FUNDAMENTALS OF
`ACOUSTICS
`
`Fourth Edition
`
`LAWRENCE E. KINSLER
`Late Professor Emeritus
`Naval Postgraduate School
`AUSTIN R. FREY
`Late Professor Emeritus
`Naval Postgraduate School
`ALAN B. COPPENS
`Black Mountain
`North Carolina
`JAMES V. SANDERS
`Associate Professor of Physics
`Naval Postgraduate School
`
`New York
`
`John Wiley & Sons, Inc.
`Chichester Weinheim
`Brisbane
`
`Singapore
`
`Toronto
`
`Page 1 of 54
`
`GOOGLE EXHIBIT 1017
`
`

`

`Fundamentals
`of Acoustics
`
`FOURTH EDITION
`
`La,,Tc11ce E. l(insler
`
`Austin R. Frey
`
`Alan B. Coppe11s
`
`Jan1es V. Sanders
`
`Page 2 of 54
`
`

`

`GLOSSARY OF SYMBOLS
`
`This list identifies some symbols that are not necessarily defined every time they
`appear in the text.
`
`a
`
`aE
`
`A
`AG
`b
`
`acceleration; absorption
`coefficient ( dB per
`distance}; Sabine
`absorptivity
`random-incidence energy
`absorption coefficient
`sound absorption
`array gain
`loss per bounce; decay
`parameter
`b(0, </)} beam pattern
`B
`magnetic field;
`susceptance
`bottom loss
`adiabatic bulk modulus
`isothermal bulk modulus
`speed of sound
`group speed
`phase speed
`electrical capacitance;
`acoustic compliance;
`heat capacity
`heat capacity at constant
`pressure
`specific heat at constant
`pressure
`heat capacity at constant
`volume
`specific heat at constant
`volume
`community noise
`equivalent level (dBA)
`detection index
`
`BL
`~
`
`~T
`C
`c~
`Cp
`C
`
`C,,p
`
`Cvp
`
`Cv
`
`cv
`
`CNEL
`
`d
`
`d'
`D
`DI
`DNL
`DT
`qJ)
`
`e
`E
`Ek
`Ep
`EL
`~
`
`~i
`
`J
`
`Jr
`Ju, Ji
`
`F
`
`Fe
`g
`
`G
`C§
`h
`H(0, </))
`
`detectability index
`directivity; dipole strength
`directivity index
`detected noise level
`detection threshold
`diffraction factor
`specific energy
`total energy
`kinetic energy
`potential energy
`echo level
`time-averaged energy
`density
`instantaneous energy
`density
`instantaneous force;
`frequency (Hz)
`resonance frequency
`upper, lower half-power
`frequencies
`peak force amplitude;
`frequency (kHz)
`effective force amplitude
`spectral density of a
`transient function;
`sound-speed gradient;
`acceleration of gravity;
`aperture function
`conductance
`adiabatic shear modulus
`specific enthalpy
`directional factor
`
`Page 3 of 54
`
`

`

`Page 4 of 54
`
`Page 4 of 54
`
`

`

`With grateful thanks to our wives,
`Linda Miles Coppens and Marilyn Sanders,
`for their unflagging support and gentle patience.
`
`ACQUISITIONS EDITOR
`
`Stuart Johnson
`
`MARKETING MANAGER
`
`Sue Lyons
`
`PRODUCTION EDITOR
`
`Barbara Russiello
`
`SENIOR DESIGNER
`
`Kevin Murphy
`
`ELECTRONIC ILLUSTRATIONS
`
`Publication Services, Inc.
`
`This book was set in 10/12 Palatino by Publication Services, Inc. and printed and bound by Hamilton
`Press. The cover was printed by Hamilton Press.
`
`This book is printed on acid-free paper.
`
`00
`
`Copyright 2000© John Wiley & Sons, Inc. All rights reserved.
`No part of this publication may be reproduced, stored in a retrieval system or transmitted in any
`form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise,
`except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either
`the prior written permission of the Publisher, or authorization through payment of the appropriate
`per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (508)
`750-8400, fax (508) 750-4470. Requests to the Publisher for permission should be addressed to the
`Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012,
`(212) 850-6011, fax (212) 850-6008, E-mail: PERMREQ@WILEY.COM. To order books or for customer
`service please call 1(800)-225-5945.
`
`Library of Congress Cataloging-in-Publication Data:
`Fundamentals of acoustics/ Lawrence E. Kinsler ... [et al.].-4th ed.
`p.cm.
`Includes index.
`1. Sound-waves. 2. Sound-Equipment and supplies. 3. Architectural acoustics. I.
`Kinsler, Lawrence E.
`
`534-dc21
`
`QC243 .F86 2000
`ISBN 0-471-84789-5
`Printed in the United States of America
`10 9 8 7 6 5 4 3 2
`
`99-049667
`
`Page 5 of 54
`
`

