Elements of Acoustics

Samuel Temkin

Originally published in 1981; Reprinted in 2001




Basic Fluid Mechanics and Thermodynamics

1.1 Indicial Notation
1.2 Continuum Model
1.3 Macroscopic Thermodynamics
First Law
Specific Heats
Second Law
Maxwell Relations
Fluids in Motion

1.4 Conservation of Mass 1.5 Equation of Motion
Stress Tensor
Navier-Stokes Equation

1.6 The Energy Equation
Second Law for a Continuum

1.7 Complete System of Equations

Basic Properties of Acoustic Waves

2.1 Ideal Fluids
2.2 Linearization
2.3 Uniform Fluids
2.4 One-Dimensional Plane Waves
Speed of Sound in a Perfect Gas
Speed of Sound in Other Fluids
Relationships between Acoustic Quantities

2.5 Monochromatic Waves
Plane, One-Dimensional Monochromatic Waves
Plane, Monochromatic Waves in Three Dimensions
Relation between Variables in a Monochromatic Wave
Time Averages

2.6 Fourier Analysis
Periodic Waveforms —Fourier Series
Nonperiodic Functions—Fourier Transform

2.7 Acoustic Energy
Energy Density
Acoustic Intensity
Reference Levels

Refliction and Transmission Phenomena

3.1 Normal Incidence
Reflection at the Interface between Two Media
Acoustic Impedance at a Boundary
Acoustical Elements having Complex Impedances
Helmholtz Resonators
Electrical Analogies

3.2 Characteristic Waves
Linearized Shock Tube

3.3 Transmission Through A Wall
3.4 Oblique Incidence
Field in Front of a Rigid Reflector

3.5 Propagation In A Two-Dimensional Channel
Excitation of Transverse Modes

3.6 Acoustic Field In A Piston-Driven Tube
Experimental Determination of a Amplitude Growth at Resonance
3.7 Some Nonlinear Effects
Distortion of a Progressive Wave
Entropy Changes
Attenuation of a Sawtooth Wave

3.8 Plane Waves In Tubes Of Varying Cross Section
Exponential Horn
Power-Law Horns
Other Shapes
Transmission Coefficient

3.9 Sudden Area Changes
Tubes with Fluids Having Different Properties
Transmission into Several Branches

3.10 Tube With Temperature Gradient

Spherical And Cylindrical Waves

4.1 Centrally Symmetric Waves
Monochromatic Case
Standing Waves in a Spherical Cavity

4.2 Problems With Spherical Symmetry
Radially Pulsating Sphere
Initial-Value Problem

4.3 Axially Symmetric Spherical Waves
Standing Waves in a Spherical Cavity
Rigid Sphere in a Sound Wave
Scattering by a Sphere
Arbitrary Spherical Waves

4.4 Circularly Cylindrical Waves
Monochromatic Waves
Waves inside Circular Tubes

4.5 Nonmonochromatic Cylindrical Waves

Sound Emission

5.1 Radiation From a Pulsating Sphere
Forces on the Pulsating Sphere
Pulsating Bubble
Simple Source

5.2 Inhomogeneous Wave Equation
Linear Array
Continuous Distributions

5.3 Emission From A Piston In An Infinite Wall
Axial Pressure
Forces on the Piston
Applications to Helmholtz Resonators

5.4 Compact Distributions of Sources
5.5 Oscillating Sphere
Force on the Sphere

5.6 Radiation From Fluctuating Forces
Point-Force Distribution
Simple-Point Forces
Arbitrary Time Dependence

5.7 Acoustic Dipoles
Line Distribution

5.8 Far-Field Of Compact Force Distribution
5.9 Acoustic Quadrupoles
5.10 Sound Emission By Heat Release
5.11 Integral Formulation For Radiation
Surface Integral Representation
Radiation Condition
Pulsating Sphere
Reduction to a Single Integral

Sound Absorption

6.1 Linearized Dissipative Equations
Physical Considerations

6.2 Attenuation Due To Viscous Effects
Translational Relaxation Time

6.3 Attenuation In Viscous, Heat-Conducting Fluid
Comparison with Experimental Data
Effects of Impurities

6.4 Energy-Dissipation Method
Unbounded Waves

6.5 Effects of Boundaries
Flow Induced by an Oscillating Plane
Thermal Waves

6.6 Attenuation In Tubes
Comparison with Experimental Data

6.7 Boundary Effects: Vector Formulation
6.8 Propagation In A Two-Dimensional Channel
Wide-Tube, Low-Frequency Approximation
Narrow-Tube, Low-Frequency Approximation

6.9 Sphere Oscillating In Viscous Fluid
Force on the Sphere

6.10 Sphere In A Sound Wave
6.11 Attenuation And Dispersion IN A Dilute Suspension
Measurements of Particle Size

6.12 Boundary Viscous And Thermal Effects
6.13 Waves Emitted By Plane Heater
Energy Considerations


General Books
Specialized Books
Review and Research Articles


A. Useful Formulas From Vector Analysis
B. Explicit Expressions For Some Vector And Tensor Quantities In Special Coordinate Systems
C. Some Properties Of The Bessel Functions
D. Some Properties Of The Spherical Bessel Functions
E. Legendre Polynomials



