hemidisc of Fig. 14-7. By orienting other one-dimensional
units, vertical hemidiscs of diffusion are easily obtained. This
is in contrast to the specular panel, which distributes reflected
energy in only a portion of half-space determined by the location of
the source and the size of the panel.
Another feature of the reflection phase grating, illustrated in Fig.
21-4, makes it especially desirable for the live end of a control room.
Let us consider the three black spots as side-wall reflections that
impinge on the rear wall, returning energy to the operator, O. If the rear
wall is specular, there is only one point on the surface returning
S S
RPG
Specular
A B
F I G U R E 2 1 - 3
A comparison of reflection of sound energy from (A) a flat surface, which yields specular reflection, and (B)
a reflection-phase grating, which yields energy dispersed in a hemidisc. Peter D’Antonio, RPG Diffusor Systems,
Inc., and Audio Engineering Society, Inc.
434 CHAPTER TWENTY-ONE
energy from each source to the operator. In contrast, each element of
the surface of the grating diffusor sends energy toward the operator.
Energy from all sound sources (direct or reflected) falling on the diffusor
are scattered to all observation positions. Instead of a single “sweet
spot” at the console, a much wider range of good listening positions
results.
Low-Frequency Resonances
in the Control Room
The Techron™ instrument continues to turn out, under expert direction,
new views of sound fields in enclosed spaces. I have devoted much
space in this book to room modes, but visualizing them is made much
easier by Fig. 21-5. This is a three-dimensional plot showing the relationship
between time, energy, and frequency at the microphone position
in a control room. The vertical scale is 6 dB between marks. This is
Specular RPG
A B
F I G U R E 2 1 - 4
The three black spots represent side-wall reflection in a control room returning energy to the operator, O. If
the rear wall is specular, only a single point on the rear wall returns energy to the operator. Each element of
a reflection-phase-grating diffusor returns energy to the operator. Peter D’Antonio, RPG Diffusor Systems, Inc.,
and Audio Engineering Society, Inc.
ACOUSTICS OF THE CONTROL ROOM 435
not sound-pressure level, but rather true energy. The frequency scale
runs from 9.64 Hz to 351.22 Hz, a region expected to be dominated by
modes. Time runs from the rear toward the reader, 2.771 msec per step,
or about 0.1 second for the entire traverse.
The dramatic ridges in Fig. 21-5A are the modal (standing wave)
responses of the control room, which together make up the low-frequency
acoustical response of the room.6 Without these modal resonance
boosts, the control room would sound like the outdoors.
In addition to the modal response, this delay contains information
concerning a second phenomenon of control rooms that must
be taken into consideration. This is the interference between the
direct low-frequency wave from the loudspeaker and its reflection
from the rear wall. If the operator position is 10 feet from the rear
wall, the reflection lags the direct sound by the time it takes the
wave to travel 20 feet. The delay is t = (20 ft) / (1,130 ft/sec) = 0.0177
Time (ms)
Energy (dB)
Frequency (Hz)
12
µ sec/
step
2771
span µS/f
111777
25874
9.64 Hz
351.22
t1
55.0
80.7 117.5
139.8 183.4
288.3
229.7
269.0
305.9
337.5
200.92 Hz/S
bandwidth
10.00 Hz
A B
F I G U R E 2 1 - 5
A three-dimensional energy-time-frequency plot of the sound field of a control room revealing the room
modes and their decay. The (A) display is the “as found” condition. Display (B) reveals the cleaning up of
modal interference by the installation of a low-frequency diffusor at the extreme rear of the control room.
Peter D’Antonio, RPG Diffusor Systems, Inc., and Audio Engineering Society, Inc.
436 CHAPTER TWENTY-ONE
second. The frequency of the first comb-filter notch is 1/2t or 1 /
(2)(0.0177) = 28.25 Hz (see chapter 17). Subsequent notches, spaced
1/t Hz, occur at 85, 141, 198, 254 Hz, etc.
The depth of the notches depends on the relative amplitudes of the
direct and reflected components. One way to control the depth of the
notches is to absorb the direct wave so that the reflection is low. This
removes precious sound energy from the room. A better way is to build
large diffusing units capable of diffusing at these low frequencies. This
is exactly what has been done in Fig. 21-5B—a 10-ft-wide, 3-ft-deep,
floor-to-ceiling diffusor behind the midband diffusors.6 The result is a
very definite improvement in the response and a much smoother
decay. These modes decay about 15 dB in the 0.1-second time sweep.
This means a decay rate of roughly 150 dB/sec, which corresponds to
a reverberation time of 0.4 second, but the variation of decay rate
between the various modes is great.
Initial Time-Delay Gaps in Practice
The realization of the general applicability of the initial time-delay
gap principle to different kinds of spaces was not long in coming.