`

`PREFACE
`
`Credit for the longevity of this work belongs to the original two authors, Lawrence
`Kinsler and Austin Frey, both of whom have now passed away. When Austin
`entrusted us with the preparation of the third edition, our goal was to update
`the text while maintaining the spirit of the first two editions. The continued
`acceptance of this book in advanced undergraduate and introductory graduate
`courses suggests that this goal was met. For this fourth edition, we have continued
`this updating and have added new material.
`Considerable effort has been made to provide more homework problems. The
`total number has been increased from about 300 in the previous editions to over
`700 in this edition. The availability of desktop computers now makes it possible for
`students to investigate many acoustic problems that were previously too tedious
`and time consuming for classroom use. Included in this category are investigations
`of the limits of validity of approximate solutions and numerically based studies of
`the effects of varying the various parameters in a problem. To take advantage of
`this new tool, we have added a great number of problems (usually marked with a
`suffix "C" ) where the student may be expected to use or write computer programs.
`Any convenient programming language should work, but one with good graphing
`software will make things easier. Doing these problems should develop a greater
`appreciation of acoustics and its applications while also enhancing computer skills.
`The following additional changes have been made in the fourth edition:
`(1) As an organizational aid to the student, and to save instructors some time,
`equations, figures, tables, and homework problems are all now numbered by chap(cid:173)
`ter and section. Although appearing somewhat more cumbersome, we believe the
`organizational advantages far outweigh the disadvantages. (2) The discussion of
`transmitter and receiver sensitivity has been moved to Chapter 5 to facilitate early
`incorporation of microphones in any accompanying laboratory. (3) The chapters
`on absorption and sources have been interchanged so that the discussion of
`beam patterns precedes the more sophisticated discussion of absorption effects.
`(4) Derivations from the diffusion equation of the effects of thermal conductivity
`on the attenuation of waves in the free field and in pipes have been added to
`the chapter on absorption. (5) The discussions of normal modes and waveguides
`
`iii
`
`Page 6 of 54
`
`

`

`iv
`
`PREFACE
`
`have been collected into a single chapter and have been expanded to include
`normal modes in cylindrical and spherical cavities and propagation in layers.
`(6) Considerations of transient excitations and orthonormality have been en(cid:173)
`hanced. (7) Two new chapters have been added to illustrate how the principles
`of acoustics can be applied to topics that are not normally covered in an under(cid:173)
`graduate course. These chapters, on finite-amplitude acoustics and shock waves,
`are not meant to survey developments in these fields. They are intended to intro(cid:173)
`duce the relevant underlying acoustic principles and to demonstrate how the funda(cid:173)
`mentals of acoustics can be extended to certain more complicated problems.
`We have selected these examples from our own areas of teaching and research.
`(8) The appendixes have been enhanced to provide more information on physical
`constants, elementary transcendental functions (equations, tables, and figures),
`elements of thermodynamics, and elasticity and viscosity.
`New materials are frequently at a somewhat more advanced level. As in the
`third edition, we have indicated with asterisks in the Contents those sections in
`each chapter that can be eliminated in a lower-level introductory course. Such a
`course can be based on the first five or six chapters with selected topics from the
`seventh and eighth. Beyond these, the remaining chapters are independent of each
`other (with only a couple of exceptions that can be dealt with quite easily), so that
`topics of interest can be chosen at will.
`With the advent of the handheld calculator, it was no longer necessary for text(cid:173)
`books to include tables for trigonometric, exponential, and logarithmic functions.
`While the availability of desktop calculators and current mathematical software
`makes it unnecessary to include tables of more complicated functions (Bessel
`functions, etc.), until handheld calculators have these functions programmed into
`them, tables are still useful. However, students are encouraged to use their desktop
`calculators to make fine-grained tables for the functions found in the appendixes.
`In addition, they will find it useful to create tables for such things as the shock
`parameters in Chapter 17.
`From time to time we will be posting updated information on our web site:
`www.wiley.com/college/kinsler. At this site you will also be able to send us
`messages. We welcome you to do so.
`We would like to express our appreciation to those who have educated us,
`corrected many of our misconceptions, and aided us: our coauthors Austin R. Frey
`and Lawrence E. Kinsler; our mentors James Mcgrath, Edwin Ressler, Robert T.
`Beyer, and A. 0. Williams; our colleagues 0. B. Wilson, Anthony Atchley, Steve
`Baker, and Wayne M. Wright; and our many students, including Lt. Thomas Green
`( who programmed many of the computer problems in Chapters 1-15) and L. Miles.
`Finally, we offer out heartfelt thanks for their help, cooperation, advice, and
`guidance to those at John Wiley & Sons who were instrumental in preparing
`this edition of the book: physics editor Stuart Johnson, production editor Barbara
`Russiello, designer Kevin Murphy, editorial program assistants Cathy Donovan
`and Tom Hempstead, as well as to Christina della Bartolomea who copy edited
`the manuscript and Gloria Hamilton who proofread the galleys.
`
`Alan B. Coppens
`Black Mountain, NC
`
`James V. Sanders
`Monterey, CA
`
`Page 7 of 54
`
`