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This book is an outgrowth of a course in acoustics I have taught for a number of years at Rutgers University. The main reason for adding one more book to an already long list of books on this subject is the lack of modern introductory texts that treat acoustics as a branch of fluid mechanics. In my view, this is the most natural approach, at least for those areas of acoustics dealing with the most common media for sound propagation, namely, air and water. This approach is, of course, not new. It was used by the authors of many of the books now considered classical, including Rayleigh, Lamb, and others. In recent times, however, many of the acoustics texts that have appeared treat the subject as a branch of electrical engineering. There are indeed many instances in which acoustic oscillations are analogous to some phenomena discussed in electrical engineering courses and the analogies are clearly advantageous to those students whose background is in that discipline. For others, the analogies may be a drawback; to them, both the acoustic equations and their electrical analogues are new.

The main subjects discussed in this book are: propagation in uniform fluids at rest; transmission and reflection phenomena; attenuation and dispersion; and emission. These are only some of the main topics in acoustics. To have attempted to cover all of them would have been presumptuous on my part. Nevertheless, there are several topics that, by some, may be considered basic enough to warrant their inclusion in a text of this nature, but that have been omitted. These include aerodynamic sound, diffraction, and propagation in nonuniform media. Some of these are mentioned in the text, but all too briefly in relation to their importance. The reasons are that some of these topics are either outside my areas of competence or are too books that have appeared recently, so that their detailed discussion in this book is unnecessary. On the other hand, sound absorption is discussed in more detail than is usual in books on acoustics. To a certain extent, this reflects my personal interest in that subject, but it is also intended to qualify the strongly held notion that dissipation effects in sound waves are unimportant. advanced compared to the general level of the book. In any event, most of them are fully treated in one or more specialized The material given here is intended primarily for a beginning graduate course in acoustics, but includes portions suitable for more advanced courses. In writing this book, I have assumed that the student's background includes the usual preparation in undergraduate physics and mathematics, as well as a course in advanced calculus and a course in basic thermodynamics. Prior acquaintance with fluid mechanics is desirable, but not required. The required material on that subject is developed in Chapter 1. Chapter 1 also includes a summary of basic thermodynamics. To make the book self-sufficient, both of these subjects are developed to a greater degree than is needed in an introductory course.

The book contains more material than is possible to cover in one semester. By deleting some of the more advanced material, it can be used in a one-semester course in basic acoustics for students in engineering or in the physical sciences. On the other hand, with some additional material, it may be used in a one-year sequence covering both basics and applications.

Because of the basic nature of the subject of this book, I have attempted to derive each result from basic principles. However, the emphasis throughout is on the physical meaning of the result's and not on the mathematics techniques that were used to derive them. On the other hand, in some of the derivations I have included more detail than customary, since all too often the student's main effort is spent in trying to fill in the mathematical steps missing between main results. Of course, this has some pedagogical value but, more often than not, it merely improves the ability of the student to manipulate equations. In my view, a better way of learning is by doing. To this end, a number of problems have been included in the text.

Each chapter contains a brief list of suggested references. A more complete list is given in the Bibliography at the end of the book. The lists are not exhaustive, their purpose is merely to direct the interested student to other general sources, or to recent articles touching on some of the material discussed in the text.

Although I have included the results of some of my own investigations, the bulk of the material presented may be considered classical. It is therefore difficult to acknowledge the sources of many of the results that are presented. I have, however, profited much from Chapter 8 of Fluid Mechanics by L.D. Landau and E.M. Lifshitz and from Chapters 1-3 of An Introduction to Fluid Dynamics by G.K. Batchelor. Other books that have influenced this work are The Theory of Sound by Lord Rayleigh, Theoretical Acoustics by P. Morse and U. Ingard, Fundamentals of Acoustic by L.E. Kinsler and A.R. Frey, and The Foundations of Acoustics by E. Skudrzyk.

A major portion of this book was written during 1974-1975 while I was on leave at ythe Technion-Israel Institute of Technology. I wish to thank Rutgers University and The Lady Davis Fellowship Trust for making this leave possible. I also owe much to the faculty of the Department of Mechanical Engineering at the Technion for their kind hospitality.

I would like to express my gratitude to Professor R.A. Dobbins of Brown University, who introduced me to the subject of this book; to my colleagues at Rutgers University for their continued encouragement; to many of my students for their valuable comments and observations; and to Mrs. Rosemarie Boysen, who typed an earlier version of this book. The final manuscript was typed by Mrs. Erma Sutton, to whom I am also indebted for improving the clarity of many passages.

To conclude, I wish to express my gratitude to my wife Judy and to my sons David and Michael, who patiently endured the writing of this book.

S. Temkin

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