Beranek coined the term to apply to concert halls. It was then studied
in recording studios, and now control rooms are being revolutionized
in order for the operator to hear the delay gap of the studio. In the
process, it was discovered that if the control room is altered to make
the studio delay gap audible, some beautiful fallout in listening benefits
are realized.
The Techron™ instrument’s energy-time display makes possible a
sophisticated, clinical evaluation of the distribution in time of acoustical
energy in a room and the presence or absence of an initial timedelay
gap. Figure 21-6 shows three energy-time displays of widely
differing spaces. The display of Fig. 21-6A shows the response for the
modern control room of Master Sound Astoria, Astoria, NY (see Fig.
21-9). The well-defined time gap (labeled ITD), the exponential decay
(a straight line on a log-frequency scale), and the well-diffused reflections
are the hallmarks of a well-designed room.
Figure 21-6B shows the energy-time display for the Concertgebouw,
Haarlem, Netherlands. A well-defined time gap of about 20
msec qualifies it as one of Beranek’s quality concert halls.
ACOUSTICS OF THE CONTROL ROOM 437
There are many high-quality concert halls in the world, and in
recent years, an ever-growing number of high-quality control rooms.
However, there have been very few, if any, small listening or recording
rooms that could be classified as quality spaces for the simple reason
6 dB
6 dB
6 dB
Energy (dB)
Energy (dB)
Energy (dB)
0
0
50
10 20
50 100 100 200
Diffuse sound
Direct
sound
ITD
Time (msec)
Time (msec)
Time (msec) A
C
B
ITD = 10 6 msec
9 7dB
24 dB RF2
F I G U R E 2 1 - 6
Illustrations of initial time-delay gaps from widely different types of spaces. (A) The quality-control room of
Master Sound Astoria, NY (see Fig. 21-9). (B) A quality concert hall. Concertgebouw, Haarlem, The Netherlands.
(C) A quality small listening room, Audio electronics Laboratory, Albertson, NY. Peter D’Antonio, RPG
Diffusor Systems, Inc., and Audio Engineering Society, Inc.
438 CHAPTER TWENTY-ONE
that they are dominated by normal modes and their associated problems.
The physics of the situation seems to doom small rooms to inferior
acoustical quality. But now physics, in another guise, comes to the
rescue. The energy-time display of Fig. 21-6B gives hope that small listening
rooms of genuine quality are now a possibility. This display is
for a small listening room belonging to Audio Electronics Laboratory
of Albertson, New York. A beautiful, initial time-delay gap of about 9
msec is made possible by a live-end-dead-end design in which the narrator
and microphone are placed in the dead end of the room facing
the live end. This end is dominated by reflection phase-grating diffusors
designed by Peter D’Antonio of RPG Diffusor Systems, Inc.7
Managing Reflections
The management of reflections is a major concern in control rooms.
Davis realized this and recommended making the entire front end
dead by applying absorbent to the surfaces. More recently, Berger and
D’Antonio have devised a method that does not depend on absorption
but rather on the shaping of surfaces to nullify the bad effect of reflections.
8–11 This sounds much simpler than it really is. The placement of
a loudspeaker close to solid boundaries can greatly affect its output. If
it is placed close to an isolated solid surface, its power output into half
space is doubled—an increase of 3 dB. If the speaker is placed close to
the intersection of two such surfaces, there is an increase of 6 dB
because the power is confined to quarter space. If placed close to the
intersection of three such solid surfaces, the power radiated is
increased 9 dB for one-eighth space.
Placing loudspeakers at some specific distance from the trihedral
surfaces of a room has been common in home listening rooms and in
control rooms. If the distances from the loudspeaker to the surfaces are
appreciable in terms of wavelength of the sound, new problems are
introduced. The overall power boost effect may be minimized, but frequency
response might be affected due to the constructive and
destructive combination of direct and reflected waves.
A point source in a trihedral corner has a flat response at an observation
point if that observation point is in a reflection-free zone. There
are no reflections to contribute to interference effects. D’Antonio
ACOUSTICS OF THE CONTROL ROOM 439
extends this observation to the idea of placing control-room-monitor
loudspeakers in trihedral corners formed from splayed surfaces. By
splaying the room boundaries, a reflection-free zone can be created
around the operator. By splaying the walls and ceiling, it is even possible
to extend the reflection-free zone across the entire console, several
feet above, and enough space behind to include the producer
behind the mix position.
The Reflection-Free-Zone Control Room
In their 1980 paper,12 Don Davis and Chips Davis specified that there
should be “...an effectively anechoic path between the monitor loudspeakers
and the mixer’s ears.” What they called “anechoic” is today
called a reflection-free zone. The most obvious way to achieve an
anechoic condition is through absorption, hence the “dead end” designation.