`

`CONTENTS
`
`I
`CHAPTER
`FUNDAMENTALS OF VIBRATION
`I.I Introduction
`1.9 Power Relations
`14
`l
`1.10 Mechanical Resonance
`1.2 The Simple Oscillator
`1.11 Mechanical Resonance
`1.3 Initial Conditions
`and Frequency
`17
`1.4 Energy of Vibration
`5
`*1.12 Equivalent Electrical Circuits
`1.5 Complex Exponential Method
`for Oscillators
`19
`of Solution
`5
`1.13 Linear Combinations of Simple
`1.6 Damped Oscillations
`Harmonic Vibrations
`22
`1.7 Forced Oscillations
`1.14 Analysis of Complex Vibrations
`by Fourier's Theorem
`24
`1.8 Transient Response
`of an Oscillator
`*l.15 The Fourier Transform
`13
`
`2
`
`3
`
`8
`
`II
`
`15
`
`26
`
`CHAPTER 2
`TRANSVERSE MOTION: THE VIBRATING STRING
`
`37
`
`2.1 Vibrations
`of Extended Systems
`2.2 Transverse Waves
`on a String
`37
`2.3 The One-Dimensional
`Wave Equation
`38
`2.4 General Solution
`of the Wave Equation
`2.5 Wave Nature
`of the General Solution
`2.6 Initial Values and
`Boundary Conditions
`
`39
`
`40
`
`41
`
`41
`
`2. 7 Reflection at a Boundary
`2.8 Forced Vibration
`of an Infinite String
`42
`2. 9 Forced Vibration of a String
`of Finite Length
`46
`(a) The Forced,
`Fixed String
`*(b) The Forced,
`Mass-Loaded String
`*(c) The Forced, Resistance(cid:173)
`Loaded String
`51
`
`46
`
`49
`
`V
`
`Page 8 of 54
`
`

`

`vi
`
`CONTENTS
`
`54
`
`2.10 Normal Modes
`of the Fixed,
`Fixed String
`52
`(a) A Plucked String
`(b) A Struck String
`54
`*2.11 Effects of More Realistic
`Boundary Conditions
`on the Freely Vibrating
`String
`54
`(a) The Fixed,
`Mass-Loaded String
`
`55
`
`CHAPTER 3
`VIBRATIONS OF BARS
`
`(b) The Fixed, Resistance(cid:173)
`56
`Loaded String
`(c) The Fixed,
`Fixed Damped String
`2. 12 Energy of Vibration
`of a String
`58
`*2.13 Normal Modes,
`Fourier's Theorem,
`and Orthogonality
`2.14 Overtones
`and Harmonics
`
`60
`
`62
`
`57
`
`83
`
`68
`
`69
`
`3.1 Longitudinal Vibrations
`of a Bar
`68
`3.2 Longitudinal Strain
`3.3 Longitudinal
`Wave Equation
`3.4 Simple Boundary
`Conditions
`71
`3.5 The Free, Mass-Loaded Bar
`*3.6 The Freely Vibrating Bar:
`General Boundary
`Conditions
`75
`*3.7 Forced Vibrations of a Bar:
`Resonance and Antiresonance
`Revisited
`76
`
`73
`
`82
`
`*3.8 Transverse Vibrations
`of a Bar
`78
`*3.9 Transverse
`Wave Equation
`80
`*3.10 Boundary Conditions
`(a) Clamped End
`82
`(b) Free End
`82
`(c) Simply
`Supported End
`82
`*3.11 Bar Clamped at One End
`*3.12 Bar Free at Both Ends
`*3.13 Torsional Waves
`on a Bar
`86
`
`84
`
`CHAPTER 4
`THE TWO-DIMENSIONAL WAVE EQUATION:
`VIBRATIONS OF MEMBRANES AND PLATES
`*4.7 The Kettledrum
`*4.8 Forced Vibration
`of a Membrane
`*4.9 The Diaphragm
`of a Condenser
`Microphone
`103
`*4.10 Normal Modes
`of Membranes
`104
`(a) The Rectangular Membrane
`with Fixed Rim
`105
`(b) The Circular Membrane
`with Fixed Rim
`106
`*4.11 Vibration
`of Thin Plates
`
`91
`
`4.1 Vibrations
`of a Plane Surface
`4.2 The Wave Equation
`for a Stretched Membrane
`4.3 Free Vibrations
`of a Rectangular Membrane
`with Fixed Rim
`93
`4.4 Free Vibrations
`of a Circular Membrane
`with Fixed Rim
`95
`4.5 Symmetric Vibrations
`of a Circular Membrane
`with Fixed Rim
`98
`*4.6 The Damped, Freely Vibrating
`Membrane
`99
`
`91
`
`100
`
`102
`
`107
`
`Page 9 of 54
`
`