To design a control room with a reflection-free zone (RFZ™) the engineer
must deal with the mathematics of image sources. The contribution
of a reflection from a surface can be considered as coming from a virtual
source on the other side of the reflecting plane on a line perpendicular to
that plane through the observation point and at a distance from the plane
equal to that to the observation point. With splayed surfaces in three
dimensions, it is difficult to visualize all the virtual sources, but this is
necessary to establish the boundaries of the reflection-free zone.
A floor plan of a reflection-free zone control room is shown in Fig.
21-7. The monitor loudspeakers are flush-mounted as close as possible
to the trihedral corner formed with the ceiling intersection. Next, both
the front side walls and the front ceiling surfaces are splayed accurately
to keep reflections away from the volume enclosing the operator.
10 It is possible to create an adequate reflection-free zone at the
operator’s position by proper splaying of walls. In this way, an anechoic
condition is achieved without recourse to absorbents.
If absorbent is needed to control specific reflections, it can be
applied to the splayed surfaces.
The rear end is provided with a complete complement of reflection
phase-grating diffusors. In Fig. 21-7 the self-similarity principle is
employed in the form of fractals. A high-frequency quadratic residue
440 CHAPTER TWENTY-ONE
diffusor is mounted at the bottom of each low-frequency diffusor well.
Wideband sound energy falling on the rear wall is diffused and
directed back to the operator with an appropriate time-delay gap. This
sound is diffused both in space and time by the hemidisc pattern of the
diffusors. An elevation view of the reflection-free zone control room is
shown in Fig. 21-8.
RPG
RFZ
F I G U R E 2 1 - 7
Plan view of a control room with a live, rather than a dead, front end. Front-end reflections
in the operator’s area are avoided by shaping the surfaces to create a reflectionfree
zone. The rear of the room is dominated by a mid- and high-band fractal diffusor
and a low-frequency diffusor. Peter D’Antonio, RPG Diffusor Systems, Inc., and the National
Association of Broadcasters.
ACOUSTICS OF THE CONTROL ROOM 441
Control-Room Frequency Range
Although this is not a treatise on control-room construction, some of the
salient constructional features will be mentioned. The range of frequencies
to be handled in the control room is very great, and every frequency-
dependent component must perform its function over that
range. The commonly accepted high-fidelity range of 20 Hz to 20 kHz is
a span of 10 octaves, or 3 decades. This represents a range of wavelengths
from about 57 feet to 5⁄8 inch. The control room must be built
with this fact in mind.
The lowest modal frequency is associated with the longest dimension
of a room, which may be taken as the diagonal. Below that frequency
there is no modal resonance support for sound, much like the
great outdoors. Needless to say, the room’s response falls sharply
below this frequency, which can be estimated by 1,130 / 2L, in which
L = the diagonal distance in feet. For a rectangular room following Sepmeyer’s
proportions having dimensions of 15.26 × 18.48 × 12 feet, the
RFZ
F I G U R E 2 1 - 8
Elevation view of the reflection-free zone of the control room of Fig. 21-7. Peter D’Antonio,
RPG Diffusor Systems, Inc., and the National Association of Broadcasters.
442 CHAPTER TWENTY-ONE
diagonal is 28.86 feet. This places the low-frequency cutoff frequency
for this particular room at 1,130/ [(2)(28.86)] = 21 Hz.
From 21 Hz up to about 100 Hz (for a room of this size), normal
modes dominate and wave acoustics must be applied. From 100 Hz to
about 400 Hz is a transition region in which diffraction and diffusion
prevail. Above 400 Hz, true specular reflection and ray acoustics take
over. These frequency zones determine construction of the control
room—a massive shell to contain and distribute the low-frequency
modal energy, and an inner shell for reflection control.
Outer Shell of the Control Room
The size, shape, and proportions of the massive outer shell of the control
room determine the number of modal frequencies and their specific
distribution as discussed in Chap. 15. There are two schools of
thought: one prefers splaying of walls of the outer shell to “break up”
modal patterns, and the other prefers the rectangular shape. Only a
modest deviation from a rectangular shape toward a trapezoidal shape
is feasible. Such a shape does not get rid of modal patterns, it just distorts
them into an unpredictable form. Others feel that symmetry for
both low-frequency and high-frequency sound better fits the demands
of stereo.
To contain the low-frequency sound energy associated with control-
room activities, thick walls, possibly 12-inch-thick concrete, are
required.
Inner Shell of the Control Room
The purpose of the inner shell is, among other things, to provide the
proper reflection pattern for the operator at the console. Consequently,
its construction can be relatively light. For the inner shell, shape is
everything.
Representative Control Rooms
An example of the application of reflection phase-grating diffusors in
a studio is shown in Fig. 21-12. This is the Blue Jay Recording Studio
of Carlisle, MA, also designed by Russell E. Berger. Figures 21-13 and
ACOUSTICS OF THE CONTROL ROOM 443
F I G U R E 2 1 - 9
Control room of Master Sound Astoria, Astoria, NY, designed by Charles Bilello Associates
utilizing RPG diffusors. Photo by Kent Howard, Peter D’Antonio, RPG Diffusor Systems, Inc.
F I G U R E 2 1 - 1 0
Control room of Red Bus Studios of London, England, designed by Neil Grant of Harris
Associates. RPG diffusors are used throughout. Peter D’Antonio, RPG Diffusor Systems, Inc.
444 CHAPTER TWENTY-ONE
21-14 are examples of low-frequency QRD™ diffusors utilizing the
Diffractal™ principle, that is, diffusors within diffusors within diffusors
to achieve wide-band effectiveness.
All of the diffusors shown in these photographs were supplied by
RPG Diffusor Systems, Inc.13 and are of the types discussed in Chap. 14.
Some European Designs
A glance at several Swiss control rooms and studios will broaden
your view of the application of Schroeder diffusors. I am indebted to
Helmuth Kolbe, a consultant in acoustics, for data on the rooms to be
described, and which he designed.7 The first example is Studio Sixty
of Lausanne, Switzerland, owned by Wolfgang Ehrlich, which is
busy around the clock on jobs from Paris and even from the United
States.
A good example of the use of diffusors in a studio is shown in
Fig. 21-15. The lower diffusor, a bit less than 7 feet high and 17.5
feet wide, is based on a primitive root-sequence. The wells are made
from medium-density board with birch veneer of 19⁄16-in width. The
F I G U R E 2 1 - 1 1
Control room of Digital Services, Houston, TX, designed by Russell E. Berger, then of Pelton-
Joiner-Rose Group of Dallas, TX. Peter D’Antonio, RPG Diffusor Systems, Inc.
ACOUSTICS OF THE CONTROL ROOM 445
maximum depth is 16.7 inches. The design frequency is taken as 400
Hz, which gives a calculated bandwidth of 300 to 4,305 Hz.
The upper diffusor is based on a quadratic residue sequence. A single
period of 19 wells per period is used. Each well is 1 inch in width.
Like the lower unit, no metal separators between the wells are used.
The design frequency of the upper unit is 375 Hz, yielding a calculated
bandwidth of 281.3 to 6,889 Hz.
Reactions of the musicians to the treatment of this studio are very
positive: They like the crisp, natural-blended, lively sound it gives.
Rear view of the control room for Studio Sixty is shown in Fig. 21-16.
F I G U R E 2 1 - 1 2
Blue Jay Recording Studio of Carlisle, MA, showing extensive use of RPG diffusors on the
ceiling and elsewhere. Designed by Russell L. Berger, then of the Pelton-Joiner-Rose
Group of Dallas, TX. Peter D’Antonio, RPG Diffusor Systems, Inc.
446 CHAPTER TWENTY-ONE
The lower diffusor in the rear of this control room is based on a primitive
root sequence. The design frequency of 400 Hz gives a calculated
bandwidth of 300 to 6,624 Hz. It is made up of 1-inch medium-density
board with a birch veneer facing. There are 67 wells per period and
only a single period is used, which gives an overall width of 67.5
inches. Aluminum sheeting separates the wells.
The upper diffusor in Fig. 21-16 is based on a quadratic residue
sequence. It has 37 wells of 3⁄4 inch width, also made of medium-density
board with birch veneer facing. The wells are separated by aluminum
dividers, each a full 13 inches in depth. The design frequency in this
case is 500 Hz, yielding a computed bandwidth of 375 to 8,611 Hz.
The plastic panels mounted on either side of the diffusors in Fig. 21-
16 are special reflectors. The specular reflections from these panels are
carefully placed in time either to shorten the initial time-delay gap, or
as Kolbe wanted in this case, to sustain the important early reflections.
F I G U R E 2 1 - 1 3
Control room of Festival Records, Sydney, Australia. The low-frequency diffusor in the
rear of the control room actually functions over a wide band because it is of the diffractal
type which incorporates a diffusor within a diffusor within a diffusor. Peter D’Antonio,
RPG Diffusor Systems, Inc.
ACOUSTICS OF THE CONTROL ROOM 447
The small control room of A + D Studio, another Swiss studio,
posed a number of problems and challenges to Kolbe. A direct view of
the rear of this control room is shown in Fig. 21-17. A quadratic
residue diffusor having a well width of 71⁄4 inches is located on either
side of the door. The design frequency of this one is 500 Hz, giving a
calculated bandwidth of 375 to 931 Hz. There are 19 wells per period
with one period used. The total width of this diffusor is 11.5 feet.
The ceiling diffusors can be seen over and behind the operator’s
head position. The close-up of Fig. 21-18 reveals a center quadratic
residue unit made up of boards of 0.45-in thickness with no metal
F I G U R E 2 1 - 1 4
The rear of the control room of Winfield Sound, Toronto, Ontario, Canada. The low-frequency
diffusor also features wide-band diffusion because of diffractals. Peter D’Antonio,
RPG Diffusor Systems, Inc.