`

`CONTENTS
`
`vii
`
`CHAPTER 5
`THE ACOUSTIC WAVE EQUATION
`AND SIMPLE SOLUTIONS
`5.1 Introduction
`113
`5.2 The Equation of State
`5.3 The Equation
`of Continuity
`116
`5.4 The Simple Force Equation:
`Euler's Equation
`117
`5,5 The Linear Wave Equation
`5.6 Speed of Sound in Fluids
`5.7 Harmonic Plane Waves
`5.8 Energy Density
`124
`5.9 Acoustic Intensity
`5.10 Specific Acoustic
`Impedance
`126
`5.11 Spherical Waves
`
`114
`
`119
`
`120
`
`121
`
`125
`
`127
`
`5.12 Decibel Scales
`130
`*5.13 Cylindrical Waves
`*5.14 Rays and Waves
`135
`(a) The Eikonal and Transport
`Equations
`135
`(b) The Equations
`for the Rc1y Path
`137
`(c) The One-Dimensional
`Gradient
`138
`(d) Phase and Intensity
`Considerations
`139
`*5.15 The Inhomogeneous
`wave Equation
`140
`*5.16 The Point Source
`
`133
`
`142
`
`149
`
`CHAPTER 6
`REFLECTION AND TRANSMISSION
`6.1 Changes in Media
`*6.6 Reflection from the Surface
`of a Solid
`160
`6.2 Transmission from
`One Fluid to Another:
`(a) Normal Incidence
`Normal Incidence
`150
`(b) Oblique Incidence
`161
`6.3 Transmission
`*6.7 Transmission Through a Thin
`Through a Fluid Layer:
`Partition: The Mass Law
`162
`Normal Incidence
`152
`6.8 Method of Images
`163
`6.4 Transmission from
`(a) Rigid Boundary
`One Fluid to Another:
`Oblique Incidence
`(b) Pressure Release
`155
`Boundary
`165
`*6.5 Normal Specific Acoustic
`Impedance
`160
`(c) Extensions
`
`161
`
`163
`
`165
`
`CHAPTER 7
`RADIATION AND RECEPTION OF ACOUSTIC WAVES
`7.1 Radiation from a
`7.5 Radiation Impedance
`184
`Pulsating Sphere
`171
`(a) The Circular Piston
`7.2 Acoustic Reciprocity
`(b) The Pulsating Sphere
`and the Simple Source
`7.6 Fundamental Properties
`7.3 The Continuous
`of Transducers
`188
`Line Source
`176
`(a) Directional Factor
`7.4 Radiation from a
`and Beam Pattern
`Plane Circular Piston
`(b) Beam Width
`(a) Axial Response
`179
`(b) Far Field
`181
`
`185
`
`187
`
`188
`
`188
`
`188
`
`(c) Source Level
`
`172
`
`179
`
`Page 10 of 54
`
`

`

`viii
`
`CONTENTS
`
`(d) Directivity
`189
`(e) Directivity Index
`(f) Estimates
`of Radiation Patterns
`*7.7 Directional Factors
`of Reversible Transducers
`
`190
`
`191
`
`193
`
`*7.8 The Line Array
`195
`*7.9 The Product Theorem
`*7.10 The Far Field Multipole
`Expansion
`199
`*7.11 Beam Patterns and the Spatial
`Fourier Transform
`203
`
`199
`
`CHAPTER 8
`ABSORPTION AND ATTENUATION OF SOUND
`*8.8 Viscous Losses
`8.1 Introduction
`at a Rigid Wall
`228
`8.2 Absorption
`from Viscosity
`*8.9 Losses in Wide Pipes
`211
`(a) Viscosity
`8.3 Complex Sound Speed
`230
`and Absorption
`213
`(b) Thermal Conduction
`232
`8.4 Absorption from
`(c) The Combined Absorption
`Thermal Conduction
`215
`Coefficient
`233
`8.5 The Classical Absorption
`*8.10 Attenuation
`Coefficient
`217
`in Suspensions
`8.6 Molecular Thermal
`(a) Fogs
`235
`Relaxation
`218
`(b) Resonant Bubbles
`8.7 Absorption in Liquids
`in Water
`238
`
`230
`
`234
`
`210
`
`224
`
`CHAPTER 9
`CAVITIES AND WAVEGUIDES
`*9.6 Sources and Transients in
`9.1 Introduction
`246
`Cavities and Waveguides
`9.2 Rectangular Cavity
`246
`*9.7 The Layer
`*9.3 The Cylindrical Cavity
`249
`as a Waveguide
`259
`*9.4 The Spherical Cavity
`250
`*9.8 An Isospeed Channel
`9.5 The Waveguide of Constant
`*9.9 A Two-Fluid Channel
`Cross Section
`252
`
`261
`
`261
`
`256
`
`CHAPTER 10
`PIPES, RESONATORS, AND FILTERS
`10.9 Acoustic Impedance
`(a) Lumped Acoustic
`Impedance
`287
`(b) Distributed Acoustic
`Impedance
`287
`IO.IO Reflection and
`Transmission of Waves
`in a Pipe
`288
`10.11 Acoustic Filters
`291
`(a) Low-Pass Filters
`(b) High-Pass Filters
`(c) Band-Stop Filters
`
`272
`
`IO.I Introduction
`272
`10.2 Resonance in Pipes
`10.3 Power Radiation
`from Open-Ended Pipes
`10.4 Standing Wave Patterns
`10.5 Absorption of Sound
`in Pipes
`277
`10.6 Behavior of the Combined
`Driver-Pipe System
`280
`10.7 The Long Wavelength
`Limit
`283
`10.8 The Helmholtz Resonator
`
`275
`276
`
`284
`
`286
`
`291
`
`293
`
`295
`
`Page 11 of 54
`
`