448 CHAPTER TWENTY-ONE
F I G U R E 2 1 - 1 5
The recording studio of Studio Sixty, Lausanne, Switzerland. The large, lower diffusor,
made of 1-9⁄16 medium-density board with birch facing, is constructed on the basis of a
primitive root sequence. The upper diffusor is based on a quadratic-residue sequence.
Acoustic Engineering H. Kolbe and Synergetic Audio Concepts.
F I G U R E 2 1 - 1 6
The rear of the control room at Studio Sixty, Lausanne, Switzerland designed by Helmut
Kolbe. The lower diffusor is based on a primitive root sequence, the upper on a quadratic-
residue sequence. The reflectors mounted on either side of the diffusors return
specular reflections to the operator’s position to support the diffused return. Acoustic
Engineering H. Kolbe.
ACOUSTICS OF THE CONTROL ROOM 449
dividers. Another quadratic residue diffusor made of 11⁄16-in boards
is placed on either side of the center unit and on the walls (Fig.
21-19). These units have bandwidths that go up to 8,000 and
15,000 Hz.
In spite of the grave problems associated with such small control
rooms and the difficulty of cramming so many diffusors into such a
limited space, operators appreciate the fact that their room sounds as
though it is much larger.
Figure 21-20 shows the control room of Studio Atmosphere
between Zurich and Bern in Switzerland. The interesting feature here
is the large concrete low-frequency diffusor out of sight behind the
wide array of quadratic residue diffusors pictured. Together, a very
wide frequency band is covered.
F I G U R E 2 1 - 1 7
Control room of A+D Studios in Switzerland, designed by Helmut Kolbe. The rear, lowfrequency
diffusor is based on a quadratic-residue sequence, as are those on the walls
and ceiling. Acoustic Engineering H. Kolbe.
450 CHAPTER TWENTY-ONE
Consultants
This chapter has only just scratched the surface of control-room design
and has not even touched on the equally complicated design of recording
studios. It is hoped that what has been covered will reveal some of
the more recent developments that have contributed so much to sound
quality in control rooms.
Anyone contemplating building, modernizing, or retrofitting a studio
or control room should enlist the services of a qualified consultant
in acoustics. How can a qualified consultant be found? One way
is to make a point of visiting some of the better, more recent installations
and evaluating them personally. Much can be learned from
these owners about the consultants involved. One earmark of a qualified
consultant is his dependence on modern instruments, such as the
Techron™ instrument, in measuring and qualifying the space. It is a
wise person who relies upon the knowledge and experience of a qualified
consultant.
F I G U R E 2 1 - 1 8
Closer view of the ceiling diffusors of the A+D control room of Fig. 17-16. Acoustic Engineering
H. Kolbe.
ACOUSTICS OF THE CONTROL ROOM 451
Endnotes
1Beranek, L.L., Music, Acoustics, and Architecture, New York, John Wiley and Sons, (1962).
2Davis, Don, The Role of the Initial Time-Delay Gap in the Acoustic Design of Control Rooms
for Recording and Reinforcing Systems, 64th Convention, Audio Eng. Soc., (Nov 1979),
preprint #1574.
3Davis, Chips and Don Davis, Live-End-Dead-End Control Room Acoustics....(etc), Recording
Eng/Prod, 10, 1 (Feb 1979) 41.
4Davis, Chips and G.E. Meeks, History and Development of the LEDE™ Control-Room
Concept, 64th Convention, Audio Eng. Soc., Los Angeles, (Oct 1982), preprint #1954.
5D’Antonio, Peter and John H. Konnert, The Role of Reflection-Phase-Grating Diffusors in
Critical Listening and Performing Environments, 78th Convention of the Audio Eng. Soc.,
Anaheim, CA, (May 1985), preprint #2255.
F I G U R E 2 1 - 1 9
Another view of the control room A+D Studio in Switzerland. Acoustic Engineering H. Kolbe.
452 CHAPTER TWENTY-ONE
6D’Antonio, Peter and John H. Konnert, New Acoustical Materials and Designs Improve
Room Acoustics, 81st Convention, Audio Eng. Soc., Los Angeles, (Nov 1986), preprint
#2365.
7Acoustic Engineering H. Kolbe, Zielacker Strasse 6, CH-8304 Wallisellen, Switzerland.
Telephone: 01/830-10-39.
8Berger, Russell E., Speaker/Boundary Interference Response (SBIR), Norman, IN, Synergetic
Audio Concepts Tech Topics, 11, 5 (Winter 1984) 6 p.
9D’Antonio, Peter, Control-Room Design Incorporating RFZ™, LFD™, and RPF™ Diffusors,
dB The Sound Eng. Mag., 20, 5 (Sept/Oct 1986) 47-55, 40 references.
10D’Antonio, Peter and John H. Konnert, The RFZ™/RPG™ Approach to Control-Room
Monitoring, 76th Convention Audio Eng. Soc., New York, (Oct 1984), preprint #2157.
11Muncy, Neil A., Applying the Reflection-Free Zone RFZ™ Concept in Control-Room
Design, dB The Sound Eng. Mag., 20, 4 (July/Aug 1986), 35-39.
12Davis, Don and Chips Davis, The LEDE™ Concept for the Control of Acoustic and
Psychoacoustic Parameters in Recording Rooms, J. Audio Eng. Soc., 28, 9 (Sept 1980), 585-
595.
13RPG Diffusor Systems, Inc., 651-C Commerce Drive, Upper Marlboro, MD 20772, Tel. 301-
249-0044, FAX 301-249-3914.
F I G U R E 2 1 - 2 0
Closer view of the ceiling diffusors of the A+D control room of Fig. 17-16. Acoustic Engineering
H. Kolbe.
In the early days of recording, artists crowded around a horn leading
to a diaphragm-driven stylus cutting a groove on a wax cylinder. In
early radio dramatics, actors, actresses, and sound-effects persons
moved in toward the microphone or faded back according to the dictates
of the script. Greater freedom came as several microphones, each
under separate control in the booth, were used. All this, of course, was
monophonic.
Monophonic is still with us in commercial form, but the advent of
stereophonic techniques has added a new dimension of realism and
enjoyment to recordings, films, radio, etc. Stereo requires, basically, a
dual pickup. This can be two separated microphones or two elements
with special directional characteristics and electrical networks
mounted close together. In broadcasting or recording a symphony
orchestra, for example, it was soon found that some of the weaker
instruments required their own microphones to compete with the
louder instruments. The signal from these was proportioned between
the left and right channels to place them properly in the stereo field.
Here, again, we see a trend from two microphones to many.
Popular music has always been with us, but its form changes with
time. Recording techniques came to a technological maturity just in
time to be clasped to the breast of new-wave musicians and musical
directors. Whether the Beatles were truly the vanguard of this new
Acoustics for
Multitrack Recording
22
453
C H A P T E R
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
454 CHAPTER TWENTY-TWO
musical development or not will be left to historians, but their style
spread like wildfire throughout the western world. “Good” sound
quality in the traditional sense was not as much sought after as was a
distinctive sound. Novelty effects, such as phasing and flanging, sold
records by the millions. A new era of studio recording, variously
called multichannel, multitrack, or separation recording, burst on the
scene. It was beautifully adapted to the production of special effects,
and the novel, distinctive sound, and it flourished.
Flexibility
The key word is flexibility. Multitrack
provides the means for recording one
instrument or soloist at a time, if desired,
as well as the introduction of special
effects along the way. A production can
be built up piece by piece and assembled
later in the mix down. In Fig. 22-1A, signals
from several microphones are combined
in a summing network and fed to a
single-track recorder. A variation is to
use a two-track recorder, distributing signals
from each microphone, partially or
wholly, between the two tracks for artificial
stereophonic recording and reproduction.
In contrast, the signal of each
microphone shown in Fig. 22-1B is
recorded on a single track of a multitrack
recorder. Many variations of this arrangement
are possible. For example, a halfdozen
microphones on the drums could
be premixed and recorded on a single
track, but mix down flexibility would be
sacrificed in the process.
Once all the component parts of a musical
production are recorded synchronously
on separate tracks, they can then be
mixed down to mono, stereo, or other mul-
Summing
network
Single
track
recorder
Multitrack recorder
A
B
F I G U R E 2 2 - 1
(A) The outputs of several microphones can be combined
by a summing network and recorded on a single-
track tape. (B) In basic multitrack recording, the
output of each microphone is recorded on a separate
track of the tape. The signals of the several tracks are
combined later in a mix down session.
ACOUSTICS FOR MULTITRACK RECORDING 455
tichannel form for release. Much attention can be lavished on each
detail in the mix down, a stage that becomes a very important part in
the production chain of events.
Advantages of Multitrack
Flexibility is the outstanding overall advantage of multitrack techniques,
but to understand the true breadth of the word some supporting
detail is offered. Multitrack recording makes possible the
conquering of space and time. Suppose the drums and electric piano
are recorded on separate tracks on Monday. On Tuesday, the guitar
player is available, so a third track is recorded as he or she listens on
headphones to a temporary mix of the first two. The tape can be
shipped across the country to another studio to pick up a big-name
female vocalist between engagements. In this way, a musical production
can be built up a piece at a time. It might not be the best way, but
it is possible.
Another big advantage of separation recording is the almost complete
control it provides over the relative levels of each instrument
and artist. Each track can be given just the equalization desired, often
after considerable experimentation. Special effects can be injected at
the mix down stage. Reverberation can be added in any desired
amount.