`

`CONTENTS
`
`ix
`
`CHAPTER 11
`NOISE, SIGNAL DETECTION, HEARING, AND SPEECH
`(a) Thresholds
`11.1 Introduction
`316
`302
`(b) Equal Loudness Level
`11.2 Noise, Spectrum Level,
`Contours
`318
`and Band Level
`302
`11.3 Combining Band Levels
`(c) Critical Bandwidth
`and Tones
`306
`(d) Masking
`320
`*11.4 Detecting Signals
`(e) Beats, Combination Tones,
`in Noise
`307
`and Aural Harmonics
`321
`*11.5 Detection Threshold
`310
`(f) Consonance and the
`(a) Correlation Detection
`Restored Fundamental
`I 1.8 Loudness Level
`(b) Energy Detection
`and Loudness
`324
`*11.6 The Ear
`312
`11. 9 Pitch and Frequency
`11. 7 Some Fundamental Properties
`*II.IO The Voice
`of Hearing
`327
`315
`
`318
`
`322
`
`326
`
`311
`
`311
`
`CHAPTER 12
`ARCHITECTURAL ACOUSTICS
`12.1 Sound in Enclosures
`333
`12.2 A Simple Model for the Growth
`of Sound in a Room
`334
`12.3 Reverberation Time-
`Sabine
`336
`12.4 Reverberation Time(cid:173)
`Eyring and Norris
`12.5 Sound Absorption
`Materials
`340
`12.6 Measurement of the Acoustic
`Output of Sound Sources
`in Live Rooms
`342
`12. 7 Direct and
`Reverberant Sound
`12.8 Acoustic Factors
`in Architectural Design
`
`338
`
`342
`
`343
`
`(a) The Direct Arrival
`343
`(b) Reverberation at 500 Hz
`(c) Warmth
`(d) Intimacy
`347
`(e) Diffusion, Blend,
`and Ensemble
`348
`*12.9 Standing Waves and Normal
`Modes in Enclosures
`348
`(a) The Rectangular
`Enclosure
`349
`(b) Damped Normal Modes
`(c) The Growth and Decay
`of Sound from a Source
`(d) Frequency Distribution of
`Enclosure Resonances
`353
`
`343
`
`345
`
`349
`
`351
`
`CHAPTER 13
`ENVIRONMENTAL ACOUSTICS
`13.1 Introduction
`359
`13.2 Weighted Sound Levels
`13.3 Speech Interference
`362
`13.4 Privacy
`363
`13.5 Noise Rating Curves
`13.6 The Statistical
`Description of
`Community Noise
`
`360
`
`364
`
`365
`
`13.7 Criteria for
`Community Noise
`*13.8 Highway Noise
`371
`*13.9 Aircraft Noise Rating
`*13.10 Community Response
`to Noise
`374
`13.11 Noise-Induced
`Hearing Loss
`
`369
`
`373
`
`375
`
`Page 12 of 54
`
`

`

`X
`
`CONTENTS
`
`13.12 Noise and
`Architectural Design
`13.13 Specification and
`Measurement
`of Sound Isolation
`379
`13.14 Recommended Isolation
`
`378
`
`382
`
`13.15 Design of Partitions
`382
`(a) Single-Leaf Partitions
`(b) Double-Leaf Partitions
`(d Doors and Windows
`(d) Barriers
`387
`
`383
`
`385
`
`387
`
`392
`
`CHAPTER 14
`TRANSDUCTION
`14.1 Introduction
`390
`14.2 The Transducer as an
`Electrical Network
`390
`(a) Reciprocal Transducers
`(b) Antireciprocal
`Transducers
`393
`14.3 Canonical Equations for
`Two Simple Transducers
`394
`(a) The Electrostatic Transducer
`(Reciprocal)
`394
`(b) The Moving-Coil Transducer
`(Antireciprocal)
`396
`14.4 Transmitters
`398
`(a) Reciprocal Source
`399
`(b) Antireciprocal Source
`14.5 Moving-Coil Loudspeaker
`*14.6 Loudspeaker Cabinets
`411
`(a) The Enclosed Cabinet
`(b) The Open Cabinet
`(c) Bass-Reflex Cabinet
`
`403
`
`406
`
`411
`
`412
`
`412
`
`414
`
`416
`
`418
`
`*14. 7 Horn Loudspeakers
`14.8 Receivers
`416
`(a) Microphone Directivity
`(b) Microphone
`Sensitivities
`417
`(c) Reciprocal Receiver
`(d) Antireciprocal
`Receiver
`418
`14.9 Condenser Microphone
`418
`14.10 Moving-Coil Electrodynamic
`Microphone
`420
`14. 11 Pressure-Gradient
`Microphones
`423
`*14.12 Other Microphones
`(a) The Carbon
`Microphone
`425
`(b) The Piezoelectric
`Microphone
`426
`(c) Fiber Optic Receivers
`*14.13 Calibration of Receivers
`
`425
`
`427
`
`428
`
`CHAPTER 15
`UNDERWATER ACOUSTICS
`15.1 Introduction
`435
`15.2 Speed of Sound
`in Seawater
`435
`15.3 Transmission Loss
`15.4 Refraction
`438
`15.5 The Mixed Layer
`440
`15.6 The Deep Sound Channel and
`the Reliable Acoustic Path
`444
`15. 7 Surface Interference
`446
`15.8 The Sonar Equations
`448
`(a) Passive Sonar
`448
`(b) Active Sonar
`
`15.9
`
`15.10
`
`15.11
`
`450
`
`453
`
`454
`
`Noise and Bandwidth
`Considerations
`450
`(a) Ambient Noise
`(b) Self-Noise
`451
`(c) Doppler Shift
`(d) Bandwidth
`Considerations
`Passive Sonar
`455
`(a) An Example
`Active Sonar
`456
`(a) Target Strength
`(b) Reverberation
`
`456
`
`457
`
`459
`
`436
`
`449
`
`Page 13 of 54
`
`