It is expensive when artists, musicians, and technical crews are
standing by for a retake of one performer or to argue the next step. The
mix down is a calmer session than recording a group on a premix
basis.
There is also a noise advantage in multitrack. In a “mix and record
as the group plays” type of session, levels of the various instruments
are adjusted, and the result, as recorded, is frozen as the final mix with
no recourse. Some potentiometers are set high and some low depending
on the requirements of the source, and the signal-to-noise ratio of
each channel is thereby fixed. In separation-type recording, the standard
practice is to record all channels at maximum level, which guarantees
the best signal-to-noise ratio on the original tape. In the
multichannel mix down, these levels are adjusted downward to
achieve the desired balance, but there is still a significant noise advantage
over the premixed case.
456 CHAPTER TWENTY-TWO
In addition to recording each track at maximum undistorted level,
the bandwidth of some channels can be reduced without affecting the
sound of the instruments. For example, violin sound has practically
no energy below 100 Hz. By cutting the low-frequency end of that
channel at 100 Hz, the noise is reduced with no noticeable degradation
of quality. The sound of some instruments is essentially unaltered by
cutting some low-frequency energy, some by cutting highs.
A pervasive argument for multichannel recording is that it is what
the client wants, expects, and gets. Customer demand, in the final
analysis, can be the greatest factor influencing the drift toward multichannel.
Of course, the customer demands it because of the inherent
flexibility, potential savings, and other virtues.
Disadvantages of Multitrack
In spite of the signal-to-noise ratio advantage, multitrack carries a disadvantage
of noise buildup as the number of tracks combined is
increased. When two tracks having equal noise levels are mixed
together, the noise on the combined track is 3 dB higher than either
original. If 32 tracks are engaged in a mix down, the combined noise is
15 dB higher than a single track. Table 22-1 lists noise buildup for
commonly used track configurations. It is simply a matter of adding
noise powers. Mixing eight tracks of equal noise powers means that
the total noise is 10 log 8 = 9.03 dB higher than the noise of one track.
If the noise of one track is 80 dB referred
to the reference level, the noise of 16
tracks is 68 dB.
The dynamic range of a system is
defined as the total usable range of audio
level between the noise at the lower
extreme and the full undistorted level at
the upper extreme. The more tracks on a
given width of tape, the narrower each
track and the lower the reproduced signal
level. Increased noise and decreased
reproduced level spell narrower dynamic
range.
Table 22-1 Multitrack noise buildup.
Number of Noise buildup above
tracks noise of one track, dB
2 3.01
4 6.02
8 9.03
16 12.04
24 13.80
32 15.05
48 16.81
ACOUSTICS FOR MULTITRACK RECORDING 457
The closer the spacing of tracks on a tape, the greater the crosstalk
between adjacent tracks. Recording circumstances determine the magnitude
of the resulting problem. For instance, if the two adjacent tracks
are of two musical instruments recorded simultaneously in the studio
while playing the same number, the congruity might make the cross
talk acceptable. The degree of separation realized between microphones
in the studio affects the judgment on the seriousness of tape
crosstalk. If the material on adjacent tracks is unrelated (usually not the
case in music recording) the crosstalk will be much more noticeable.
Artistic responsibility might become diffused in multitrack
recording unless the musical director is intimately involved in both
recording and mix down. The very nature of the mix down technique
dictates seemingly endless hours of detailed comparison of tracks,
recording pickups, and overdubs, which are the basic creative steps
in a production. Often this meticulous duty falls on the recording
engineer with only an occasional check by the music director. In contrast,
the old style of premix recording session ends with an essentially
completed product with the music director in full charge all
the way.
Some separation recording sessions tend to separate musicians to
the extent that spontaneous interaction is lost. Musicians respond
to each other, and this desirable effect may or may not be maintained
in the face of cueing by foldback headphones and as the musicians are
physically isolated by baffles, screens, and isolation booths.
While we are considering the negative aspects of multitrack recording,
the degradation of quality as the tape is run and rerun scores of
times must be added to the rest. In what other endeavor does the original
recording receive such treatment? With 2-inch magnetic tape,
contact with the head becomes a problem as the tape is passed through
the machine, especially the outer tracks. The wise recording engineer
reserves for outside tracks those sounds least affected by loss in highfrequency
response.
Achieving Track Separation
Achieving 15- to 20-dB intertrack separation requires intelligent
effort and attention to detail. Without such separation, the freedom of
458 CHAPTER TWENTY-TWO
establishing relative dominance in the mix down is sacrificed. The
following methods are employed to yield the required separation:
 Adjusting the acoustics of the studio.
 Spacing the artists.
 Using microphone placement and directivity.
 Use of physical barriers.
 Use of gating techniques.
 Use of contact transducers or electrical instruments.