`

`CONTENTS
`
`xi
`
`(c) Detection Threshold
`for Reverberation-Limited
`Performance
`463
`(d) An Example
`464
`*15.12 Isospeed Shallow-Water
`Channel
`465
`(a) Rigid Bottom
`
`467
`
`467
`
`(b) Slow Bottom
`(c) Fast Bottom
`467
`*15.13 Transmission Loss Models
`for Normal-Mode
`Propagation
`468
`(a) Rigid Bottom
`(b) Fast Bottom
`
`470
`
`470
`
`CHAPTER 16
`SELECTED NONLINEAR ACOUSTIC EFFECTS
`16.1 Introduction
`16.5 Nonlinear
`478
`Plane Waves
`484
`16.2 A Nonlinear Acoustic
`Wave Equation
`(a) Traveling Waves
`478
`in an Infinite
`16.3 Two Descriptive
`Half-Space
`Parameters
`484
`480
`(b) Traveling Waves
`(a) The Discontinuity
`in a Pipe
`Distance
`485
`481
`(c) Standing Waves
`(b) The Goldberg Number
`in a Pipe
`487
`16.4 Solution by Perturbation
`Expansion
`16.6 A Parametric Array
`483
`
`483
`
`488
`
`CHAPTER 17
`SHOCK WAVES AND EXPLOSIONS
`17.3
`The Reference Explosion
`17.1 Shock Waves
`494
`(a) The Rankine-Hugoniot
`(a) The Reference Chemical
`Equations
`Explosion
`495
`501
`(b) Stagnation
`(b) The Reference Nuclear
`and Critical Flow
`Explosion
`496
`502
`(c) Normal Shock Relations
`The Scaling Laws
`(d) The Shock Adiabat
`498
`Yield and the
`Surf ace Effect
`17.2 The Blast Wave
`500
`
`497
`
`17.4
`17.5
`
`503
`
`504
`
`501
`
`APPENDIXES
`Al Conversion Factors
`and Physical Constants
`A2 Complex Numbers
`509
`A3 Circular and
`Hyperbolic Functions
`A4 Some Mathematical
`Functions
`510
`(a) Gamma Function
`(b) Bessel Functions,
`Modified Bessel Functions,
`and Struve Functions
`511
`
`508
`
`510
`
`510
`
`513
`
`(c) Spherical Bessel
`Functions
`513
`(d) Legendre Functions
`A5 Bessel Functions:
`Tables, Graphs, Zeros,
`and Extrema
`514
`(a) Table: Bessel and
`Modified Bessel
`Functions of the
`First Kind of Orders
`0, 1,and 2
`514
`
`Page 14 of 54
`
`

`

`xii
`
`CONTENTS
`
`(b) Graphs: Bessel Functions
`of the First Kind of Orders
`o, 1, 2, and 3
`516
`(c) Zeros: Bessel Functions
`of the First Kind,
`Jm(jmn) = 0
`516
`(d) Extrema: Bessel Functions
`of the First Kind,
`J 1m(jmn) = 0
`516
`(e) Table: Spherical Bessel
`Functions of the First Kind
`of Orders 0, 1, and 2
`517
`(f) Graphs: Spherical Bessel
`Functions of the First Kind
`of Orders 0, 1, and 2
`518
`(g) Zeros: Spherical Bessel
`Functions of the First Kind,
`jm({mn) = 0
`518
`(h) Extrema: Spherical Bessel
`Functions of the First Kind,
`j 1mC{mn) = 0
`518
`A6 Table of Directivities
`and Impedance Functions
`for a Piston
`519
`A7 Vector Operators
`520
`(a) Cartesian Coordinates
`
`520
`
`526
`
`527
`
`529
`
`(b) Cylindrical Coordinates
`(c) Spherical Coordinates
`AS Gauss's Theorem
`and Green's Theorem
`521
`(a) Gauss's Theorem in Two(cid:173)
`and Three-Dimensional
`Coordinate Systems
`521
`(b) Green's Theorem
`521
`A9 A Little Thermodynamics
`and the Perfect Gas
`522
`(a) Energy, Work, and the
`First Law
`522
`(b) Enthalpy, Entropy, and the
`Second Law
`523
`(c) The Perfect Gas
`524
`AIO Tables of Physical Properties
`of Matter
`526
`(a) Solids
`(b) Liquids
`(c) Gases
`528
`Al 1 Elasticity and Viscosity
`(a) Solids
`(b) Fluids
`531
`Al2 The Greek Alphabet
`
`520
`
`521
`
`529
`
`533
`
`ANSWERS TO ODD-NUMBERED PROBLEMS
`
`INDEX
`
`534
`
`543
`
`Page 15 of 54
`
`

`

`Chapter7
`
`RADIATION
`AND RECEPTION
`OF ACOUSTIC WAVES
`
`7.1 RADIATION FROM A PULSATING SPHERE
`
`The acoustic source simplest to analyze is a pulsating sphere-a sphere whose
`radius varies sinusoidally with time. While pulsating spheres are of little practical
`importance, their analysis is useful for they serve as the prototype for an important
`class of sources referred to as simple sources.
`In a medium that is infinite, homogeneous, and isotropic, a pulsating sphere
`will produce an outgoing spherical wave
`p(r, t) = (A/ r)ej(wt-kr)
`
`(7.1.1)
`
`where A is determined by an appropriate boundary condition.
`Consider a sphere of average radius a, vibrating radially with complex speed
`Uo exp(jwt), where the displacement of the surface is much less than the radius,
`Uo I w << a. The acoustic pressure of the fluid in contact with the sphere is
`given by (7.1.1) evaluated at r = a. (This is consistent with the small-amplitude
`approximation of linear acoustics.) The radial component of the velocity of the
`fluid in contact with the sphere is found using the specific acoustic impedance for
`the spherical wave (5.11.10) also evaluated at r = a,
`z(a) = /JOC cos 0a ej8•
`where cot0a = ka. The pressure at the surface of the source is then
`
`(7.1.2)
`
`p(a, t) = pocUo cos 0a ej(wt-ka+B.)
`
`Comparing (7.1.3) with (7.1.1) gives
`
`A = PoCUoa cos 0a ei(ka+B.)
`
`so the pressure at any distance r > a is
`p(r, t) = PoCUo(a/r) cos 0a ej[wt-k(r-a)+B.]
`
`(7.1.3)
`
`(7.1.4)
`
`(7.1.5)
`171
`
`Page 16 of 54
`
`

`

`172
`
`CHAPTER 7 RADIATION AND RECEPTION OF ACOUSTIC WAVES
`
`The acoustic intensity, found from (5.11.20), is
`
`(7.1.6)
`
`If the radius of the source is small compared to a wavelength, 8a _, 1r /2 and the
`specific acoustic impedance near the surface of the sphere is strongly reactive. (This
`reactance is a symptom of the strong radial divergence of the acoustic wave near a
`small source and represents the storage and release of energy because successive
`layers of the fluid must stretch and shrink circumferentially, altering the outward
`displacement. This inertial effect manifests itself in the mass-like reactance of the
`specific acoustic impedance.) In this long wavelength limit the pressure
`
`p(r, t) = jpocUo(a/r)ka ej(~t-kr)
`
`ka << 1
`
`(7.1.7)
`
`is nearly 1r /2 out of phase with the particle speed (pressure and particle speed are
`not exactly 1r /2 out of phase, since that would lead to a vanishing intensity), and
`the acoustic intensity is
`
`ka << 1
`
`(7.1.8)
`
`For constant Uo this intensity is proportional to the square of the frequency and
`depends on the fourth power of the radius of the source. Thus, we see that sources
`small with respect to a wavelength are inherently poor radiators of acoustic energy.
`In the next section, it will be shown that all simple sources, no matter what their
`shapes, will produce the same acoustic field as a pulsating sphere provided the
`wavelength is greater than the dimensions of the source and the sources have the
`same volume velocity.
`
`7.2 ACOUSTIC RECIPROCITY
`AND THE SIMPLE SOURCE
`
`Acoustic reciprocity is a powerful concept that can be used to obtain some very
`general results. Let us begin by deriving one of the more commonly encountered
`statements of acoustic reciprocity.
`Consider a space occupied by two sources, as suggested by Fig. 7.2.1. By
`changing which source is active and which passive, it is possible to set up different
`sound fields. Choose two situations having the same frequency and denote them
`as 1 and 2. Establish a volume V of space that does not itself contain any sources
`but bounds them. Let the surface of this volume be S. The volume V and the
`surface S remain the same for both situations. Let the velocity potential be <1>1 for
`situation 1 and~ for situation 2. Green's theorem (see Appendix A8) gives the
`general relation
`
`(7.2.1)
`
`where ft is the unit outward normal to S. Since the volume does not include
`any sources, and since both velocity potentials are for excitations of the same
`
`Page 17 of 54
`
`

`

`7.2 ACOUSTIC RECIPROCITY AND THE SIMPLE SOURCE
`
`173
`
`I\ n
`
`Figure 7.2.1 Geometry used in deriving
`the theorem of acoustical reciprocity.
`
`frequency, the wave equation yields
`
`v2-i>1 = -k2-i>1
`v2-i>2 = -k2-i>2
`
`(7.2.2)
`
`so that the right side of (7.2.1) vanishes identically throughout V. Furthermore,
`recall that the pressure is p = -jwpo4> and the particle velocity for irrotational
`motion is ii = V-i>. Substitution of these expressions into the left side of (7.2.1)
`gives
`
`(7.2.3)
`
`This is one form of the principle of acoustic reciprocity. This principle states that, for
`example, if the locations of a small source and a small receiver are interchanged in
`an unchanging environment, the received signal will remain the same.
`To obtain information about simple sources, let us develop a more restrictive
`but simpler form of (7.2.3). Assume that some portion of S is removed a great
`distance from the enclosed source. In any real case there is always some absorption
`of sound by the medium so the intensity at this surface will decrease faster than
`1 / r2. Since the area of the surf ace increases as r2, the product of intensity and area
`vanishes in the limit r -+ oo. In addition, if each of the remaining portions of S is
`either (1) perfectly rigid so that ii · n = 0, (2) pressure release so that p = 0, or
`(3) normally reacting so that p/(ii · ft) = Zn, then the surface integrals over these
`surfaces must vanish. Under these conditions, (7.2.3) reduces to an integral over
`only those portions of S that correspond to sources active in situations 1 or 2:
`
`I (p1ii2 · ft - p2ii1 · n) dS = 0
`
`sources
`
`(7.2.4)
`
`This simple result will now be applied to develop some important general proper(cid:173)
`ties of sources that are small compared to a wavelength.
`Consider a region of space in which there are two irregularly shaped sources,
`as shown in Fig. 7.2.2. Let source A be active and source B be perfectly rigid in
`
`Page 18 of 54
`
`

`

`174
`
`CHAPTER 7 RADIATION AND RECEPI'ION OF ACOUSTIC WAVES
`
`Figure 7.2.2 Reciprocity
`theorem applied to simple
`sources.
`
`situation 1, and vice versa in situation 2. If we define p1 as the pressure at B
`when source A is active with iii the velocity of its radiating element, and pz as the
`pressure at A when source B is active with u2 the velocity of its radiating element,
`application of (7.2.4) yields
`
`(7.2.5)
`
`If the sources are small with respect to a wavelength and several wavelengths
`apart, then the pressure is uniform over each source so that
`
`(7.2.6)
`
`Assume that the moving elements of a source have complex vector displacements
`
`where §: gives the magnitude and direction of the displacement and cf, the temporal
`phase of each element. If fl is the unit outward normal to each element dS of the
`surface, the source will displace a volume of the surrounding medium
`
`V = Is Sej(wt+</J) • fl dS = Vej(wt+e)
`
`(7.2.8)
`
`(7.2.7)
`
`Page 19 of 54
`
`

`

`7,2 ACOUSTIC RECIPROCITY AND THE SIMPLE SOURCE
`
`175
`
`where V is the complex volume displacement, V the generalization of the volume
`displacement amplitude discussed in Section 4.5, and fJ the accumulated phase
`over the surface of the element. The time derivative av/ at, the complex volume
`velocity, defines the complex source strength Q
`
`Qei"'t = av = ( u. . n dS
`Js
`at
`where u = a~/ at is the complex velocity distribution of the source surface. The
`complex source strength of the pulsating sphere has only a real part,
`
`(7.2.9)
`
`Q = Q = 41ra2Uo
`
`Substitution of (7.2.9) and p = P(r) expU(wt - kr)] into (7.2.6) gives
`
`Qi/P1 (r) = Q2/P2(r)
`
`(7.2.10)
`
`(7.2.11)
`
`which shows that the ratio of the source strength to the pressure amplitude at
`distance r from the source is the same for all simple sources (at the same frequency)
`in the same surroundings. This allows us to calculate the pressure field of any
`irregular simple source since it must be identical with the pressure field produced
`by a small pulsating sphere of the same source strength. If the simple sources are
`in free space, (7.1.7) and (7.2.10) show that the ratio of (7.2.11) is
`
`Q/P(r) = -j2Ar/poc
`
`(7.2.12)
`
`This is the free field reciprocity factor.
`Rewriting (7.1.7) with the help of (7.2.10) results in
`
`p(r, t) = hPDc(Q/ Ar)ej(wt-kr)
`
`(7.2.13)
`
`which, from the above, must be true for all simple sources. The pressure amplitude
`is
`
`(simple source)
`
`(7.2.14)
`
`and the intensity is
`
`I = lpoc(QI Ar)2
`
`(7.2.15)
`
`Integration of the intensity