Studio Acoustics
Heretofore in considering studio acoustics, the criterion has been quality
(naturalness and freedom from colorations) of the recorded signal. In
multitrack recording, the emphasis is shifted to track separation, and
the very meaning of quality tends to be dissipated in the process. Reflective
surfaces in the studio are not ruled out, but they are generally localized
for specific instruments while the general studio acoustics are
made quite dead and absorptive. The number of musicians to be accommodated
is limited, among other things, by the size of the studio. If the
walls are highly absorptive, musicians can be placed closer to them, and
more artists can be accommodated in a given space. Reverberation time
rather loses its meaning in a studio specializing in separation recording,
but if measured, it would tend to be quite short.
Distance between Artists
In an absorptive studio, increasing distance between the various
instruments is a step toward track separation. Sound level falls off at a
rate of 6 dB for each doubling of the distance in free field. While the
falloff rate is less indoors, this is still a fair rule to use in estimating
the separation that can be realized through spacing of musicians.
Microphone Management
This principle of separation by distance also applies to microphones.
The placement of the microphone for musician A and the microphone
for musician B must be considered along with the actual relative positions
of A and B. In some cases, there is a directional effect associated
ACOUSTICS FOR MULTITRACK RECORDING 459
with certain musical instruments that can be used to advantage, certainly
the directional properties of microphones can be used to
improve separation. The distance between adjacent musicians and the
distance between microphones are obvious factors as well as the distance
between each musician and his or her own microphone. There is
an interplay between all these distance effects and microphone directivity.
The nulls of a cardioid or bidirectional microphone pattern may
save the day in controlling a troublesome crosstalk problem.
Barriers for Separation
Physical separation of musicians, absorbent studios, and proper selection,
placement, and orientation of microphones still are limited in the
degree of acoustical separation they can produce. Baffles (or screens,
as they are sometimes called) are used to increase isolation of the
sound of one musician from that of another. Baffles come in a great
variety of forms: opaque and with windows, reflective and absorbent,
large and small. And now, with the availability of quadratic residue
diffusors, sound diffusion can become a feature of baffles. Extreme
forms of barriers are nooks and crannies for certain instruments or a
separate booth for drums or vocals.
The effectiveness is very low for baffles of any practical size at low
frequencies. Once more we come up against the basic fact of physics
that an object must be large in terms of the wavelength of the sound to
be an effective obstacle to the sound. At 1 kHz, the wavelength of
sound is about 1 ft, hence a baffle 6-ft wide and 4-ft tall would be reasonably
effective. At 100 Hz, however, the wavelength is about 11 feet
and a sound of that frequency would tend to flow around the baffle,
regardless of the thickness or material of the baffle.
Electronic Separation
Some use, although not extensive, has been made of electronic gating
circuits to improve separation between sources. These circuits reject
all signals below an adjustable threshold level.
Electronic Instruments and Separation
Contact pickups applied to almost any musical instrument with a special
adhesive can transform an acoustical instrument to an amplified
instrument. In addition, there are many electrical instruments that are
460 CHAPTER TWENTY-TWO
completely dependent on the pickup transducer and amplification.
The electrical output from such instruments can be fed to the console,
providing dependence on an electrical signal rather than a microphone
pickup of an acoustical signal. The separation between two
such tracks can be very high. Amplified instruments with their own
loudspeakers in the studio can be picked up by microphones placed
close to the loudspeakers. Even though the quality of the sound is
degraded, this approach has its enthusiastic followers.
The Future of Multichannel
Multichannel recording techniques have production advantages that
promise to be a permanent part of the recording scene of the future to one
degree or another. The audio control room of a network television production
center probably has a mixing console with at least 48 channels
and multitrack recording facilities to match. Recording a symphony
orchestra today commonly requires as many channels and as many
tracks. In this case, it is not true separation recording, but rather singlepoint
recording for ambience with augmentation of certain instruments,
sections, and soloists as required. At this time, there seems to be a trend
away from the artificialities of strict separation recording in some areas.
Automation
The number of knobs, switches, buttons, and VU meters on the average
recording console is enough to dazzle the uninitiated and impress
the musically oriented client. In fact, it becomes something of a problem
for an operator equipped with only one pair of eyes, one brain,
two hands, and a normal reaction time to operate all these controls and
still have time for the more creative aspects of his or her job. Minicomputer
automation control relieves the operator of much of the
tedium and releases the operator for more creative work. Automatic
computer-controlled adjustment of pot settings, equalization, etc., in a
mix down, especially, is a boon to the operator, making him or her a far
more productive worker. Development in the direction of greater
automation is inevitable for the future.
Post recording; digital sampling; midi; editing; sound effects; (foley)
dialog replacement; voice over; sound processing; digital/analog
recording; synthesizers; composing; video production; equipment
testing/evaluation.
A separate room for each function? Hardly practical. One room for
all these functions? Possible, with some compromises, but this is the
working hypothesis for this chapter.
With so many functions, the only recourse is to consider the
acoustical factors common to all of the functions: