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Full text of "Vibration and Sound"

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INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS 
Cat. P. HARNWELL, Consulting Editor 

Advisory Editorial Committee: E. U. Condon, Goorge R. Harrison, 
Elmer HutcMasoo, K. EL Darrow 



VIBRATION AND SOUND 



The quality of Uie materials used in ike manufacture 
of this book is governed hy continued postwar shortage*. 



INTERNATIONAL SERIES IN 
PURE AND APPLIED PHYSICS 

G. P. HarkwbI/L, Consulting Editor 



Rachhb and QoTOtMn— ATOMIC ENERGY STATES 
Bktbb— INTRODUCTION TO FERROMAGNETISM 
Brillouin— WAVE PROPAGATION IN PERIODIC STRUCTURES 

Cauy— piezoelectricity 

Cla-rk— APPLIED X-RAYS 
Curtis ELECTEICAX MEASUREMENTS 
Davey-CRYSTAL STRUCTURE AND ITS APPLICATIONS 
Edwards— ANALYTIC AND VECTOR MECHANICS 
Haujjv and PaaWH— TH3B PRINCIPLES OF OPTICS 
IIauxwell-ELECTRICITY AND ELECTROMAGNETISM 
HABNWBU. and Lmxcoon— EXPERIMENTAL ATOMIC PHYSICS 
Houston— PRINCIPLES OF MATHEMATICAL PHYSICS 
Hughes and DuBridgk— PHOTOELECTRIC PHENOMENA 
HTOO>-- HIGH-FREQUENCY MEASUREMENTS 

PHENOMENA IN HIGH-FREQUENCY SYSTEMS 
Keuble-PRIXCIPLES OF QUANTUM MECHANICS 
Keknabd— KINETIC THEORY OF GASES 
Kollbb— THE PHYSICS OF ELECTRON TUBES 
Morse— VIBRATION AND SOUND 

Paulino and Goudsiot- THE STRUCTURE OF LINE SPECTRA 
Richtmyerand Kshhab©— INTBODtTCTlONTO MODERN PHYSICS 
Ruark and Urey— ATOMS, MOLECULES AND QUANTA 
Seitz— THE MODERN THEORY OF SOLIDS 
Slatuib— INTRODUCTION TO CHEMICAL PHYSICS 
MICROWAVE TRANSMISSION 

Slater and Frank— ELECTROMAGNETISM 

INTRODUCTION TO THEORETICAL PHYSICS 

MECHANICS 
Smythe-STATIC AND DYNAMIC ELECTRICITY 
Stratton— ELECTROMAGNETIC THEORY 
White— INTRODUCTION TO ATOMIC SPECTRA 
Williams— MAGNETIC PHENOMENA 



Dr. Lrso A. Du Bridge was consulting editor of the series from H139 
to 1046. 



VIBRATION 
AND SOUND 



By 
PHILIP M, MORSE 

Director, Broukhaven National Laboratory 



SECOND EDITION 



NEW YORK TORONTO LONDON 

McGRAW-HILL BOOK COMPANY, INC, 
1948 



VIBRATION AND SOUND 

Copyright, 1936, 1048, by the 
McCiuAW-HiLL Book Company, Inc. 

PRINTED IN THE UNITED STATES OF AMERICA 

All riykts reserved. This book, or 

parts thereof, may not be reproduced 

in any form without per mission of 

(he publishers, 



THE MAPLE PRESS COMPANY, YORK, PA. 



TO 
PROFESSOR DAYTON C. MILLER 



PREFACE TO THE SECOND EDITION 



The recent war induced a considerable development in the science 
of acoustics and in the mathematical techniques that are particularly 
useful in theoretical acoustics. Studies in ultrasonics have quickened 
interest in problems of radiation and scattering and in transient 
phenomena; and the rapid development of microwave techniques, 
which parallel acoustic techniques, has stimulated interest in the 
general theory of wave motion. 

Reflecting these developments, the present edition of this volume 
includes more detail than the first edition on radiation problems and 
introduces the important subject of transient phenomena and the 
technique of the operational calculus. Both of these subjects arc 
usually shunned because of their reputed difficulty. The writer is 
convinced that they are not particularly difficult conceptually, 
although they usually necessitate calculations of rather wearisome 
extent. They are here discussed with the intent to bring out funda- 
mental ideas, rather than to ensure mathematical rigor of treatment. 

In keeping with the plan of the first edition, the more difficult sub- 
jects are segregated at the terminations of each chapter, so that they 
need not be assigned if the book is to be used for a beginning course. 

The writer is pleased to acknowledge his indebtedness to a large 
number of friends for valuable suggestions as to improvements in 
exposition, grammar, arithmetic, and aJgebra. Some of the most 
obvious errors in the first edition have been corrected. 

Especial thanks are owing J. R. Pel lam, who has expended much 
time and effort in checking the manuscript and mathematics, and to 
Dr. Cyril Harris for his many helpful suggestions concerning subject 
matter. 

Philip M. Mohsk 
Upton, n. Y. 

January, 1948 



rii 



PREFACE TO THE FIRST EDITION 



The following book on the theory of vibrations and sound is 
intended primarily as a textbook for students of physics and of com- 
munications engineering. After teaching the introductory course in 
this subject at the Massachusetts Institute of Technology for several 
years, the author has become persuaded that there is need for a new 
textbook in the field. 

There are, of course, many other boobs on the theory of sound. 
The author's excuse for adding another to the list is that in the past 
ten years the rapid growth of atomic physics has induced a complete 
reorganization of the science of acoustics. The vacuum tube and the 
other applications of electronics have provided immensely powerful 
tools for the measurement, recording, and reproduction of sound; 
tools which have revolutionized acoustic technique. Another useful 
tool, perhaps not so obvious, is the new mathematical technique which 
has been developed for the working out of quantum mechanics, and 
which is capable of throwing light on all problems of wave theory. 
The last chapter of this book is an example of the utility of these 
methods. In it the mathematical methods developed for the study of 
the radiation of light from an atom are applied to the theory of the 
acoustic properties of rooms. 

During the recent rapid change in the science of sound, certain 
parts of the subject have gained and other parts have lost importance. 
The present book attempts to follow this change in emphasis and to 
discuss the new development as well as those portions of the older 
theory which are still import-ant. 

The book has been planned as a textbook with a twofold aim 
in view. The first aim, of course, is to give the student a general 
introduction to the theory of vibration and sound. An introduc- 
tory course in this subject must of necessity be more theoretical than 
practical. In no other branch of physics are the fundamental measure- 
ments so hard to perform, and the theory relatively so simple; and in 
few other branches are the .experimental methods so dependent on a 
thorough knowledge of theory. Since this is so, the student must 
first be given a physical picture of the fundamental theory of the 

ix 



x PREFACE TO THE FIRST EDITION 

vibration of solid bodies and the propagation of sound waves before 
he can appreciate the techniques used in the measurements of sound, 
and before he can begin to design acoustical apparatus. 

The second aim is to give the student a series of examples of the 
method of theoretical physics; the way a theoretical physicist attacks 
a problem and how he finds its solution. This subject is too often 
neglected, especially in engineering courses. The student is usually 
given a series of formulas to use in standardized cases, the formulas 
sometimes introduced by a cursory derivation and sometimes with no 
derivation at all. After such a course the student is capable of using 
the formulas on standard problems, but he is unable to devise a new 
formula to use in unusual cases. 

In this book the author has tried to derive every formula from 
the fundamental laws of physics {there arc a few exceptions to this 
procedure) and to show in some detail tin: steps in these derivations 
and their logical necessity. This docs not mean I. hut the mathematical 
machinery is given in excessive detail, but that the steps in the physical 
reasoning are brought out. Often generality and mathematical rigor 
have been sacrificed to make the chain of logic more distinct. It has 
been the author's experience that once the student can grasp the 
physical picture behind a mathematical derivation, he can himself add 
what extra generality and rigor he may need. Often, too, the author 
has supplemented or replaced the rigid and esoteric technical vocab- 
ulary by more colloquial phrases, in order to make vivid a concept, 
or to suggest a new point of view. 

It is assumed that the student has a thorough knowledge of 
calculus, and some acquaintance with the fundamental laws of 
mechanics. A knowledge 6f differential equations is helpful but is 
not necessary, for the solutions of the various differential equations 
encountered are worked out in the text. Tables of the functions used 
are given in the back of the book. 

Although the book is designed primarily as a textbook, a certain 
amount of material of an advanced nature has been introduced. In 
this way, it is hoped, the volume will be useful as a fairly complete 
reference work for those parts of the theory of sound which seem at. 
present to be most important for the acoustical scientist. The 
advanced material has been included in the form of extra sections 
placed at the end of various chapters. The instructor may assign the 
first few sections of these chapters for the introductory course, and the 
student may refer to the other sections for further details when he 
needs them. 



PREFACE TO THE FIRST EDITION » 

The author wishes to express his gratitude to Professor R. D. 
Fay and to Dr. W. M. Hall, whose help in choosing subject matter 
and methods of presentation has been invaluable. Ho is also indebted 
to Dr. J. B. Fisk for his willing and painstaking aid in correcting proof, 
and to many other colleagues in the Department of Physics at the 
Massachusetts Institute of Technology, for their many helpful 

suggestions. 

Philip M. Morse 

Cambridge, Mass. 
August, 1936 



CONTENTS 

Preface to the Second Edition vii 

Preface to the First Edition. . -, ix 

Chapter I 
INTRODUCTORY 

1. Definitions and Methods. 1 

Units. Energy 

2. A Little Mathematics 3 

Tho Trigonometric Functions. Beasel Functions. The 

Exponential, Conventions as to Sign. Other Solutions. 

Contour Integrals. Infinite Integrals. Fourier Trans- 
forms 

Problems . • 1? 

Chapter II 

THE SIMPLE OSCILLATOR 

3. Free Oscillations 20 

The General Solution. Initial Conditions. Energy of 
Vibration 

4. Damped Oscillations 23 

The General Solution. Energy Relations 

5. Forced Oscillations 27 

The General Solution. Transient and Steady State. 
Impedance and Phase Angle. Energy Relations. Electro- 
mechanical Driving Force. Motional Impedance. Piezo- 
electric Crystals 

xiii 



xiv CONTENTS 

Section 

6. Response to Transient Forces 42 

Representation by Contour Integrals, Transients in a 
Simple System. Complex Frequencies. Calculating the 
Transients. Examples of the Method, The Unit Func- 
tion. General Transient, Some Generalizations. La- 
place Transforms 

7. Coupled Oscillations . 52 

The General Equation. Simple Harmonic Motion. Nor- 
mal Modes of Vibration, Energy Relations. The Case of 
Small Coupling. The Case of Resonance. Transfer of 
Energy. Forced Vibrations. Resonance and Normal 
Modes. Transient Response 
Problems . . 66 

CfiAl'TEK III 

THE FLEXIBLE STRING 

8. Wares on a Siring 71 

The Wave Velocity, The General Solution for Wave 
Motion. Initial Conditions. Boundary Conditions. Re- 
flection at a Boundary. Strings of Finite Length 

9. Simple Harmonic Oscillations 80 

The Wave Equation. Standing Waves. Normal Modes. 
Fourier Series. Initial Conditions. The Scries Coeffi- 
cients, Plucked String, Struck String. Energy of 
Vibration 

10. Forced Vibrations. . . , 91 

Wave Impedance and Admittance. General Driving 
Force. String of Finite Length. Driving Force Applied 
Anywhere. Alternative Scries Form. Distributed Driv- 
ing Force, Transient Driving Force. The Piano String. 
The Effect of Friction. Characteristic Impedances and 
Admittances 

11. Strings of Variable Density and Tension , . . , 107 

General Equation of Motion. Orthogonality of Character- 
istic Functions. Driven Motion. Nonuniform Mass. 
The Sequence of Characteristic Functions. The Allowed 



CONTENTS XV 

Section 

Frequencies. Vibrations of a Whirling String, The Al- 
lowed Frequencies. The Shape of the String. Driven 
Motion of the Whirling String 

12. Perturbation Calculations. 122 

The Equation of Motion. First-order Corrections. Ex- 
amples of the Method. Characteristic Impedances. 
Forced Oscillation. Transient Motion 

13. Effect of Motion of the End Supports. . . 133 

Impedance of the Support. Reflection of Waves. Hyper- 
bolic Functions. String Driven from One End. Shape of 
the String. Standing Wave and Position of Minima. 
Characteristic Functions. Transient Response. Reca- 
pitulation 
Problems M7 

Chafer TV 

THE V JURATION OF BARS 

14. The Equation of Motion 151 

Stresses in a Bar. Bending Moments and Shearing Forces. 
Properties of the Motion of I he Bar. Wave Motion in an 
Infinite Bai- 
ls. Simple Harmonic Motion 156 

Bar Clamped at One End. The Allowed Frequencies. 
The Characteristic Functions. Plucked and Struck Bar. 
Clamped-clamped and Free-free Bars. Energy of Vibra- 
tion. Nonuniform Bar. Forced Motion 

16. Vibrations of a Stiff String 1G6 

Wave Motion on a Wire. The Boundary Conditions. 
The Allowed Frequencies 
Problems 170 

ClI.Vl'TKlt V 

MEMBRANES AND PLATES 

17. The Equation of Motion ', 172 

Forces on a Membraue. The Laplacian Operator. Bound- 



xvi CONTENTS 

Section 

ary Conditions and Coordinate Systems. Reaction to a 
Concentrated Applied Force 

IS. Thz Rectangular Membrane. 177 

Combinations of Parallel Waves. Separating the Wave 
Equation. The Normal Modes. The Allowed Fre- 
quencies. The Degenerate Case. The Characteristic 
Functions 

19, The Circular Membrane . , 183 

Wave Motion on an Infinite Membrane. Impcrmanence 
of the Waves, Simple Harmonic Waves. Bessel Func- 
tions. The Allowed Frequencies. The Characteristic 
Functions. Relation between Parallel and Circular 
Waves. The Kettledrum. The Allowed Frequencies 

2U. Forced Motion, The Condenser Microphone . 195 

Neumann Functions. Unloaded Membrane, Any Force. 
Localized Loading, Any Force. Uniform Loading, Uni- 
form Force. The Condenser Microphone. Electrical Con- 
nections. Transient Response of Microphone 

21. The Vibration of Plates 208 

The Equation of Motion. Simple Harmonic Vibrations. 
The Normal Modes. Forced Motion 

Problems , 213 

Chapter VI 

PLANE WAVES OF SOUND 

22. The Equation of Motion. 217 

Waves along a Tube. The Equation of Continuity. 
Compressibility of the Gas, The Wave Equation. Energy 
in a Plane Wave. Intensity. The Decibel Scale. Inten- 
sity and Pressure Level. Sound Power. Frequency Dis- 
tribution of Sounds. The Vowel Sounds 

23. The Propagation of Sounds in Tubes . ........... 233 

Analogous Circuit Elements. Constriction. Tank. Ex- 
amples. Characteristic Acoustic Resistance. Incident 
and Reflected Waves, Specific Acoustic Impedance. 



CONTENTS xvii 

Snrnont 

Standing Waves. Measurement of Acoustic Impedance, 
Damped Waves. Closed Tube. Open Tube. Small- 
diameter Open Tube. Reed Instruments. Motion of the 
Reed, Pressure and Velocity at the Reed. Even Har- 
monics. Other Wind Instruments. Tube as an Analo- 
gous Transmission Line. Open Tube, Any Diameter. 
Cavity Resonance, Transient Effects, Flutter Echo 

24. Propagation of Sound in Horns , . 265 

One-parameter Waves. An Approximate Wave Equation. 
Possible Horn Shapes. The Conical Horn. Transmission 
Coefficient- A Horn Loud-speaker. The Exponential 
Horn. The Catenoidal Horn, Reflection from the Open 
End, Resonance. Wood-wind Instruments. Transient 
Effects 

Problems 288 

CnAI'TKlt VII 

TBS RADIATION AND SCATTERING OF SOUND 

25. The Wave Equation, . 294 

The Equation for the Pressure Wave. Curvilinear Coordi- 
nates 

26. Radiation from Cy folders , 297 

The General Solution. Uniform Radiation. Radiation 
from a Vibrating Wire. Radiation from an Element of a 
Cylinder. Long- and Short-wave Limits. Radiation from 
a Cylindrical Source of General Type, Transmission inside 
Cylinders. Wave Velocities and Characteristic Imped- 
ances. Generation of Wave by Piston 

27. Radiation from Spheres 311 

Uniform Radiation. The Simple Source. Spherical Waves 
of General Form. Legendre Functions. Bessel Functions 
for Spherical Coordinates, The Dipole Source. Radiation 
from a General Spherical Source. Radiation from a Point 
Source on a Sphere. Radiation from a Piston Set in a 
Sphere 



xviii CONTENTS 

Section 

28. Radiation from a Piston in a Plane Wall 326 

Calculation of the Pressure Wave. Distribution of In- 
tensity. Effect of Piston Flexure on Directionality, Radi- 
ation Impedance, Rigid Piston. Distribution of Pressure 
over the Piston, Nonuniform Motion of the Piston. 
Radiation out of a Circular Tube. Transmission Coeffi- 
cient ■ for a Dynamic Speaker. Design Problems for 
Dynamic Speakers. Behavior of the Loud-speaker, 
Transient Radiation from a Piston 

29. The Scattering of Sound 346 

Scattering from a Cylinder. Short Wavelength Limit. 
Total Scattered Power. The Force on the Cylinder. 
Scattering from a Sphere. The Force on the Sphere. 
Design of a Condenser Microphone. Behavior of the 
Microphone 

30. The Absorption of Sound at a Surface 360 

Surface Impedance, Unsupported Panel. Supported 
Panel. Porous Material. Equivalent Circuits for Thin 
Structures. Formulas for Thick Panels. Reflection of 
Plane Wave from Absorbing Wall 

31. Sound Transmission through Ducts 308 

Boundary Conditions. Approximate Solutions. Principal 
Wave, Transient Waves. The Exact Solution. An Ex- 
ample 

Problems , , . . 37(3 

Chapter VIII 

STANDING WAVES OF SOUND 

32. Normal Modes of Vibration 381 

Room Resonance, Statistical Analysis for High Fre- 
quencies. Limiting Case of Uniform Distribution. Ab- 
sorption Coerneinnt. Reverberation. Reverberation Time. 
Absorption Coefficient and Acoustic Impedance. Standing 
Waves in a Rectangular Room. Distribution in Frequency 
of the Normal Modes. Axial, Tangential, and Oblique 
Waves. Average Formulas for Numbers of Allowed Fre- 
quencies. Average Number of Frequencies in Band. 



CONTENTS xix 

SECTION 

The Effect of Room Symmetry. Nonrect angular Rooms. 
Frequency Distribution for Cylindrical Room 

33. Damped Vibrations, Reverberation 401 

Rcct tmgular Room, Approximate Solution. Wall Coeffi- 
cients and Wall Absorption. Reverberation Times for 
Oblique, Tangential, and Axial Waves. Decay Curve for 
Rectangular Room. Cylindrical Room. Second -order 
Approximation. Scattering Effect of Absorbing Patches 

34. Forced Vibrations. 413 

Simple Analysis for High Frequencies. Intensity and 
Mean-square Pressure. Solution in Series of Characteristic 
Functions. Steady-state Response of a Room, Rec- 
tangular Room. Transmission Response. The Limiting 
Case of High Frequencies. Approximate Formula for 
Response. Exact Solution. The Wall Coefficients, 
Transient Calculations, Impulse Excitation. Exact Solu- 
tion for Reverberation 

Problems 429 

bibliography 433 

Glossary of Symbols .,,,,.,.,,......... 435 

Tables of Functions 438 

I and II, Trigonometric and Hyperbolic Functions. Ill 
and IV, Hyperbolic Tangent of Complex Quantity. V, VI 
and YU, Bessel Functions. VTII, Impedance Functions 
for Piston. IX, Legendre Functions. XII, General Im- 
pedance Functions for Piston. XIII, Absorption Coeffi- 
cients 

Plates 453 

I and II, Hyperbolic Tangent Transformation.. Ill, 
Magnitude and Phase Angles of sinh and cosh. IV, 
Standing Wave Ratio and Phase vs. Acoustic Impedance. 
V, Hxact Solutions for Wave Modes in Rectangular Ducts 
and Rooms. VI, Absorption Coefficient vs, Acoustic 
Impedance 

Index 459 



VIBRATION AND SOUND 



CHAPTER I 
INTRODUCTORY 

1. DEFINITIONS AND METHODS 

The discussion of any problem in science or engineering has two 
aspects: the physical side, the statement of the facts of the case in 
everyday language and of the results in a manner that can be checked 
by experiment; and the mathematical side, the working out of the 
intermediate steps by means of the symbolized logic of calculus. 
These two aspects are equally important and are used side by side in 
every problem, one checking the other. 

The solution of the problems that we shall meet in this book will, 
in general, involve three steps: the posing of the problem, the inter- 
mediate symbolic calculations, and the statement of the answer. 
The stating of the problem to be solved is not always the easiest part 
of an investigation. One must decide which properties of the system 
to be studied are important and which can be neglected, what facts 
must be given in a quantitative manner and what others need only a 
qualitative statement. When all these decisions are made for problems 
of the sort discussed in this book, we can write down a statement some- 
what as follows: Such and such a system of bodies is acted on by such 
and such a set of forces. 

We next translate this statement in words into a set of equations 
and solve the equations (if we can). 

The mathematical solution must then be translated back into 
the physical statement of the answer: If we do such and such to the 
system in question, it will behave in such and such a manner. It is 
important to realize that the mathematical solution of a set of equa- 
tions is not the answer to a physical problem; we must translate the 
solution into physical statements before the problem is finished. 

Units.— The physical concept that force causes a change in the 
motion of a body has its mathematical counterpart in the equation 

F = Jt (mv) (1.1) 



2 INTRODUCTORY [LI 

In order to link the two aspects of this fact, we must define the physical 
quantities concerned in a quantitative manner; we must tell how each 
physical quantity is to be measured and what standard units of 
measure are to be used. The fundamental quantities, distance, mass, 
and time, can be measured in any arbitrary units, but for convenience 
we use those arbitrary units which most of the scientific world is using: 
the centimeter, the gram, and the second. (The units of the few other 
quantities needed, electrical, thermal, etc., will be given when we 
encounter them.) 

The units of measure of the other mechanical quantities are denned 
in terms of these fundamental ones. The equation F = d(mv)/dt 
is not only the mathematical statement of a physical law, it is also the 
definition of the unit of measure of a force. It states that the amount 
of force, measured in dynes, equals the rate of change of momentum in 
gram centimeters per second per second. If force were measured in 
other units than dynes, this equation would not be true; an extra 
numerical factor would have to be placed on one side or the other of the 
equality sign. 

Energy. — Another physical concept which we shall often use is 
that of work, or energy. The wound-up clock spring can exert a force 
on a gear train for an indefinite length of time if the gears do not move. 
It is only by motion that the energy inherent in the spring can be 
expended. The work done by a force on a body equals the distance 
through which the body is moved by the force times the component 
of the force in the direction of the motion; and if the force is in dynes 
and the distance is in centimeters, then the work is given in ergs. The 
mathematical statement of this is 

W = fP-ds (1.2) 

where both the force F and the element of distance traveled ds are 
vectors and their scalar product is integrated. 

If the force is used to increase the velocity of the body on which 
it acts, then the work that it does is stored up in energy of motion 
of the body and can be given up later when the body slows down 
again. This energy of motion is called kinetic energy, and when 
measured in ergs it is equal to mv 2 /2. 

If the force is used to overcome the forces inherent in the system — 
the "springiness" of a spring, the weight of a body, the pull between 
two unlike charges, etc. — then the work can be done without increasing 
the body's velocity. If we call the inherent force F (a vector), then 



1.2] A LITTLE MATHEMATICS 3 

the work done opposing it is W = — JF • ds. In general, this amount 
depends not only on the initial and final position of the body but also 
on the particular way in which the body travels between these posi- 
tions. For instance, owing to friction, it takes more work to run an 
elevator from the basement to the third floor, back to the first, and 
then to the fourth than it does to run it directly to the fourth floor. 

In certain ideal cases where we can neglect friction, the integral W 
depends only on the position of the end points of the motion, and in 
this case it is called the potential energy V of the body at its final 
position with respect to its starting point. If the body moves only in 
one dimension, its position being given by the coordinate x, then 



F--j>* or ,--g) 



(1.3) 



In such a case we can utilize Newton's equation F = m(dv/dt) to 
obtain a relation between the body's position and its velocity. 

V = — I Fdx = — ra I (-rrjdx = — ra I vdv = — %mv 2 + constant 



or 



imv 2 + V = constant (1.4) 

This is a mathematical statement of the physical fact that when 
friction can be neglected, the sum of the kinetic and potential energies 
of an isolated system is a constant. This proof can be generalized 
for motion in three dimensions. 

2. A LITTLE MATHEMATICS 

With these definitions given we can go back to our discussion of 
the twofold aspect of a physical problem. Let us take an example. 

Suppose we have a mass m on the end of a spring. We find that 
to keep this mass displaced a distance x from its equilibrium position 
requires a force of +Kx dynes. The farther we push it away from its 
equilibrium position, the harder the spring tries to bring it back. 
These are the physical facts about the system. The problem is to 
discuss its motion. 

We see physically that the motion must be of an oscillatory nature, 
but to obtain any quantitative predictions about the motion we must 
have recourse to the second aspect, the mathematical method. We 
set up our equation of motion F — m(d 2 x/dt 2 ), using our physical 
knowledge of the force of the spring F = — Kx. 



4 INTRODUCTORY [1.2 

^ + n 2 x = (2.1) 

where n 2 stands for (K/m). 

The physical statement corresponding to this equation is that 
the body's acceleration is always opposite in sign and proportional 
to its displacement, i.e., is always toward the equilibrium point 
x = 0. As soon as the body goes past this point in one of its swings, 
it begins to slow down; eventually it stops and then returns to the 
origin again. It cannot stop at the origin, however, for it cannot 
begin to slow down until it gets past the origin on each swing. 

The Trigonometric Functions. — Equation (2.1) is a differential 
equation. Its solution is well-known, but since we shall meet more 
difficult ones later, it is well to examine our method of obtaining the 
solution, to see what we should do in more complicated cases. We 
usually just state that the solution to (2.1) is 

x = a cos (nt) + fli sin (nt) (2.2) 

but if we had no table of cos or sin, this statement would be of very 
little help. In fact, the statement that (2.2) is a solution of (2.1) 
is really only a definition of the symbols cos and sin. We must have 
more than just symbols in order to compute x for any value of t. 

What we do — what is done in solving every differential equation — 
is to guess the answer and then see if it checks. In most cases the 
guess was made long ago, and the solution is familiar to us, but the 
guess had nevertheless to be made. 

So we shall guess the solution to (2.1). We shall make a pretty 
general sort of guess, that re is a power series in t, and then see if some 
choice of coefficients of the series will satisfy the equation: 

x = a + ait + a 2 t 2 + a s t s + a^ A + • • • 

For this to be a solution of (2.1), we must have d 2 x/dt 2 , which equals 
2a 2 + Qa 3 t + I2a4 2 + 20a 5 t 3 + • ■ • plus n 2 x, which equals n 2 a + 
n 2 a x t + n 2 arf, 2 + n 2 a z t z + • • • equal zero for every value of t. The 
only way that this can be true is to have the sums of the coefficients 
of each power of t each zero by themselves; i.e., a 2 = —n 2 a /2, a 3 = 
— n 2 ai/6, a 4 = — ra 2 a 2 /12 = n 4 a /24:, etc. Therefore the series that 
satisfies (2.1) is 



( i _ VlIl a- Vl!l _ Vl!l 4- ^ 

V 2 + 24 720 + ' ' ' / 

V 6 + 120 ' ' / 



X = Go 

+ aAt- 



1.2] A LITTLE MATHEMATICS 5 

By comparing this with (2.2) we see that what we actually mean by 
the symbols cos, sin is 

»2 «4 «*> 

cos(*) = l-- 2 - + 2l-^+ t 

sm(z) = 2 _._ + _ _ ^q + 

(in the problem 2 = nt) and that when we wish to compute values of 
cos or sin we use the series expansion to obtain them. For instance, 

cos (0) = 1 - + • • • = 1 

cos(i) = 1 - 0.125 + 0.003 • • • = 0.878 

cos (1) = 1 - 0.5 + 0.042 - 0.002 • • • = 0.540, etc. 

We see that the mathematical solution involves certain arbitrary 
constants «o and a\. These must be fixed by the physical conditions 
of the particular experiment we have made and will be discussed 
later (Sec. 3). 

Of course, we usually mean by the symbols cos (z), sin (z) certain 
ratios between the sides of a right triangle whose oblique angle is z. 
To make our discussion complete, we must show that the trigonometric 
definitions correspond to the series given above. In any book on ele- 
mentary calculus it is shown that the trigonometric functions obey 
the following relations: 

^ cos (z) = -sin (z), -£ sin (z) = cos (z) 
cos(0) = 1, sin(0) = 

By combining the first two of these relations, we see that both the 
sine and cosine of trigonometry obey the equation d 2 y/dz 2 = — y, 
which is equivalent to (2.1). Taking into account the third and 
fourth relations, we see that the series solutions of this equation which 
correspond to the two trigonometric functions must be the ones given 
in (2.3). 

Once this is known, we can utilize the trigonometric properties of 
the sine and cosine to simplify our discussion of the solution (2.2) 
[although, when it comes to computing values, the series (2.3) are 
always used]. For instance, we can say that a; is a periodic function 
of time, repeating its motion every time that nt increases by 2ir or 
every time t increases by (2w/ri). The value of {2k /n) is called the 
period of oscillation of the mass and is denoted by the symbol T. 
The number of periods per second {n/2ir) is called the frequency of 



6 INTRODUCTORY [1.2 

oscillation and is denoted by v. Remembering the definition for n, 
we see that 



t =**4> -(1)4 



Bessel Functions. — More complicated differential equations can 
be solved by means of the same sort of guess that we used above. 
For instance, we can solve the equation 

s+iS+0-?)'- ™ 

by again guessing that y = a + a x x + a 2 x 2 + a 3 x z + • • • and set- 
ting this guess in the equation. Thus: 

a + dix + a 2 x 2 + a 3 x s + • • • 

ai — a 3 x — aix 2 — a&x s — • • • 

x 

— + 2a 2 + 3a 3 x + 4a 4 x 2 + ba h x z + • • • 
x 

2a 2 + Qa 3 x + 12a 4 ;c 2 + 20a 5 £ 3 + ■ • ■ 

must equal zero. Equating coefficients of powers of x to zero as 
before, we have 

a = 0, o 2 = — 7^ = 0, a 3 = — g- 

rt a 3 ai , 

a 4 = 0, a 5 = ~ 24 = g . 24 * etc - 

Therefore, 



2/ = 




1 

minus ^ y = 


X 2 


, 1 rfw 
plus - ~r = 
x ax 




1 d 2 y 
P lus ^i = 





/ z 3 

^"2^4 



2/ = a x [ x - ^ + 2 . 4 . 4 . 6 - 2-4-6-4-6-8 + 



We shall call this series a Bessel function, just as we call each of the 
series in (2.3), a trigonometric function. To save having to write 
the series out every time, we shall give it a symbol. We let 



1/ x s 

2\ X 2-4 



Ji(x) = 2V X " 2^1 + 2-4-4-6 ' ) (2 ' 5) 

just as we represent series (2.3) by the symbols cos and sin. 

Essentially the series (2.5) is no more complicated than series (2.3). 
We can compute values of J\(x) in the same manner as we computed 
values of cos (2). 



1.2] A LITTLE MATHEMATICS 7 

J x (0) = - • • • = 

JiG) = 0.250 - 0.008 ■ • • = 0.242 

Ji(l) = 0.5 - 0.063 + 0.003 • • • = 0.440, etc. 

Once someone has obtained series (2.5), given us the symbol Ji(x) 
and computed its values, we can use the symbol with as great freedom 
as we use the symbol cos (z). We can say that the solution of (2.4) 
is y = AJi(x), where the arbitrary constant A is to be determined 
by the physical conditions of the problem, as we shall discuss later. 

We notice that the solution of (2.4) involves only one function and 
one constant A, whereas solution (2.2) involved two constants and 
two different functions cos (nt) and sin (nt). Actually, there is another 
solution of (2.4), designated by the symbol Ni(x) and called a Neumann 
function; but Ni(x) becomes infinite at x = 0, and so we shall not be 
able to use it for many of the problems discussed in this book. We 
can say that the complete solution of (2.4) is y = AJi(x) + BNx(x) 
but that, since a solution representing physical facts must not become 
infinite, the constant B must be made zero in all cases where we let x 
become zero. These functions will be dealt with more fully in Chap. V. 

The Exponential. — Another very useful way of dealing with the 
solution of (2.1) can be obtained by the following chain of reasoning. 
We utilize the series method to show that a solution of the equation 
(dy/dz) = y is the series called the exponential: 

z 2 z 3 z 4 z 5 

«,= !+* + _ + _ + _ + __+... (2.6) 

By repeated differentiation we see that x = Ce at is a solution of the 
equation (d 2 x/dt 2 ) — a 2 x = 0. If a 2 were to equal — n 2 , this equation 
would be the same as (2.1). Therefore we can say that a solution of 
(2.1) is Ce~ int , where i = \/—l (we could also use e +int ). 

The function e~ iz is a complex number, with real and imaginary 
parts, and can be represented in the usual manner by a point on the 
"complex plane" whose abscissa is the real part of the function and 
whose ordinate is the imaginary part. It is also represented by the 
vector drawn from the origin to this point in the complex plane. 

If we expand e~ iz in its series form 

'*-( 1 -5 + H----)- < (*-£ + i35---') 

we see immediately that the relation between the imaginary expo- 
nential and the trigonometric functions is 



INTRODUCTORY 

e -i z _ cog (^ _ £ gm ^ 

cos (2) = i(e i!S + e~ iz ) 
sin (2) = — %i (e iz — e~ iz ) 



[L2 



(2.7) 



This shows that the number e~ iz can be represented on the complex 
plane (see Fig. 1) by a vector of unit length inclined at an angle — z 



Imag. 
Axis 



C=cio+icv 




ContouiG. 




Direction 
of Rotation 




Direction 
of Rotation 

Fig. 1. — Representation of complex numbers on the complex plane. Integration 
around a contour G in the complex plane. Relation between i and j. 

with respect to the real axis. It also shows th at the co mplex number 
a + ib can be expressed as Ae'*, where A = Va 2 + b 2 and tan (<3?) = 
(b/a). For Ae™ = A cos($) + iA sin(<i>); and since cos(3>) = (a/A) 



1.2] A LITTLE MATHEMATICS 9 

and sin ($) = (b/A), we have Ae i# = a + i&. The factor e** signifies 
a rotation of the vector A through an angle $ in the complex plane. 

Thus we see that Cer int is a solution of (2.1) and can be expressed 
as a combination of cos (nt) and sin (nt), as is demanded by (2.2). 
But this new solution is a complex number, and the results of physics 
are, in general, real numbers, so the new solution would seem to have 
little value to us. What can be done, however, is to write down the 
solution as being Ce~ int and make the convention that we use only the 
real part of this solution when we use it to check physical measure- 
ments. All through this book we shall be writing down complex 
solutions of differential equations, with the convention that we use 
only their real parts. It is possible to do this with the solution of 
any linear differential equation (i.e., equations containing only the 
first power of the unknown function and its derivatives) ; for if a com- 
plex function is a solution of a linear differential equation, then both 
its real and imaginary parts by themselves are also solutions. (Why 
is this not true for nonlinear differential equations?) We could, of 
course, make a convention that we use only the imaginary part of 
the function, for the imaginary part is also a solution of the equation, 
but the usual convention is to 'take the real part. 

We can therefore express solution (2.2) as the real part of 

x = Ce~ int , C = a + iax (2.8) 

This can be checked by the use of (2.7), for 

Ce~ int = a cos (nt) + ia x cos (nt) — ia sin (nt) + ai sin (nt) 

and the real part of this corresponds to (2.2). The advantages of 
this method will become apparent as we use it. For instance, since 
C = Ae { *, A 2 = al + a\, tan(f>) = (ai/o ), we see that the real part 
of Ce~ int = 4 e -*'c»*-*) is A cos (nt + <£>). Therefore we can express the 
solution of (2.1) in any of four ways: 

x = a cos (nt) + ai sin (nt) = A cos (nt — $) (2.9) 

or 

x = Ce~ int = Aer-t^-v 

where our convention requires that we take only the real part of the 
last expressions. The constant A is called the amplitude of oscillation 
of the mass, since it is the value of the maximum displacement of the 
mass from equilibrium. 

Our conventions as to the use of complex quantities must be used 
with tact when we come to compute powers and energies, where squares 



10 , INTRODUCTORY [1.2 

Of quantities enter. For instance, in a given problem the power radi- 
ated may turn out to be Rv 2 , where v is the velocity of a diaphragm 
and R is a real number. The velocity may be represented by the 
expression v= Ce~ 2 * ivt , C = a + ia h just as x is in Eq. (2.9), But 
to compute the average power radiated, we must take the real part 
of the expression before squaring and averaging, 

Rv 2 = RA 2 cos 2 (2irvt - $) 

according to Eq. (2.9) . Since the average value of cos 2 is £, the average 
power radiated will be 

iRA 2 = iR(a 2 + a 2 ) = i#|C| 2 = %R\v\ 2 
\v\ 2 = (real part of v) 2 + (imaginary part of v) 2 

If z — x + iy is any complex number, the quantity \z\ is called the 
magnitude of z. It is the distance from the origin to the point in the 
complex plane corresponding to z, and it equals the square root of 
the sum of the squares of the real and imaginary parts of z. There- 
fore another important rule concerning the use of complex quantities 
in physical problems is the following: the average value of the square 
of a quantity represented by the complex function Ce~ iat is equal to 
one-half of the square of the magnitude of C. 

The angle <i> in Eq. (2.9) is called the phase angle of the complex 
quantity C, since it measures the angle of lag of the quantity Ce~ int 
behind the simple exponential e~ int . We notice that the phase angle 
of i is 90 deg, that of (-1) is 180 deg, and that of (-i) is 270 deg or 
90deglead. If a; = Ce' int , the velocity v = (dx/dt) = - inCer*" leads 
the displacement x by 90 deg, as is indicated by the fact that v = 
( — i)nx. 

Conventions as to Sign. — In the present volume we use the letter i 
to stand for V^-l and the symbol e~ int to express simple harmonic 
dependence on time. Many books on electrical engineering use the 
symbol j instead of i and e jnt with the positive sign in the exponential. 
Since we intend to take the real part of the result, the choice of symbol 
and sign is but a convention; either convention is satisfactory if we 
are only consistent about it. 

As long as we are studying simple systems, with displacements a 
function only of time, there is little to choose between the two con- 
ventions; as a matter of fact the positive exponential e int would be 
slightly preferable. However, as soon as we come to study wave 
motion, with displacement a function of position as well as of time, 
it turns out that the form involving the negative exponential e~ int is 



1.2] A LITTLE MATHEMATICS 11 

rather more satisfactory than the positive. This will become appar- 
ent later in this book. 

Since we are going to deal with problems of wave motion in this 
book we shall consistently use the negative exponential e~ int . And 
since this convention differs from the positive exponential used in 
most electrical engineering books, we shall use the symbol i instead 
of j, to emphasize the difference. It will, however, always be possible 
to transform any of the formulas developed in this book into the usual 
electrical engineering notation by replacing every iin the formulas by —j. 
We might, if we wish, consider i and j as the two roots of ( — 1), so 
that 

i = -j; (iy = (j)2 = _i 

In this notation, the impedance of a circuit with resistance and induct- 
ance in series will be R — iuL. 

Other Solutions. — Power series are not always the best guesses for 
the solutions of differential equations. Now that we have denned the 
exponential function, we can sometimes express solutions of other 
equations in terms of exponentials. Consider the equation 

-^ + n 2 x = ae~ ipt (2.10) 

where, to represent a physical problem, we must use our convention 
on the right-hand side of the equation. This could be solved by 
expanding the exponential in a series and solving for the series for x 
as before. But since an exponential is in the equation, and since we 
know that Ce~ int is the solution of (2.10) when a is zero, it will be 
simpler to guess that 

x = Ce~ int + Be~ lpt 

Setting this in the equation, we have that n 2 x = n 2 Ce~ int + n 2 Be~ ipt 
plus d 2 x/dt 2 = -n 2 Ce~ int - p 2 Be~ ipt must equal ae~ ipt . This means 
that B = a/(n 2 — p 2 ). ,If we have used our convention on the term 
ae -i P t m (2.10), we must use it on the answer. The real part of 

x = Ce~ int + 



.i 2 — p 2 
is 

x = a cos (nt) + a x sin (nt) + -^ — ^ cos (pt) (2.11) 

if a is a real quantity. If a is complex and equals De**, then we can 



12 INTRODUCTORY [1.2 

write 

x = A cos (nt - 4> ) + n2 _ 2 cos (p£ - tfO 

We notice again that in (2.11) our solution has two arbitrary con- 
stants a and a x . The arbitrariness corresponds to the fact that this 
solution must represent all the possible sorts of motion which the 
system can have when it is acted on by the forces implied by Eq. 
(2.10). A mass on a spring can have different motions depending 
on how it is started into motion at time t = 0. Therefore the par- 
ticular values of the arbitrary constants a and a x in (2.2) are deter- 
mined entirely by the physical statements as to how the system was 
started into motion. These physical statements are called initial 
conditions and are usually stated by giving the position and velocity 
of the system at t = 0. More will be said on this point in the next 
section. 

Contour Integrals. 1 — In a number of cases discussed in this volume 
it will be necessary to use integrals of complex quantities of the general 
form $F(z) dz, where F = U + iV is a function of the complex varia- 
ble z = x + iy. Since z can vary over the x,?/-plane, it will in general 
be necessary to specify the path over which the integration is carried 
out. A natural extension of our usual definition of integration indi- 
cates that the integral is the limit of a series of terms, a typical term 
being the product of the value of F{z) for a value of z on the path, and 
a quantity dz = \dz\e i& , with length \dz\ and with phase angle & deter- 
mined by the direction of the tangent to the path at the point z. 
The integral is the limit of such a sum as [dz\ goes to zero. This is 
shown in Fig. 1. Another definition is that jF dz = j[U + iV][dx 
-f- i dy] where x and y are related in such a manner that the integral 
follows the chosen path. 

When the path chosen is a closed one, the integral is called a 
contour integral (labeled f) and the chosen path is called a contour. 
Contour integrals have certain remarkable properties, proved in 
standard texts on theory of functions of a complex variable. We 
have space here to outline only some of these properties. For sim- 
plicity it will be assumed that the function F{z) to be dealt with is a 
smoothly varying, reasonable sort of function over most of the z- 
plane. 

The value of a contour integral depends largely on whether the 
integrand F(z) becomes zero or infinity for some value or values of z 

1 This section need not be studied unless the later sections on transient phe- 
nomena are to be studied. 



1.2] A LITTLE MATHEMATICS 1,3 

inside the contour. Near such points F(z) would take on the form 
(z — z ) n R(z) where R(z ) is neither zero or infinity and R(z) is not 
discontinuous near z . When n is not an integer, positive or negative, 
the point z is called a branch point and the problem becomes more 
complicated than is necessary to discuss here. When n is a negative 
integer, the point z is called a pole of F(z), the pole f or n = — 1 being 
called a simple pole and that for n = —2,-3 • • • being called a poZe 
of second, third, etc., order. The statement at the beginning of this 
paragraph can now be made more specific: the value of the contour 
integral $F(z) dz depends on the behavior of F at the poles and 
branch points that happen to be inside the contour. 

When F has no branch points or poles inside the contour, the value 
of the contour integral is zero. This can be verified by a tedious bit 
of algebraic manipulation for those cases where F can be expressed 
as a convergent power series in z. 

When F has a simple pole at z = z , but has no other pole or branch 
point inside the contour (see Fig. 1), then the contour integral can 
be shown to be equal to that around a vanishingly small circle drawn 
about z . Since z is specified as being a simple pole, F(z) has the 
form R(z)/(z — z ) near z where R(z) is continuous and finite in 
value near z . We can write the equation for the resulting circle as 
z — z = ee iv and dz = ue iv d<p, where e is vanishingly small. The 
contour integral then reduces to 

£F(z) dz = R(z ) f ' l * e% \ d<P = iR(z ) I d<p = 2iriR(z ) 
Jo &* Jd 

where R(z ) — lim [(z — z )F(z)) is called the residue of F{z) at its 

z—*zo 

simple pole zo. 

When F(z) has N simple poles, at z , z x • • • zn, and no other poles 
or branch points, then the contour integral j>F{z) dz equals (27ri) 
times the sum of the residues of F at the poles that are inside the chosen 
contour. We note that the direction of integration around the con- 
tour is counterclockwise; changing the direction will change the sign 
of the result. 

This is a remarkably simple result; in fact, it seems at first too 
simple to be true, until we realize that the function F(z) we are dealing 
with is a very specialized form of function. F is not just any complex 
function of x and y; its dependence on these variables is severely 
limited by the requirement that it be a function of z = x + iy. It 
can have the form bz 2 + (C/z) or z sin(z), but not the form x sin(z) -f 
iy or \z\. For such a specialized function F(z), which is called an 



14 INTRODUCTORY (1.2 

analytic function, the positions of its branch points and poles, and its 
behavior near these branch points and poles, completely determine 
its behavior everywhere else on the complex plane. When this is under- 
stood, the result we have quoted for the value of a contour integral 
does not seem quite so surprising. 

As an example of the rule of residues, we can take the integral 
<f sin (2) dz/(z 2 — a 2 ), where the contour is a circle of radius greater 
than a, with center at the origin 2 = 0. The integrand has two 
simple poles, at 2 = a and z = —a, with residues (l/2a) sin (a) and 
— (l/2a) sin ( — a). The result is therefore 



*^*-OW>[*f] 



A special case of the residue theorem can be stated in equation 
form 



§ 



z 



dz = 2tiR(z ) (2.12) 



where the contour of integration does not enclose a branch point or 
pole of R(z). By differentiating once with respect to 2 on both sides 
of the equation, we obtain the equation 



§ 



, R(Z \ 2 dz = 2riR'(z ) (2.13) 

\Z Zoj 



where R'(z) = dR(z)/dz. This indicates how we can evaluate con- 
tour integrals around poles of second order. Cases for poles of higher 
order may be dealt with by further differentiation of Eq. (2.13). 

Infinite Integrals. — An oft-encountered class of integral may be 
evaluated in terms of contour integrals. Consider the integral 



/-". w 



dz 



where the integral is taken along the real axis. If F(z) has no branch 
points or poles of higher order, if the simple poles of F are not on the 
real axis, and if F(z) goes to zero as \z\ goes to infinity, then the integral 
can be made into a contour integral by returning from + 00 to — ».." 
along a semicircle of infinite radius. When t is positive (to make it 
more general we should say when the real part of t is positive, but let 
us assume that t is real), the exponential factor ensures that the integral 
along this semicircle is zero if the semicircle encloses the lower half 
of the 2 plane. For along this path, 2 = pe iv = p cos <p + ip sin <p, 



1.2] A LITTLE MATHEMATICS 15 

with p infinitely large and with ^ going from to — -ny so that sm<p 
is negative. Therefore along the semicircle 

pr-izt ^ g— ipt cos <P + fit sin (P 

As long as sin<p is negative this vanishes, owing to the extremely large 
value of p. Therefore the addition of the integral around the semi- 
circle turns it into a contour integral but does not change its value. 

The final result is that the integral equals (— 2xi) times the sum of 
the residues of F(z)e~ izt at every one of its poles located below the real 
axis. The negative sign is due to the fact that the contour in this 
case is in the clockwise direction. 

When t is negative, we complete the contour along a semicircle 
enclosing the upper half of the z-plane, which now has the vanishing 
integral, and the resulting value is (27rc) times the sum of the residues 
at each of the poles above the real axis. The limitations on F have 
already been given. 

As an example, we can write down the result 

« 2 f " e ~** _ | (a/2)e at (t < 0) 

%r J- . z 2 + a 2 dZ \ (a/2)e~ at (t > 0) 

practically by inspection. Another result of interest later is 



ia f °° 
2^ J_ 



^ *-{•?_ <!<2 (2-14) 



z + ia [ ae~ at (t > 0) 



In some cases of interest, F has poles on the real axis. These cases 
can also be treated by making the convention that the integration is 
not exactly along the real axis but is along a line an infinitesimal 
amount above the real axis. With this convention, and for functions 
F(z) that have only simple poles and that vanish at \z\ — * « , we have 
the general formula 



sum of residues of \ 

(2ti) I ^'^ at aU itS I it > 0) 
\ poles on and below J 



/ 



+ » „ , , . ± , I \the real axis / 



F{z)e->»dz=( pi«~~» (2l5) 

/sum of residues ot\ 

+ (2wi)( Fe ~ Ut at aU itS | (t < 0) 
\ poles above the I 

\real axis / 






./^ f^^^eU 



16 INTRODUCTORY [1.2 

As an example, we can write down the very useful formulas 

_^p e -i, (t<0) 

2tJ-„z* - a*™ \ sin (<rf) (< > 0) C ^ 17; 
_J_p^T 0>O) 

Fourier Transforms. — When the quantities / and F are related by 
the formula 

/(0 = j^ a) F{z)e- i " t dz (2.19) 

then / is said to be the Fourier transform of F. A great deal of abstruse 
mathematical reasoning, which we shall not go into here, is needed to 
prove that if / is the Fourier transform of F, according to Eq. (2.19), 
then 

F{z) = ± J ^ f(f)e** dt (2.20) 

so that, contrariwise, 2tF is the Fourier transform of /. Naturally 
Eqs. (2.19) and (2.20) are valid only when F and / are reasonable sorts 
of functions, which approach zero in a proper manner as \z\ approaches 
infinity. 

When f(t) is zero for t < 0, as it is in Eqs. (2.14), (2.16), (2.17), 
and (2.18), this reciprocal integral may be written 



2irJo 



FHV) =2^J d f( l ) e ~ Pt dt (2.21) 

where p = — iz. This integral converges satisfactorily as long as the 
real part of p is positive, i.e., as long as the imaginary part of z is 
positive. 

Formula (2.21) provides a simple means of checking Eqs. (2.14) 
to (2.18). 

The usefulness of these integral formulas will become apparent 
in the succeeding chapters, as we deal with problems of the response of 
vibrating systems to transient {i.e., nonperiodic) forees. If, instead 
of Eq. (2.10), we had put 



d 2 x C " 

^f + n 2 x = I F(p)e-^ dp 



Ml' A LITTLE MATHEMATICS 17 

then the solution for x could have been expressed in terms of a con- 
tour integral. This problem will be considered in the next chapter. 

Problems 

1. A mass m. slides without friction on a horizontal table. It is attached to a 
light string which runs, without friction, through a hole in the table. The other 
end of the string is pulled downward by a constant force F. The mass (which is 
too large to drop through the hole) is held at rest a distance D from the hole and 
then let go Set up the equation of motion of the mass, and solve it. Is the 
motion periodic? If so, what is its frequency, and how does the frequency depend 
onZ)? 

2. A bead of mass m slides without friction on a straight wire that is whirling 
in a horizontal plane, about one end, with a constant angular velocity w radians 
per sec. It is found that the centrifugal "force" on the bead is mw 2 r, where r is 
the distance of the bead from the center of rotation. The direction of the force is 
away from the center. Set up the equation of motion of the bead, and solve it. 
Is the motion periodic, and, if so, what is the frequency? 

3. Show that a solution of the equation 



g+*g+-»-° 



can be represented by the series 

_ c A _ n 2 x 2 ntx* n 6 x" \ 

\ 2 2 ~*~ 2 2 • 4 2 2 2 -4 2 -6 2 ' " / 

The series inside the bracket i3 called the Bessel function J (nx), where J (z) 
= 1 - (2 2 /2 2 ) + • • • . Compute values of /,(0), Jo(l), and J (l) to three 
places of decimals. 

4. Show that the solution of the equation 

g + (1 - kx 2 ) y = 

is 

y = a D e (k, x) + aiD„(k, x) 

where D e and D are symbols for the following series: 

nri.~\ . * s 7k . , Ilk + 15k 2 , 
D e (k,x) -ooB* + i5 **-3g 6 *« + ^j )Mr - ** 

n ti. \ • , * « 13* , , 17* + 63* 2 „ 
D (k,x) - sinx + 25 *• - 2 52 q *' + -^720" * 

Compute values of D e and D , to three places of decimals, for * = 1 and for x =» 0, 
i, and 1. 

6. Show that the solution of the equation 

Tx [ (1 ~ x2) 2] + Xy = ° is y = a <>P e (\x) + ai P (\,x) 



18 INTRODUCTORY 0L2 

where P, and Pt, are symbols for the following series: 

P.(X,x) = 1 - ±x> - ^^ x« - M6-X)(20-A ) ;c< , 

P,(M)-» + V' , + <a "" X) 5 ( ? 2 "" X) "' 



| (2-X)(12-A)(30-\) x , { 



Compute values of P e and P , to three places of decimals, for X = 0, 1 and 2, for 
x = 0, £, and 1. 

6. Show that the solution of the equation 

J^(* 2 |0 + ( 1 -§) y = O is »-**<*>+«»*»<*> 

where ji and ni are symbols for the following functions: 

. , . sin a; cosx . N cosx sin a; 

Ji(x) = — ; , n x (x) = 5 

x % x ' x 2 x 

What are the values of j t and rti at x = 0, £ and 1 ? What is the solution of the 
equation 



?s(-£)+»- M 



7. What is the length of the line drawn from the origin to the point on the 
complex plane represented by the quantity (a + ib)~*? What is the angle that 
this line makes with the real axis? What are the amplitudes and phase angles of 
the following quantities: 

(a - to)-*; (a + t&) -1 + (c + id)- 1 ; (a + ib) n e- iat 

8. What are the real and imaginary parts, amplitudes, and phase angles of 

Va + ib, log (a + ib), e - 2Ti( - v+i ^ t , e~ ix (l + e^ 5 )" 1 

9. The hyperbolic functions are defined in a manner analogous to the trigo- 
nometric functions [see Eq. (2.7)]. 

cosh (z) = §(e* + e - *); sinh (z) = §(e* — e~ g ) 

tanh (z) = , ,{ = — ,. . . (see Tables I and II) 
cosh (z) coth (z) v ' 

Show that cosh (iy) = cos(y); cos (iy) =cosh(y); sinh (iy) =i sin(y); sin (iy) 
= i sin (y). Find the real and imaginary parts of cosh (x + iy), cos (x + iy), 
sinh (x + iy), sin (x + iy). What are the magnitudes and phase angles of these 
quantities? 

10. Find the real part R and imaginary part X of tanh (x + iy). 

Show that 

B. + ^ + ootCW-j^ 



1.2] A LITTLE MATHEMATICS 19 

Plot the curves on the complex plane for tanh (x + iy) when y is allowed to vary 
but x = 0, (ir/10), oo ; when x is allowed to vary but y = 0, (tt/4), (x/2). 

11. Where on the complex plane are the poles of the functions 1 /cosh (z) ; tan (z) ; 
e «/ z 2( z 2 _ a 2). tan(«)/z(z 2 +a 2 )? Are they all simple poles? Calculate the 
residues at all the simple poles of these functions. 

12. Compute the values of the integrals 



/» g—izt /• 00 

, , , — 5T dz; I tan (z)e 
- «. z(z 2 + a 2 ) J- * 



rfz 



z 



X2x 
(1 — 2p cos + p 2 ) -1 d0, to a contour 

integral over z, by setting e i9 = z. What is the shape of the contour? Where are 
the poles of the integrand, when p is real and smaller than unity? Calculate the 
value of the integral when < p < 1. 
14. Calculate the value of the integral 



16. Show that 
_ il f°° -■ 



f- 



x 2 dx 



-» (1 + x 2 )(l - 2x cos + x 2 ) 



l — (_l)n e i2(nir/u)-| I (^ < 0) 

g2 _ m2 \dz = ^ sin {at) [0 < t < (nw/o>)] 

( [(n7r/a>) < t] 



CHAPTER II 
THE SIMPLE OSCILLATOR 

3. FREE OSCILLATIONS 

Now that we have discussed, to some extent, the mathematical 
methods that we shall need in our work, we shall come back to the 
physics. 

The whole study of sound is a study of vibrations. Some part 
of a system has stiffness; when it is pulled away from its position of 
equilibrium and then released, the system oscillates. We shall first 
study the simplest possible sort of vibrations for the simplest sort of 
system, a mass m fastened to some sort of spring, so that it can oscil- 
late back and forth in just one direction. A very large number of 
vibrators with which we deal in physics and engineering are of this 
type or are approximately like it. All pendulums (the "spring" here 
is the force of gravity) are like this, and all watch balance wheels. 
Loud-speaker diaphragms which are loaded so that their mass is con- 
centrated near their center are approximately like this (at least at 
low frequencies), as are loaded tuning forks, etc. Even when an 
oscillating system is more complex than the simple oscillator, many of 
its properties are like it. Later, when we study these more compli- 
cated systems, we can simplify our discussion considerably by point- 
ing out first the properties wherein the system behaves like the simple 
case and then showing where it differs. 

The forces exerted by the various springs in the examples of the 
simple oscillator mentioned above have one property in common: 
The restoring force opposing the displacement of the mass from its 
position of equilibrium is proportional to this displacement if it is 
small enough (we remember the discussion of Hooke's law for a spring 
and the discussion of the force on a single pendulum given in the ele- 
mentary physics course). In some problems it is not very accurate to 
assume that the displacements are small enough, but in our work it is 
a good assumption, for the vibrations with which we deal in sound have 
displacements that are very small indeed. For instance, an air mole- 
cule needs to vibrate with an amplitude of motion of only about a 
tenth of a millimeter to do its full part in transmitting away the racket 

20 



H.3] FREE OSCILLATIONS 21 

generated at Times Square on New Year's Eve. Seldom does the amp- 
litude of oscillation of a loud-speaker diaphragm exceed a millimeter. 
The General Solution. — We shall assume, then, that the "springi- 
ness" force acting on the simple oscillator can be expressed by the 
equation 

F = -Kx (3.1) 

where x is the displacement of the mass from equilibrium, K is called 
the stiffness constant (its value depends on the sort of spring we use), 
and the minus sign indicates that the force opposes the displacement. 
This will be a very good assumption to make for most of the cases 
dealt with in this book. The reciprocal of the stiffness constant, 
C m = (1/K), is called the compliance of the spring. 

To start with, we shall consider that no other forces act on the 
oscillator. This is, of course, not a good assumption in many cases; 
usually there are frictional forces acting, and sometimes external 
forces come in. There are many cases, however, where the frictional 
force is negligible compared with the springiness force, and we shall 
treat these first, bringing in friction and external forces later. 

This brings us to the equation of motion (2.1) discussed in Sec. 2: 

^ = -co 2 z, w 2 = (K/m) = (l/mC m ) (3.2) 

We have already seen that the solution of this equation can be ex- 
pressed, in terms of our convention, as 

x = Ce-™ 1 , C = a + ia x 
or as 

x = a ccs {u)t) + «i sin (ut) (3.3) 

We must now discuss the physical implications of this solution. 

It has been shown in Sec. 2 that the moti on of the body is periodic, 
having a frequency v = (o)/2ir) = (l/27r) s/K./m. This frequency is 
larger for a stiffer spring (for a larger K) and is smaller for a heavier 
mass. 

Initial Conditions. — We have seen that the specific values of the 
constants a and ai are determined by the way that the mass is started 
into motion. Ordinarily, we start an oscillator into motion by giving 
it a push or by pulling it aside and letting it go, i.e., by giving it, at 
t = 0, some specified initial displacement and initial velocity. Once 
we have fixed these two initial conditions, the motion is completely 
determined from then on, unless we choose to interfere with it again. 
It is not hard to see that a is the value of the initial displacement 



22 THE SIMPLE OSCILLATOR [IL3 

and ojai that of the initial velocity. Solution (3.3) can then be rewrit- 
ten in the following forms: 

x = x cos (2rvot) + ( ~- J sin (2rvot) 

= 4 cos (2™ * -<*>); v = Q^^;} (3.4) 

*- * + &)'.■ tan * = fek) - 

where ar is the initial displacement and v the initial velocity of the 
mass. 

These equations reemphasize the fact that only the initial value of 
x and of (dx/dt) need be given to determine the subsequent motion of 
an oscillator completely. Once x and v are specified, then, even if 
we did not have a completely worked-out solution of (3.2), we 
could find the initial value of the body's acceleration by inserting x 
on the right-hand side of (3.2). The initial value of the third deriva- 
tive can be obtained by differentiating (3.2) with respect to t and 
placing the value of v on the right-hand side in place of (dx/dt); 
and so on. If it is recalled that the behavior of a function is com- 
pletely specified by its Taylor's-series expansion 



(dj\ ,?(d?f\ ,t2(<pf\ 

W*-o 2 W/«-o "*" 6 \dt*/ t 



m = /co) + 1 y; (=o + - 2 ^ + ff ^ + - . . 

(within a certain range of t whose limits are of interest in specific 
problems, but which need not bother us here), we see that once x 
and v are given, thus fixing the values of all the higher derivatives 
at t — 0, the future motion is determined. 

It is not hard to generalize this reasoning, to see that a body, acted 
on by any sort of force that depends on x and on v, will have its motion 
specified completely just by assigning definite values to its initial 
position and velocity. The mathematical counterpart to this state- 
ment is the rule that the solution of any second-order differential 
equation (one having a second derivative term in it but no higher 
derivative) has two arbitrary constants in it. 

Another very important physical fact which can be deduced from 
(3.4) is that the frequency of the oscillation depends only on K and 
m and not at all on xo or v . This means that for a given mass and 
a given spring, as long as the law of force of the spring is F = —Kx, 
then the frequency of oscillation will be the same no matter how we 



U.4] DAMPED OSCILLATIONS 23 

start the system to oscillate, whether it oscillates with an amplitude 
of motion of 1 cm or 0.001 mm. This is a very important fact in its 
practical applications, for if the. law of force of actual springs were not 
nearly F — —Kx, or if this property did not hold for solutions of 
Eq. (3.2), then no musical instrument could be played in tune. Imag- 
ine trying to play a piano when the frequency of each note depended 
on how hard one struck the keys! Compare this with Prob. 1, page 
17. 

We might have found this fact for a number of cases by a long 
series of experimental observations, but our mathematical analysis 
tells us immediately that every mass acted on by a force F = —Kx 
has this property. Oscillations of this type are called simple harmonic 
oscillations. 

Energy of Vibration. — We shall need an expression for the energy 
of a mass oscillating with simple harmonic motion of amplitude A and 
frequency v . The energy is the sum of the potential and kinetic 
energies 

W = \mv 2 + JT* Kx dx = %mv 2 + iKx 2 

= 2-K 2 mv\A 2 sin 2 (2W - $) + \KA 2 cos 2 (2W - *) 
But 4t 2 v% = (K/m) so that 

W = i£A 2 [sin 2 (2W - $) + cos 2 (2W - $)] 



or 



W = \KA 2 = 2-K 2 mv\A 2 = \mU 2 (3.5) 



where U = 2ttv A is the velocity amplitude of the motion. 

The total energy is thus equal to the potential energy at the body's 
greatest displacement (iKA 2 ) or is equal to the kinetic energy at the 
body's greatest speed (kmll 2 ). Expressed in terms of v and A, we 
see that W depends on the square of these two quantities. 

4. DAMPED OSCILLATIONS 

So far, we have not considered the effects of friction on oscillating 
systems. In general, friction does not play a very important role in 
the problems that we shall consider in the first part of the book. If 
we show its effects on the simplest system with which we deal, we can 
deduce its effect, by analogy, on more complicated systems; therefore, 
we shall mention friction only occasionally, later on in this book. 

The friction that is most important in vibrational problems is the 
resistance to motion which the air surrounding the body manifests. 



24 THE SIMPLE OSCILLATOR [TLA 

Energy in the form of sound waves is sent out into the air. From the 
point of view of the vibrating system, this can be looked on as friction, 
for the energy of the system diminishes, being drained away in the 
form of sound. This resisting force depends on the velocity of the 
vibrator, and unless the velocity is large (much larger than those with 
which we shall deal), it is proportional to the velocity. It can be 
expressed mathematically as —R m (dx/dt), where the constant R m is 
called the resistance constant. The total force on a simple oscillator 
acted on by both friction and springiness is therefore — R m (dx/dt) — 
Kx, and the equation of motion becomes 

dt 2 + dt + 47r VoX " °' 4tt 2 . 2 = (K/m) (4>1; 

The value of v is the frequency that the oscillator would have if the 
friction were removed (R m were zero) and is called the natural frequency 
of the oscillator. 

It is interesting to notice that the equation for the free oscillation 
of charge in a circuit containing inductance, resistance, and capacitance 
has the same form as (4.1). The inductance is analogous to the mass 
m, the resistance to the resistance factor R m , and the inverse of the 
capacitance to the stiffness constant K. 

The General Solution. — To solve Eq. (4.1) we make use of the 
exponential function again. We guess that the solution is Ce bt and 
solve for b. Substitution in (4.1) shows that (6 2 + 2kb + 4r 2 v$)e bt 
must be zero for all values of the time. Therefore 



6 2 + 2kb + 4tt 2 ^ = 0, or b = -k ± Vk 2 - 4tt 2 ^ 

In all the problems that we shall consider, the stiffness constant K 
is much larger than the resistance constant R m {i.e., friction will not 
be big enough to make the motion much different from that discussed 
in the last section). Since 4r 2 vl is supposed to be larger than k 2 , 
then the square root in the expression for b is an imaginary quantity, 
and we had better write 

b = -k ± %riv f , v f = vo Vl ~ (k/2rv ) 2 (4.2) 

This means that, following our convention, we can write 

x = Ce- fc <- 2 ™"< (4.3) 

or 

x = e- ki [a cob (2rv ft) + «i sm(2irv f t)] = A e- kt cos(2ttj7J - $) 

The values of a and a x are again fixed by the initial conditions for 



H.4] DAMPED OSCILLATIONS 25 

the oscillator; a must equal the initial displacement x , and the initial 
velocity in this case is equal to %cv } a x — ka , so that 



eti 



( t»o + kx<\ 
2rv f . J 



The solution is not periodic, since the motion never repeats itself, 
each swing being of somewhat smaller amplitude than the one before 
it. However, if A- is quite small compared with v f , we can say that 
it is very nearly periodic. In any case, the frequency of the oscilla- 
tions is v f , which is very nearly equal to v if k is small. It again turns 
out that the frequency is independent of the amplitude of the motion. 
Of course, strictly speaking, we should not use the word frequency in 
connection with nonperiodic motion. But when the damping is small, 
the motion is almost periodic, and the word will have some meaning, 
although a rather vague one. 

There are several respects in which the motion of the damped oscil- 
lator differs from that of the simple oscillator. The most important 
difference is that the amplitude of motion of the damped oscillator 
decreases exponentially with time; it is A e~ kt , instead of being just A 
(A is the initial value of the amplitude). The amplitude decreases 
by a factor (1/e) in a time (1/k) sec (e = 2.718). This length of time 
is a measure of how rapidly the motion is damped out by the friction 
and is called the modulus of decay r of the oscillations. The fraction 
of this decrease in amplitude which occurs in one cycle, i.e., the ratio 
between the period of vibration and the modulus of decay, is called 
the decrement 8 of the oscillations. Another method of expressing this 
is in terms of the "Q of the system," where Q = (co m/# m ) is the num- 
ber of cycles required for the amplitude of motion to reduce to (1/e*) 
of its original value. If these quantities are expressed in terms of 
the constants of the system (the small difference between v f and v 
being neglected), it turns out that 

«-fe)-(?> -**-$ *-£) r441 

r=l = -^; 5 = l = l = J- ( ( } 

k irvn' Q v Q tvq J 

The smaller R m is, the larger Q and r are, indicating that it takes 
a longer time for the oscillations to damp out, and the smaller 5 is, 
indicating that the reduction in amplitude per cycle is smaller. These 
properties, of course, are independent of the way the oscillator is 
started into motion. 

Another difference between the damped and the undamped oscil- 



26 THE SIMPLE OSCILLATOR [II.4 

lator is the difference in frequency. When (k/2rv ) is small, the 
expression for v f can be expanded by means of the binomial theorem, 
and all but the first two terms can be neglected. 



Vf 



=-v^(sy = "-(§^) + -'- (4 - 5) 

In most of the cases with which we shall be dealing, k and v have such 
values that even the second term in the series is exceedingly small, so 
that the change in frequency is usually too small to notice. 

Energy Relations. — The subject of damped oscillations can be 
considered from a quite different point of view — that of energy loss. 
We must first develop an expression for the average energy of the 
system at any instant. We cannot use the formulas (3.5) because 
the amplitude of oscillation in the present case is not constant. The 
sum of the kinetic and potential energies of a body of mass m acted 
on by a springiness force Kx, whose displacement is given by the 
formula A(t) cos{2irvt — <£), is 

W(t) = \mv 2 + iKx 2 

= 2-K 2 mv 2 A 2 - 2mmv (^7 J A sm(2rvt - <3>) cos(2tt^ - <S>) 

+ im i^£\ cos 2 (2rvt - *) 

When averaging this value over a single oscillation, the second term 
on the right drops out. If A is very slowly varying, so that (dA/dt) 
is small compared with vA, then the third term can be neglected, 
and we have the approximate formula for the energy of motion and 
position {i.e., the energy that can be recovered, that is not yet irre- 
vocably lost in heat) 

W(t) ~%R[A(t)] 2 = 2T 2 mv 2 [A(t)Y = im[U(t)] 2 (4.6) 

where the symbol ~ means "is approximately equal to." In the 
case of the damped oscillator, this "free energy" is 2-K 2 mv%A\e~ 2U \ 
which diminishes exponentially with time. 

The rate of loss of energy due to friction is equal to the frictional 
force opposing the motion R m v multiplied by the velocity v (since force 
times distance is energy, rate of change of energy is force times 
velocity). The rate of loss of energy is 

P = R mV 2 = [Air 2 R m v f 2 sin 2 (2jrvft - 4>) 

+ 4;irR m Vfk sin (2irv f t — 3>) cos (2irv f t — 3>) 

+ Rmk 2 cos 2 (2rv ft - $)]A§e~ 2fct 



D.6] FORCED OSCILLATIONS 27 

Using the same approximations as before, we have for the average loss 
of energy per second • 



--&) 



Wv\R m A\e-™ = %R m [U(t)]* (4.7) 



If we had started out without formula (4.3) for the details of the 
motion but had simply said that the free energy at any instant was 
given by (4.6) and that the energy loss was given by (4.7), where R m 
is small, we could have found the dependence of W on the time by 
means of (4.7). Eliminating U from Eqs. (4.6) and (4.7) results in 
P ~ (R m /m)W = 2kW, so that we have 

p = ~ (t) - 2kW > or w - +2kW ( 4 - 8 ) 

The solution of this is 

W~ W e~ 2kt = imU%e- 2kt = %c i v\mA\e-' lu 

which checks with Eq. (4.6). We see from this that the damping out 
of the motion is required by the fact that the energy is being lost by 
friction. We might point out here that the fraction of free energy lost 
per cycle is just (2k/v f ) ~ (2k/v ) = (2w/Q), where Q is given in 
Eq. (4.4). 

In nearly every more complicated case of vibrations, the effect of 
friction is the same as in this simple case. The amplitude of vibration 
slowly decreases, and the frequencies of natural oscillation are very 
slightly diminished. Usually, the change of frequency is too small to 
be of interest. 

5. FORCED OSCILLATIONS 

It often happens that a system is set into vibration because it is 
linked in some way with another oscillating system (which we shall 
call the driving system). For instance, the diaphragm of a microphone 
vibrates because it is linked, by means of sound waves, to the vibra- 
tions of a violin string; and a loud-speaker diaphragm vibrates because 
it is linked to the current oscillations in the output circuit of an ampli- 
fier. The system picks up energy from the driving system and oscil- 
lates. In the two instances mentioned, and in many others, the driven 
system does not feed back any appreciable amount of energy to the 
driving system, either because the linkage between the two is very 
weak (as is the case with the violin and microphone) or else because 
the driving system has so much reserve energy that the amount fed 



28 THE SIMPLE OSCILLATOR [II.6 

back is comparatively negligible (as is the case with the amplifier 
and loud-speaker). In these cases the only property of the driving 
system that we need to know is that it supplies a periodic force which 
acts on the driven system. The more complicated case, where the 
feedback of energy cannot be neglected, will be considered in the next 
section. 

The General Solution. — For the present, we ask what happens to 
a simple oscillator when it is acted on by a periodic force F cos (2-irvt), 
or Fe~ 2lcivt , according to our convention. We wish to know what its 
motion is just after the force has been applied and, more important, 
what its motion is after the force has been acting for a long time. We 
are also interested in how this behavior depends on the frequency v 
of the driving force (which does not have to be the same as the natural 
frequency v of the oscillator). 

The total force on the oscillator is a combination of the springiness 
and the frictional and the driving force -R m v - Kx + Fe~ 2 * ivt . The 
equation of motion is 

^§ + 2k d 4i + 4ttV .t = ae- 2 * M , a = (F/m), k = (R m /2m), 
at 2 at 

co = 2rv, 4tt 2 ^ = (K/m) (5.1) 

We discussed a similar equation in Sec. 2, where we showed that a 
choice of two exponentials, one corresponding to the free vibration 
of the oscillator and the other to the forced motion, was a good guess 
for the solution. Substituting x = Ce~ u '^ iv t t + Der 2 * M in (5.1), we 
find that it is a solution if 



n _ a = v(vl - v 2 ) + ivk = R itf 

U ~ 4tT 2 (^ - V 2 ) - ^ivk ±TC\V% ~ V 2 ) 2 + \KV 2 k 2 



where 



B = (<*/4ir) _ (F/2tp) 



\Ar 2 ("o ~ v 2 ) 2 + v 2 ¥ 



4 



RI.+ 



o K 

ZTV 



and 
tan# = 



Rr 



*{v\ - v 2 ) (K/2-kv) - 2Tvvm 



The solution can be thrown into a semblance of simplicity, and 
the analogy with electric circuits can be made more apparent, if we 
define a mechanical impedance for the system 



II.5] FORCED OSCILLATIONS 29 

Z m = R m - dam - — J = R m - iX m = \Z m \e~ i -» (5.2) 



co = 2wv t \Z m \ = ^Jr* + U m _ ±) 
„ = * - 90° = tan- [ " m ~^ /c ° ] 



We recall that the usual electrical-engineering notation is obtained 
by substituting —j for i, so we can see that the equation for Z m is 
exactly analogous to the equation for the complex electrical impedance 
of a series circuit, with the mechanical resistance R m analogous to the 
electrical resistance, the mass m analogous to the electrical inductance 
and the mechanical compliance C m = (l/K) analogous to the electrical 
capacitance. The quantity X m = urn — (K/co) can be called the 
mechanical reactance of the system. The units in which this mechanical 
impedance are expressed are not ohms, for the quantity is a ratio 
between force and velocity rather than between voltage and current. 
The symbolic analogy is close enough, however, to warrant the use 
of the same symbol Z with the subscript m to indicate "mechanical." 
The units of mechanical impedance are dyne-seconds per centimeter, 
or grams per second. 

The solution of Eq. (5.1) can therefore be written in either of the 
two alternate forms 

F 

X = Q e -hi-1riv t t _ -«-"•-' 



lirivZn 



or 



\2Tv\Z m \J 



x = e~ kt [a cos (2irv f t) + a t sin (2irv f t)] + I - — r T? — x ) cos (2irvt — &) 

\ZTV\Z/ m \/ 

The constants a and ai are determined, as before, by the initial posi- 
tion and velocity of the mass. 

Transient and Steady State. — When the force is first applied, the 
motion is very complicated, being a combination of two harmonic 
motions of (in general) different frequencies. But even if the friction 
is small, the first term, representing the free, or "transient," vibra- 
tions, damps out soon, leaving only the second term, which represents 
simple harmonic motion of frequency equal to that of the driving 
force (see examples of this in Fig. 2). 



30 



TEE SIMPLE OSCILLATOR 



IIL6 



x --*■ — 



F 



2irivZ r , 



e - *"', or x — » 



i? 



2tv\Z„ 



sin (%irvt — ■ <p) 



where 



y ~ * \ 7" ) e ~ iat ' 0r U — * \ l7~T ) C0S ^^ ~ ^ 

(com) - (X/co) 



(5.3) 



tan co = — cot# = 



Rt! 



) co = 2^^ 



This part of the motion is called the steady state. We see that it is 
completely independent of the way in which the oscillator is started 




V=" 2 A> f 



V=Vf 



\=2v* 



Time-*- 

Fig. 2. — Forced motion of a damped harmonic oscillator (A; = weff/10)- Curve a 
shows free oscillations, and curves b, c, and d show forced oscillations due to sudden 
application of force at t = 0. Dotted curves give force as function of time; solid curves 
give displacement. Effect of transient is apparent at the left side of the curves; at the 
right side the steady state is nearly reached. 

into motion, its amplitude, phase, and frequency depending only on 
the constants F and v of the force and on the oscillator constants m, 
R m , and K. No matter how we start the oscillator, its motion will 
eventually settle down into that represented by (5.3). 



II.5] FORCED OSCILLATIONS 31 

Steady-state motion is motion of a system that has forgotten how 
it started. 

Impedance and Phase Angle. — The amplitude and velocity ampli- 
tude are proportional to the amplitude of the driving force and are 
inversely proportional to the magnitude of the mechanical impedance 
Z m . The analogy with electric circuits is therefore complete. The 
velocity corresponds to the current, the mass to the inductance, the 
frictional constant to the resistance, and the stiffness constant to 
the reciprocal of the capacitance. 

The impedance is large except when v = v , but at this frequency, 
if the friction is. small, it has a sharply defined minimum. Therefore, 
the amplitude of motion in the steady state is small except when 
v = v , where it has a sharp maximum. The case where the frequency 
v of the driving force equals the natural frequency v Q of the oscillator, 
when the response is large, is called the condition of resonance. The 
peak in the curve of amplitude against v is sharp if R m is small and is 
broad and low if R m is large, as is shown in Fig. 3. This figure also 
shows that the steady-state motion of the oscillator is not very sensi- 
tive to the value of the frictional constant except in the range of fre- 
quencies near resonance. The dotted curve for the amplitude of 
motion is for a value of R m eleven times that for the solid curve, yet 
the two are practically equal except in this frequency range. 

The motion is not usually in phase with the force, the angle of lag 
of the displacement behind the force being given by the angle &, 
which is zero when v = 0, is (x/2) when v = v , and approaches x as 
v approaches infinity (indicating that the displacement is opposite in 
direction to the force). The angle of lag of the velocity behind the 
force, <p = + (x/2) + #, is analogous to the phase angle in a-c theory. 
It is —k/2 when v = 0, zero when v = v , and +7r/2 when v is very 
large. 

In other words, when the frequency v of the driving force is much 
smaller than the natural frequency v of the oscillator, then the ampli- 
tude is small, and the displacement is in phase with the force. As v 
is increased, the amplitude increases and gets more and more out of 
phase with the force, until at resonance the amplitude is very large 
(if R m is small), and the velocity is in phase with the force. As v is 
still further increased, the amplitude drops down and eventually 
becomes very small. For these large values of v the displacement 
is opposed to the force. Figures 2 and 3 illustrate this behavior 

We use systems driven by periodic forces in two very different ways. 
One type of use requires the system to respond strongly only to par- 



32 



THE SIMPLE OSCILLATOR 



[II.6 



ticular frequencies (examples of this are the resonators below the bars 
of a xylophone, the strings of a violin, and the human mouth when 
shaped for a sung vowel). In this case we must make the friction 
as small as possible, for then the only driving frequency that produces 
a large response is that equal to the natural frequency of the driven 
system. The other type of use requires the system to respond more 
or less equally well to all frequencies .(examples are the diaphragms 
of microphones and loud-speakers and the sounding board of a violin). 




Fig. 3. — Amplitude and phase of forced motion, as functions of the frequency v 
of the driving force. The frictional constant R m for the dotted curve is eleven times 
that for the solid curve, all other constants being equal. 

In some cases we wish the steady-state amplitude to be independent of 
the frequency; in others we wish the amplitude of the velocity to be 
constant; and in still other cases we would like the acceleration to 
have a constant amplitude. One or another of these requirements can 
be met within a certain range of frequencies by making the stiffness, 
the friction," or the mass large enough so that its effects outweigh those 
of the other two in the desired range of frequency. 

These three limiting types of driven oscillators are called stiffness- 
controlled, resistance-controlled, and mass-controlled oscillators, respec- 



(5.4) 



II.6J FORCED OSCILLATIONS 33 

tively. Their properties and useful ranges of frequency are 

Stiffness controlled: K large ; Z m ~ i f - — h x ~ I -=z \ e _iat 

v considerably less than both ( ?r- I -» /— and ( n „ I 

\27r/ \ m \2trR m / 

Resistance controlled: R m large ; Z m ~ R m ; ( — J ~ ( — J e~ iut 

v considerably less than I -p-^- \ larger than ( _ „ J 

\2irm/ \2TR m / 

Mass controlled: m large; Z m ~ — Qnrivm; ( -r^ I ~ ( — ) e~ iwt 
v considerably larger than both \w-j -*/— and ( ^-^- ] 

It is to be noticed that every driven oscillator is mass controlled in 
the frequency range well above its natural frequency v , is resistance 
controlled near v (though this range may be very small), and is 
stiffness controlled for frequencies much smaller than v . It simply 
requires the proper choice of mechanical constants to place one or 
another of these ranges in the desired place in the frequency scale. 
It is also to be noted that there is always an upper limit to the fre- 
quency range over which an oscillator is stiffness controlled, a lower 
limit to the range over which one is mass controlled, and both an 
upper and a lower limit to the range over which an oscillator can be 
resistance controlled. We can move these limits about by changing 
the mechanical constants, but we never remove the limits entirely. 

Energy Relations. — The average energy lost per second by the 
oscillator due to friction, when in the steady state, is P = %R m U 2 = 
iR m (F/\Z m \) 2 [see Eq. (4.7) of the last section]. The rate of supply of 
energy from the driver to the driven oscillator is 

F 2 
vF cos (2irvt) = j=-, cos (2irvt) cos (2irvt — <p) 



\Z n 
F 2 
\Z m \ 



cos {2-irvt) [cos <p cos (2-Kvt) + sin <p sin (2Tcvt)] 



The average value of this energy supplied per second by the driver 
is HF 2 /\Z m \)cos<p = %R m (F/\Z m \) 2 (since cob* = RJ\Z m \), which 
equals the loss of energy to friction P. One can say that the amplitude 
and phase of the driven oscillator so arrange themselves that the energy 
delivered by the driver just equals the energy lost by friction. 



34 THE SIMPLE OSCILLATOR [II.5 

Electromechanical Driving Force.— There is one particular form of 
coupling between the driver and driven systems that is of particular 
interest to us in the study of sound: the electromechanical coupling 
between a vibrating diaphragm and an electrical driving circuit. In 
practically all acoustical work at present, the sound vibrations dealt 
with are transformed into electrical oscillations in order to amplify 
them. After amplification they are then changed back into sound 
waves, if need be, and an electromechanical coupling between the 
amplifier and the sound generator is used. Such coupling can have a 
variety of forms. For instance, the amplifier current can be sent 
through a coil wound on a magnet, varying the magnetic field and thus 
varying the force on a steel diaphragm (this is the coupling used in 
the ordinary telephone receiver). Or the current can be sent through 
a coil, attached to the diaphragm, which is placed in a magnetic field 
whose direction is perpendicular to the coil winding (this coupling is 
used in the so-called dynamic loud-speakers). Or the electric voltage 
may be impressed across a piezoelectric crystal, causing it to change 
size and shape (this coupling is used in crystal microphones and 
loud-speakers). 

In the magnetic forms of coupling the mechanical force exerted on 
the driven oscillator is proportional to the instantaneous current I 
through the coil, F = DI dynes, where the factor of proportionality D 
has a value that depends on the particular sort of coupling system 
that is used. For instance, if the coupling coil is fastened to the 
diaphragm and consists of n turns, each of radius b cm, and if the coil 
is placed in a radial magnetic field whose intensity is B gauss at the 
position of the winding, then D = 2irnbB/10, if the current is measured 
in amperes. In certain cases D may be a complex quantity, as, for 
example, in an ordinary telephone receiver when hysteresis must be 
taken into account. 

If the current is an alternating one, represented by loe -2 "*', the 
force on the oscillator is DIoe~^ ivt , and if the driven system behaves 
like the simple oscillator discussed in this section, its steady-state dis- 
placement and velocity will be, according to Eq. (5.2), 

x = ^° e- 1 '**, v = §^-° <r*"«, a = 2™ (5.5) 

The motion of the driven system produces a back emf in the coil. 
Owing to the reciprocal relation between the force on a current in a 
magnetic field and the emf produced by motion in the same field, we 
can immediately say that the back emf produced in the coil by moving 



II.6] FORCED OSCILLATIONS 35 

the driven system with a velocity v is vD/10 7 volts, where D is the 
same constant that appeared in the expression for F. The back emf 
produced in the coil due to its motion is therefore 

Em = y~ he-*"' = 15-1 

where the constant r = D 2 /10 7 is called the electromagnetic coupling 
constant. The effective electrical impedance in the amplifier circuit due 
to the motion of the driven system is the ratio between Em and I. This 
complex quantity Z M 

r 

Zm = Rm — iXM = 77- — TY m = T(G m — iB m ) 

<6m 



where 

G m — 



( R m \ B _ _ (X m \ Y _ _1_ iv \ (5.6) 

toj' Bm ~ \\z m \*)' Ym ~ \z m \ e ( 

\Z m \*=-Kl + Xl; Z m = com-^ 

Z M is called the motional impedance of the coil. It is a true electrical 
impedance, measured in ohms, in contrast to the mechanical impedance 
Z m . The quantity Y m = {\/Z m ) = G m — iB m is called the mechanical 
admittance, with units seconds per gram or centimeters per dyne- 
second, having a real part G m , called the mechanical conductance, and 
an imaginary part B m , called the mechanical susceptance. For the case 
of electromagnetic coupling the electrical impedance due to the motion 
of the coil is proportional to the mechanical admittance of the coil. 
The constant of proportionality r has the units of ohm-grams per 
second. We label all electrical impedances with a capital-letter sub- 
script to distinguish them from the mechanical impedances Z m , Z r , etc. 

Equations (5.6) indicate that the larger the mechanical impedance 
of the coil (i.e., the harder it is to move), the smaller the motional 
impedance of the coil (the less the back emf due to its motion). A 
little thought will show that this is a reasonable result. 

Motional Impedance. — It is interesting to notice that an electrical 
circuit with an inductance whose magnitude equals (T/K), a resistance 
of magnitude (T/R m ) and a capacitance (m/T), all in parallel, will give 
an electrical impedance of just the value given in (5.6). As far as the 
electrical circuit is concerned, the loud-speaker mechanism is com- 
pletely equivalent to the circuit shown in Fig. 4. If the complex 
impedance of the coil when the loud-speaker is rigidly clamped is 
Re — 2irivLe, then its impedance when the loud-speaker is allowed to 



36 



THE SIMPLE OSCILLATOR 



[H.5 



move is Z E — Z c + Z M , where Z c = Rc — iwLc and Z M = T/Z m . Z c is 
called the clamped or blocked impedance of the coil and Z E its total 
electrical impedance. 

When a current Z = Joe - ""* is sent through the coil and no external 
force F e is applied, there will be a total voltage drop E = Z E I across 
its terminals. This voltage can be considered to be the sum of the 
drop across the blocked impedance Z C I and the motional emf E M — 
Z m Im, which is the voltage drop across the part of the circuit enclosed 
in a dotted line in Fig. 4. Once the equivalent electrical circuit is 
determined, and the mechanical behavior of the coil, its velocity and 



r 



M echanical System _ 
Current /^Source F e /P 



10 



Im 



<r/K) 

^O'OOWO'OOWH 

(r/Rm) 

-wwwvww- 



(m/r) 



E m . 



— ^OOOOO'OOW* — vwwwwv ' 

L c R c 

Fig. 4. — Electrical circuit equivalent to the driving coil of a loud-speaker or 
"dynamic" type of microphone. Effective electrical impedance of mechanical system 
is equal to that of circuit enclosed in dashed line. Effect of external force F e applied 
to coil is equivalent to constant-current source of strength (F e /D) applied as shown. 

displacement, the power Po lost owing to heating the coil and the power 
used in moving the coil P M can all be determined in terms of the voltage 
Em across and the current Im through the mechanical part of the 
circuit : 



E M = ImZm volts 

r 

V 



~f 



X = 



(= Z M IMF e = 0) 
Em = VlO^T \~\ cm/sec 



cm; 



2iriv 

Pm = %Rm(\Im\ 2 ) — -^rj?, 



Rc 



watts 



(5.7) 



\Z n 



\v\ 



iR m tkv watts 



In the last of these equations, we see the relation between the electrical 
and mechanical units. On page 26 we showed that the expression 



II.6] FORCED OSCILLATIONS 37 

ii2 m |v| 2 is equal to the power dissipated by th'e mechanical resistance 
R m . Mechanical power is measured in ergs per second, however, and 
it takes 10 7 ergs per sec to equal 1 watt. The relationship between T 
and D 2 provides just this factor, so that Pm, computed from the equiva- 
lent electrical circuit, comes out in watts. 

There are several interesting corollaries from Eq. (5.7) which are 
worth mentioning. The mechanical impedance discussed in the 
example can be represented by a sum of impedances -torn, R m , and 
(K/—iw), as in a series circuit. The equations are more general than 
this, however: if the mechanical load on the coil can be represented 
by a mechanical admittance Y m = (1/Z TO ) which is any complex func- 
tion of co, then the motional impedance of the coil due to this load is 
TY m . For instance, if the mechanical load consists of a spring and a 
resistance in parallel, the mechanical admittance is (l/R m ) — (ioo/K) 
and the motional impedance is Z M = (T/R m ) — (iwT/K) ohms, as 
though an electrical resistance (T/R m ) and inductance (T/K) were in 
series. Later in the book we shall consider impedances due to radia- 
tion loads. Here again, to find the motional impedance Z M) we multi- 
ply the total mechanical admittance of diaphragm plus air load by I\ 

If the coil is short-circuited and is then moved by a mechanical 
force, the mechanical load is greater than the mechanical impedance, 
because the motion of the coil induces a current in the coil which pro- 
duces an additional reaction force. The additional mechanical imped- 
ance, due to current in the short-circuited coil, is Z s = T/Z c g per sec 
ana is the counterpart, for the mechanical system, of the motional 
impedance Z M for the electrical circuit. The total mechanical imped- 
ance of the diaphragm and shorted coil is Z m + Z 8 . The quantity 
Z a can be called the magnetomotive impedance. 

This magnetomotive impedance can be utilized to obtain an alter- 
native formula for the velocity of the diaphragm-coil system when 
driven by an emf E = Etfr™ % impressed across the coil 

DIm DE DE 

v = 



Z m z m (Zc + Zm) Z m z c + r 

= DE / Z c = Die 

Zm ~T" Z s Z m -\- Z s 

where I c is the current that would flow through the coil if it were 
clamped. This formula is of use in computing the change in velocity 
of a diaphragm when the mechanical load is changed but the electrical 
circuit is kept the same. In such a case D, I c , and Z„ would be 
unchanged and only Z m would change. 



38- THE SIMPLE OSCILLATOR [II.5 

When an external force F e is applied to the coil, as occurs when the 
system is used for a microphone, an additional term must be included 
in the equations. It turns out that the corresponding addition to the 
equivalent circuit is a constant-current generator, of strength (F e /D) 
amp, applied across the "mechanical" part of the circuit, as shown in 
Fig. 4. 

For mechanical resonance the mechanical reactance arm — {K/w) 
is zero, the magnitude \Z m \ of the mechanical impedance is a minimum, 
and the magnitude Y/\Z m \ of the motional impedance is a maximum. 
Other quantities being equal, at resonance \E M \ has its maximum value, 
and consequently the velocity magnitude |v| is maximum. At still 
higher frequencies there may be a series resonance between the induc- 
tance L c and the capacitance (jn/T). 

So many kinds of impedance have been discussed in this section 
that it might be well to list them together to contrast their meaning 
and the units in which each are expressed: 

Z E = total electrical impedance, in ohms 

Z c = clamped electrical impedance, in ohms 

Z M — motional impedance, electrical, in ohms 

Z m = mechanical impedance, in g per sec [ w-8) 

Z r = radiation impedance, in g per sec 

Z 8 = magnetomotive impedance, in g per sec 

Throughout the book, whenever there is danger of confusion, we shall 
use capital-letter subscripts for electrical impedances, ratios between 
volts and amperes; and we shall use lower-case subscripts for mechanical 
impedances, ratios between forces and velocities. 

Piezoelectric Crystals. — Rochelle salt crystals are also electro- 
mechanical transducers, transforming electrical into mechanical energy, 
and vice versa. For the "a>cut" type, for instance, the crystal is cut 
in a rectangular form. A force is applied between two faces (between 
face S and the rigid table T in the figure) by means of some mechanical 
system (such as a diaphragm) with effective mechanical impedance 
Z m . Voltage then develops between two other parallel faces, which 
can be measured across the terminals A and B. Alternately, a voltage 
applied across A and B will cause a force to be exerted on the mechanical 
system attached to face S. The dimensions of the crystal a, b, and 
d are shown in the figure. 

In this case the displacement from equilibrium of face S, which 
we. can label x, is due to the crystal expanding lengthwise. The 
expansion per unit length ij = (x/q) is called the strain. The mechan- 



n.6] 



FORCED OSCILLATIONS 



39 



ical force causing the expansion or contraction is the external stress X in 
dynes per sq cm. This stress, in a piezoelectric crystal, causes an 
electric polarization n, in_ this case perpendicular to the condenser 
plates. The polarization in turn causes a charge to be formed On the 
plates, in addition to that caused by the external applied field. For 
ease in initial calculation we shall use electrostatic units for the first 
steps. The relations between these internal quantities £, X, and n 



External Force F» 




echanical Load 



I Y't'/'/W i 





Source 




J 

Mechanical System 

Fig. 5. — Typical arrangement of piezoelectric crystal, showing electrical connections 
and direction of application of force F e - An equivalent electrical circuit is shown. For 
other crystal cuts the metal plates would be on the top and bottom, rather than on a 
pair of sides. 

and the external quantities F e , x, the potential difference E between the 
plates and the total charge Q on the plates are given by the set of 
equations 



F e = bdX + Z m v, v = — io)X = —iu£a 



(5.9) 



40 THE SIMPLE OSCILLATOR [II.6 

where ki is the dielectric susceptibility of the unloaded crystal, su is 
one of its elastic moduli, and du is the appropriate piezoelectric con- 
stant. The constants /c, s, and d vary markedly with temperature. 

The first of the fwe equations (5.9) gives the relation between the 
driving force F e , the internal stress, and the motion of the mechanical 
load. The second is the usual relation between velocity and displace- 
ment for simple harmonic motion. The third is the usual electrical 
relationship between surface charge density, applied field E/d, and 
polarization II. The last two equations are peculiar to piezoelectric 
materials. The equation for the polarization in the particular direc- 
tion we have chosen is related both to the applied field (which is 
usual) and also to the mechanical stress. The strain in the vertical 
direction depends in the usual manner on the stress; but there is an 
additional dependence on the applied field. Our equations are given 
for one particular cut of crystal, with force and field applied in a par- 
ticular way. Other arrangements with respect to the crystal axes 
will change the last two of Eqs. (5.9), the modified equations involving 
other constants k, d, and s labeled by different subscripts. 

We have also assumed that the frequency of oscillation is smaller 
than the lowest resonance frequency of the crystal; otherwise the 
stress, strain, polarization, etc., will not be uniform throughout the 
crystal. The more complicated case of high frequencies will be taken 
up later. 

After a large amount of algebra, we can obtain an expression for 
the current I = —iuQ in terms of the applied voltage 

t . n /o6\ /1 , ,, n ■ (bdg/su ZmjFe + ioi(abd\J sud)E 

1 = " UaE \&Td) (1 + 4TKl) + ' 1 - (l/ia,)(db/asuZ m ) 

Still further manipulation can be applied to show that this relation 
between I and E is equal to that due to the equivalent circuit shown 
in Fig. 5, where 

^rff 4 io" radg { (510) 

v d s 4 4 9 ' 

t = 300 (r^y 1 ) volts/dyne, a = 10 7 r £ ohm-sec/g 

and where the transition from electrostatic to practical units has been 
made. 

The constant a can be called the piezoelectric coupling constant, 



II.6] 



FORCED OSCILLATIONS 



41 



analogous to r in the electromagnetic case. It changes the mechanical 
impedance of the load into electrical impedance in the equivalent cir- 
cuit. Values of these constants for Rochelle salt, for various tem- 
peratures in degrees centigrade, are given in Table 1. 

Table 1.— Piezoelectric Constants for Rochelle Salt (See Fig. 5) 



Tempera- 


Capacitance, nni 






ture °C 






t, juv/dyne 


a, ohms/g 
tor 


T 


(Cd/ab) 


(Ccd/ab) 


(C p d/ab) 


br X 10 6 


-10 


35.6 


15.9 


19.7 


315 


0.992 


- 5 


25.6 


13.3 


12.3 


370 


1.369 





19.4 


10.5 


8.9 


425 


1.806 


5 


16.7 


9.2 


7.5 


462 


2.134 


10 


16.7 


8.8 


7.9 


454 


2.061 


15 


22.2 


10.9 


11.3 


396 


1.568 


20 


42.2 


16.7 


25.5 


291 


0.847 


23.7 




Upper Curie poi 


nt 




24.7 


132.3 


24.4 


107.9 


209 


0.437 


25.7 


71.5 


19.7 


51.8 


240 


0.576 


28.2 


38.1 


15.4 


22.7 


303 


0.918 


31.0 


23.4 


12.2 


11.2 


382 


1.459 



From H. Mueller, Properties of Rochelle Salt, Phys. Rev., 57, 829, and 68, 
565 (1940). These constants depend strongly on temperature, particularly near 
the Curie points. The corresponding constants for quartz are much less dependent 
on temperature. 

We notice several important differences between this equivalent 
circuit and the one in Fig. 4. In the first place, the part of the circuit 
corresponding to the mechanical load is a series circuit, not a parallel 
one; so that if m, R m , or K is very large, the equivalent electrical 
impedance is very large, instead of very small, as is the case in Fig. 4. 
A very large value of Z m corresponds to loading or clamping the crystal 
so that its upper face cannot move. In this case the equivalent elec- 
tric circuit reduces to the condenser C c (the subscript c indicating 
"clamped"). If the crystal is unloaded and the external force F e is 
zero, the equivalent circuit consists of the two condensers in parallel, 
having a combined capacitance of C farads. 

In the present case the external applied force F e appears as a voltage 
source, in series with the impedance equivalent to the mechanical 
load, instead of as a current source parallel to this impedance, as in 
Fig. 4. 

The velocity of the top face of the crystal is proportional to the 
current in the "mechanical" arm of the equivalent circuit. The equa- 



03 



42 THE SIMPLE OSCILLATOR [H.6 

tions giving this velocity and the power used in moving the connected 
mechanical system are 

v = 10 7 tI p , x = - 10 7 ( ^ j I P 

•*"• ' = 300 fe) i (511) 

f\v\ 2 \ 
Pm = i(rR m \I P \ 2 = iR m I t^7 ) watts 

We note that the equivalent capacitance C p corresponds to an addi- 
tional stiffness, which is the stiffness of the crystal. The effective 
series capacitance of the mechanical arm of the circuit is Cm = [(1/C P ) 
+ Kt 2 ]~ x , and the corresponding total stiffness of the mechanical 
system is 

Mechanical resonance will occur when the equivalent reactance 
t 2 [ojw — (K t /a))] is zero. At this frequency, the maximum current 
will flow in the mechanical arm for a given applied voltage E. At a 
higher frequency, parallel resonance will occur with the shunt capaci- 
tance C c . 

6. RESPONSE TO TRANSIENT FORCES 

In the previous section we dealt with the response of various simple 
mechanical and electromechanical systems to simple harmonic driving 
forces. In particular, we studied the steady-state response to such 
forces after the transient effects, due to the force starting, have died 
out. In this section 1 we wish to follow a little further the study of the 
transient effects, particularly when the driving force itself is transient 
and the system never does settle down into steady driven oscillation. 
To carry this out we shall use the techniques of complex integration 
discussed in Chap. I. 

Representation by Contour Integrals. — The general method of 
attack is as follows : we express our transient driving force as an integral 
of the form of Eq. (2.19) 

f(t) = f^° M F(w)er^ t da 

such as the ones given in Eqs. (2.15) to (2.18), where F can be deter- 
mined by the use of Eq. (2.20). We next assume that the solution x 

1 This section need not be studied unless the other sections in the book on 
transient phenomena are to be studied. 



II.6] RESPONSE TO TRANSIENT FORCES 43 

is also of this-form : ^ v : - 

x ® = f-„ "X(fi>)e~ iut 'da> (6.1) 

so that if X is known x can be determined; or, vice versa, if x is known, 
X can be obtained by the related equation 



2xJ_ 



X(<a) = y I x(t)e iu>t dt 

By this mathematical trick we can change from a differential equation 
relating x and / to an algebraic equation relating X and F. Finally, 
Eq. (6.1) can be used to obtain x. 

This method may seem rather mysterious and long winded. 
Actually, it is a mathematical restatement of a rather simple principle. 
Equation (2.19) says that a transient force can usually be expressed 
as a limiting sum {i.e., an integral) of a whole series of component 
simple harmonic forces of differing frequency (w/2tt) and amplitude 
F((a) dca. The individual equations for each of the component forces 
are solved to find the steady-state amplitudes X(a)) du>, and these are 
then finally combined by the limiting sum of Eq. (6.1) to give the 
required displacement x(t). To obtain the transient motion, we 
analyze it into its component simple harmonic steady-state motions 
and then recombine at the end. 

In most problems of practical interest, we can assume that the 
system is at rest at equilibrium until t = 0, when the transient is 
suddenly applied. Therefore, we can assume (at least for the time 
being) that the functions fit) and x(t) differ from zero only f or t > 0. 
In this case we can use Eq. (2.21) to suggest a simpler form for X 

X(ip) = 2^1 x(t)e-*" dt (6.2) 

where p = —i<a and the real part of p is positive. There is a similar 
equation giving F(ip) in terms of f(t). 

One general relation of great usefulness is the expression for the 
Fourier transform of (dx/dt). This can be obtained from Eq. (6.2) 
by integration by parts, or from Eq. (6.1) by differentiation inside 
the integral sign, if this is permissible. 



2xJ \dt) 



e~ pt dt = pX '= — icoX 

(6.3) 



Additional terms must be added if x is not zero for t < 0, 



44 THE SIMPLE OSCILLATOR [II-6 

Transients in a Simple System.— By the use of these formulas we 
can transform the general differential equation 

m w + Rm % + Kx = m * £» F(o})e ' iwt do} 

into the algebraic equation 

{K - ioiR m - co 2 m)X(co) = F(u) 

by multiplying both sides of the differential equation by (l/2ir)e~ pt 
and integrating over t. The differential equation, of course, corre- 
sponds to a simple mechanical system of mass m, resistance R m , and 
stiffness K, acted on by force f(t). 

The equation relating X and F is a familiar one, dealt with earlier 
in our discussion of steady-state response to a simple harmonic force. 
The equation 

X(co) = F(w) = F{0,) (6.4) 

K — iwRm — co 2 ra —ioiZ{u>) 

where 



Z(co) = R m — i (com J 



is simply the expression for the amplitude of the component X(co), 
having frequency co, in terms of the component i^(co) entering into 
the integral for the total transient force. The final expression for the 
displacement and velocity of the simple system acted on by force 
/(0 is 



x O = J -TTn^ y w = ty x du (6.5) 

J- =<, — 2coZ(co) J- «» Z(co) 



where 



F(p) = ^j Q fV) e - pt d *> V = -t« 



These equations are valid if the imaginary part of co is always positive 
in the integration and if x, v; and / are all zero for negative values of 
t [this incidentally means that F has no poles above the real axis; 
see Eq. (2.15)]. In order that these integrals converge there must be 
certain obvious restrictions on the way Z behaves as co goes to infinity- 
To solve the contour integrals for x and v we must investigate the 
poles of the function F/(—iaZ). The poles of F(w) cannot be investi- 
gated until we choose a form for f(t), but we can discuss the factor 
l/(— icaZ). For this factor to have poles, the function — z'coZ(co) 
must have zeros for some values of u>, real or complex. 



11.6} 



RESPONSE TO TRANSIENT FORCES 



45 



Complex Frequencies. — We have thus arrived at a broadening of 
our concept of impedance. Originally Z was defined as the complex 
ratio between a simple harmonic force of frequency (co/2r) = v and 
the corresponding steady-state velocity. We now are keeping to the 
same definition, but are broadening the concept by allowing o> to be 
a complex quantity, with real and imaginary parts. In particular, 
we are searching for those complex values of co for which (— zcoZ) is 
zero. 

For a. simple mechanical system ( — ioZ) = — co 2 ra — iu>R m -f- K. 
This can be factored to make the zeros apparent 

(—iooZ) = — m(co - (0/ + ik)(ca + co/ + ik) (6.6) 

where k = (R m /2m), u> f = co 2 , — k 2 , co§ = (K/m), in accordance with 
Eqs. (4.1) and (4.2). The values of co for which this function goes 
to zero therefore have for a real part plus or minus (2x) times the 
frequency of free vibration of the mechanical system and have for an 
imaginary part minus the damping constant of the system. At these 
values of co the function F/(—io)Z) will have simple poles, unless by 
extraordinary coincidence F has zeros or poles at these same values 
of oj (which eventuality we shall not consider). 

Calculating the Transients. — The poles of the function F/(—ioiZ) 
that are due to Z are therefore closely related to the transient oscilla- 
tions of the system discussed in Sec. 5. The points on the complex 
plane where these poles occur are symmetrically placed with respect 
to the origin, are both a distance co = \/K/m from the origin, and 
are both a distance k below the real axis. The residues of Fe-' Mt / 
(—iwZ) at these poles are 



and 



2mo)f 



g — Utft 



_ kt F(o> f -i k) 
2mco/ 



at co = — oif — ik 



> at w = co/ — ik 



As long as jP(co) has only simple poles, and as long as none of these 
coincides with the poles of l/(— iuZ), we can utilize Eq. (2.15) to 
write down the result for the displacement of the system 



x(t) = 

ir 1 
x(t) = — — tf-*'[e*»/«iP(— co/ - ik) — er^'WCa/ - 

(2iri) times the sum of the residues 
oiFer^t/i—iwZ) at all simple poles 
of F on and below the real axis of co 



(for t < 0) 
ik)] 

(for t > 0) 



(6.7) 



46 THE SIMPLE OSCILLATOR [II.6 

The first term is the transient motion of the system due to the sudden 
onset of F(t) at t = 0. The terms due to the poles of F(w) are a sort 
of generalized steady-state motion and depend on the specific form 

<*/(*). 

The corresponding expression for v{t) is 

v{t) = (for t < 0) 

v(t) = JL- e- kt [(a f + i^e^Fi-oif - ik) 

+ (w/ - itye-^f'Ficof - ik)] ^ (6.8) 

{2iri) times the sum of the residues of 

F e -iut/z at al i po i es f p on or b e i ow (for t > 0) 

the real axis of o> 

Further discussion requires choosing specific forms for / and F. 

Examples of the Method. — We shall first take a case that we have 
touched on earlier, the case where a simple harmonic force of frequency 
v = (a/2r) is turned on at t = 0. The force function chosen is 

f /0 (*<0) 

J \F e- iat (t > 0) 

According to Eq. (2.18), the corresponding Fourier transform is 

(F /2ti) 



F(co) = - 



co — a 



To find the resulting displacement of the system we insert the 
expression for F into Eq. (6.7) and turn the algebraic crank. After a 
number of turns, the first part, due to the poles of l/(— icaZ), turns 
out to be 

F e- kt ( e™' 1 e~ iaft \ 

2mo)f \co/ + a + ik o> f — a — ik) 

while the term due to the pole of F is F e- iat /[ — iaZ(a)]. Further sub- 
stitution of expressions for co , oi f , and k in terms of K, m, and R m 
results in the final equation for a; 

... F e- iat F e~ kt , '... . , 

— iaZ(a) 2a) f [—iaZ(a)] ' 

+ («/ + a + ik)e-™*\ (for t > 0) 

which is to be compared with the results of page 29. The final 
numerical results can be obtained by taking the real part of the expres- 
sions for f(t) and x (t). 



II.6] RESPONSE TO TRANSIENT FORCES 47 

No doubt this result could be obtained with less trouble by the 
simpler methods outlined in Sec. 5. However, we shall find later that 
the present method is more useful in discussing transient effects in 
more complicated systems, where the simpler method breaks down. 
The present examples are discussed so that we can become familiar 
with the method. 

The Unit Function. — Another force function which will be useful to 
study is that of a force suddenly applied at t = and maintained at a 
steady value thereafter. From Eq. (2.16) we see that the Fourier 
transform of 

*«-««-{? J > g » ™ - " (as) (6.9) 

The function u{t) is called the unit step function, often used in transient 
problems. The resulting expression for the displacement of the same 
simple system due to the application of the unit step force is therefore 



(6.10) 



1 1 

x u (t) = I — — ~- e~ kt cos (<j) f t — 




(*<0) 


a) 


(*>0) 


tan a — — 






<*f \/w§ — k 2 





The first term is the "steady-state" displacement from equilibrium 
caused by the unit force after t = 0, which comes from the residue at 
the pole of F(«) (at o> = 0). The second term is the transient oscil- 
lation due to the sudden application of the force at t = 0, which comes 
from the residues at the poles of l/( — icaZ). 

One sometimes wishes to know what happens to a system that is 
held aside from equilibrium for a long time and then is suddenly 
released at t = 0. A force producing this behavior would be f(t) = 
1 — u(t), a unit force applied steadily until t = and then released. 
The response for such a force would be 

*w = (A) - «.<« = i w* ? , „ , ('<« 

\ K J I (wo/Ka> f )e- kt cos (<a f t -a) (t > 0) 

If the force holding the system away from equilibrium is F instead of 
unity, the corresponding motion is x = F x h (t). 

If the force is an impulsive one, applied instantaneously at t = 0, 
the integral of this force over time (the impulse of the force) must be 
proportional to the unit step function. We can define a unit impulse 
function 8(t) as one that is zero for every value of t except zero and 



48 THE SIMPLE OSCILLATOR [H.6 

that has there such a value that 

jl ^ 5(r) dr = u(t) (6.11) 

The function 8(t) is sometimes called the Dirac delta function. Look- 
ing back at Chap. I, we see that the function defined in Eq. (2.14) 
approaches the function 8(t) as a approaches infinity. 

It is therefore useful to compute the response of the simple mechan- 
ical system to a force of the type defined by Eq. (2.14). The corre- 
sponding Fourier transform is 

n/ \ a 1 

^(co) = - 



2ri co + ia 

and the expression for the displacement for t > is 
ae~ at 



a 2 m + aR m + K 



aer kt r e %w f t e -u, f t i 

2mco/ Leo/ + i(k — a) to/ — i(k — a) J 



The limiting case for a — > °° gives the response of the system to an 
impulsive force 8(t) 

(0 (*<0) (6.12) 

XsKl) \ {l/mo> f )e- kt sin (a f t) (t > 0) 

The responses of the system to the two unit functions u{t) and 3(0 
might be considered to be the basic transient responses. They are 
closely related as Eq. (6.11) indicates. If it were not that u and 5 
are discontinuous functions, we could say that 8(t) is equal to (du/dt). 
From a formal point of view this is correct, since the limiting value of 
the quantity 



© 



|J Mfl - <t ~ A)] 

as A goes to zero has the properties of the function 8{t). The relation- 
ship is also apparent between the two transient responses, for x$ = 
dx u /dt, as a little close reasoning can prove. 

General Transient. — The two basic transient responses can be 
utilized to compute the response of the system to any more compli- 
cated force function. The delta function has the general property 
that, for any function /, 



f~J<?) W -r)dr= f(t) 



H.6] RESPONSE TO TRANSIENT FORCES 49 

and, by integrating by parts, the step function has the property 



/. 



_ M ^-r)^tfr=/(*) 



The meaning of these integrals (if they have any!) is that a smooth 
continuous function of time can be considered as being built up of an 
infinite sequence of impulse functions, the one at time r having ampli- 
tude /(t), and so on. Or the function can be built up stepwise, the 
increase at time t being proportional to df/dr. By utilizing the func- 
tions xs(t) and x u (t), the responses to the unit functions 8(t) and u(t), 
we can write down the response of the system to any force f(t) as either 

x(t) = j^ J{r)xi{t - r) dr 
or ■ > (6.13) 

x{f) = J ^ %j& Xu (t - r) dr 

The second form should not be used with discontinuous force functions 
(where df/dr is infinite) unless one adds terms proportional to x u (t — t„), 
where r n is the location of the discontinuity in time and the propor- 
tionality factor equals the magnitude of the discontinuous step in /. 
In a manner of speaking, Eqs. (6.13) form an alternative method to 
Eq. (6.7) for obtaining the displacement function x{t) — although Eq. 
(6.7) must be used to obtain the functions x s , x u used in Eqs. (6.13). 
Actually, as might be suspected, these equations are two equivalent 
ways of writing the same fundamental equation. This can be seen 
by writing out the integral for x u in terms of Eq. (6.5) 



^ ) = 2^J_ -^^ do) 



C — Kilt 



and, by differentiation, 



o — lut 



if" e 

Substituting this into Eqs. (6.13) and changing the order of integra- 
tion yields 

and, similarly, ^ (6.15) 



50 THE 'SIMPLE OSCILLATOR [II.6 

These expressions might be -considered to be fundamental to both 
Eqs. (6.7) and (6.13), the final form depending on whether we inte- 
grate first with respect to r or first with respect to w. 

If we integrate initially with respect to w, we obtain the first of 
Eqs. (6.13), and the second equation is obtained by integration by 
parts. If we integrate initially with respect to t, Eq. (2.20) shows that 
the factor in brackets is F(u>), and the result is just Eq. (6.5), from 
which we have obtained Eq. (6.7). Which of the forms for calculation 
of x(t) is to be used in a given problem depends entirely on the relative 
ease of integration of the two forms. 

Some Generalizations. — We can now review the method of com- 
plex integration (sometimes called the method of operational calculus), 
so as to point the way to the application to the analysis of more com- 
plex systems. Suppose that we have a force f(t), which is zero for 
t < 0, applied at some point to a mechanical system. To obtain the 
response to the general force, we first obtain the steady-state response 
of the system to a simple harmonic force of frequency v = (co/2ir). 
All that is needed is the ratio between the force (applied at point 1) 
and the steady-state velocity of some point of the system (either the 
same point 1 or another point 2) which is called the impedance Zn(co) 
or Zu(eo), a function of co. The ratio between the force and the corre, 
sponding displacement is -to2n(w) or —io)Z 12 (a). 

Often it is more useful to obtain the reciprocal of the impedance, 
the mechanical admittance Fn(«) or Y 12(a), the ratio between the 
velocity and the force. The ratio between displacement and force 
would then be Y (&>)/(—•&'«). This is a minor point, however, the 
main objective being to obtain the relation between the response of the 
system and the force for steady-state motion of frequency (co/27r). 

A possible second step is to analyze the transient f(t) into its 
simple harmonic components by obtaining the Fourier transform F(cj), 
according to Eq. (2.21). The final response to force f(t) is then 
obtained in terms of contour integrals of the form given in Eqs. (6.5), 
where Z(«) may be either Zn or Z i2 , depending on whether the response 
of the point of application or that of another point is required. 

An alternative form of the second step is to find the response of the 
system to an impulsive force 5(t), applied at point 1 

where, again, Z may be Z n or Z 12 . The response to any force f(t) 



II.6] RESPONSE TO TRANSIENT FORCES 51 

applied to point 1 is then given by 

x ® = Jl „ /( T )**(* ~ T ) dr, v(t) = f_ n f(T) Vi (t - t) dr (6.17) 

These equations are of very general usefulness, as we shall see 
later in this volume. The important steps in the process, for any 
type system, are finding the impedance or admittance function, giving 
the ratio between the simple harmonic force applied at some point 
of the system and the corresponding steady-state response of the same 
or another point, and computing the resulting contour integral, in 
the form of Eqs. (6.5) or Eqs. (6.16). 

It should be pointed out that the integrals of Eq. (6.16) do not 
converge unless Z and coZ go to infinity faster than the first power 
of co as co goes to infinity; otherwise Eqs. (6.15) must be used to find 
the correct solution. For instance, if Z = R, a constant, Eqs. (6.16) 
diverge. However, insertion of f(t) = 8(t) in Eq. (6.15) and utiliza- 
tion of the definitions of the Fourier transform would indicate that 
v$(t) = (l/R)8(t) and, by integration, x&(t) = (\/R)u(t) in this case. 

Sometimes a few tricks must be used to obtain results. For 
instance, if Z were R + (iK/u>), we could utilize the obvious equations 

1 = (co/fl) = 1_ _ (iK/R 2 ) 
Z co + i(K/R) R co + (iK/R) 

The integral for x« then becomes (using the "definition" of u) 
This integral is well-behaved, and we finally obtain, for t > 0, 



Xi(t) = 4 u(t) - 4 + 4 e-twt = ( ^ J e-c*/*)« 



R " w " R "*" R 



The corresponding velocity can be obtained by differentiation, or by 
using the same trick 

vs{t) = G) m ~ {§*) e ~ {K/m > {t > o) 

Later in the book we shall have further occasion to utilize this trick. 
Laplace Transforms. — Another way of describing the procedure 
utilizing the impulsive response is applicable to all systems we shall 
encounter in this book. We first compute the steady-state behavior 
of the system under consideration (simple oscillator, string, diaphragm, 



52 THE SIMPLE OSCILLATOR [H.7 

or room full of air) to a driving force (1/2tt) e~™ 1 applied at the point 
(or area or volume) we wish to subject to transient force. After com- 
puting the behavior (displacement, velocity, shape of string, etc.) 
we obtain the corresponding response to a unit impulsive force by 
integrating the expression for this behavior over a> from — oo to + » .. . 
The response for a general transient force can then be obtained by 
using Eq. (6.17). This technique is of general utility, and we shall 
refer to it from time to time throughout this book (keeping our eyes 
open to be sure that the integrals converge, of course). 
Equations (6.16) are of the general form 

f(t) =7^J_ n Qi-iah-^da (6.18) 

By Eq. (2.20) we see that $> can be expressed in terms of an integral 
of /. Whenever f(t) is zero for t < 0, we obtain the following integral 
for 3>(p), after substituting p for — iw. 

*(P) = f ~ e~ pt f(t) dt (6.19) 

which is related to Eq. (6.2). The function 3>(p) is said to be the 
Laplace transform of f{t) 

<*> = £(/) 

In many cases where it is difficult to evaluate Eq. (6.18) directly in 
terms of residues [when $ has other infinities besides simple poles, as 
for instance a factor Vco + a, or where the convergence of (6.18) is 
questionable] it turns out to be better to try to find an f(t) which, 
when substituted in Eq. (6.19), gives the required form for $. In 
order to aid in this procedure of working the problem backward, 
several tables of Laplace transforms [tables of f(t) with their corre- 
sponding $(£>)] have been published; these can be used just as tables 
of integrals are used. Such tables must be used if 3>(p) has terms with 
p = —iu involving radicals. This will be discussed again in Chaps. 
VI and VII./ /" - --- l ) 

7. COUPLED OSCILLATIONS 

We must now treat the case that we avoided in the last section, 
the behavior of two oscillators coupled tightly enough together so 
that we cannot neglect the feedback of energy from the driven system 
to the driver. ■ In this case both oscillators are on an equal footing; 
we cannot call lone the driven and the other the driver, since each is 
affected by the other. We shall expect that the results obtained 



n.7i 



COUPLED OSCILLATIONS 



53 



from our analysis will not be much different from those of Sec. 5 except 
when the frequencies of the two oscillators are nearly equal, for in 
any other case the amplitude of motion of one oscillator will be much 
smaller than that of the other, and the feedback will not be large. 

The General Equation. — The general case will be treated first, 
however. We shall call one of the coupled oscillators No. 1, call its 
mass mi and its displacement from equilibrium #i,.and call the other 
oscillator No. 2, with mass m 2 and displacement x%. When x\ and x 2 
are both zero, the system will be in equilibrium. We neglect friction 

, Equilibrium , 
I Positions \ 



\ rC"" 1 



f 



fill ■ K 3\ ..; 

'"' Masses -■' \ 
"'Stiffness Constants 





m, 



\\ 



K 3 



k; 



_|| — r^TR^p — I 




k; 



Fio. 6. — Simple example of two coupled oscillators; two masses connected by 
springs. The effective stiffness constant for mi is K\ = K\' + K%; that for m, is 
Kt = Kt' + Ks. The analogous electrical circuit is also shown. A force applied to 
mi is analogous to the voltage source F\, a force on mt to the source ^2. 

for the time being, since it will only confuse the problem. It can 
be considered later if need be. 

Suppose it turns out that if we keep x% equal to zero, the force 
on mass wii is equal to — K\X X (this will be true if Xi is small enough). 
Then if we clamp oscillator 2 at x 2 — 0, oscillator 1 will vibrate with 
a frequency v\ = (1/2jt) V-Ki/wi. Similarly, if we clamp mass mi 
at xi = 0, the restoring for ce on m 2 is — K^x%, and its frequency of 
vibration is v z = 0-/2w) \jK%/m^. A displacement of mass mi, 
however, produces a force on m 2 , for this is what we mean by coupling. 
Suppose that this force is Kzpc\. Then owing to symmetry of the 
system, the force on mi due to a displacement of m 2 is K^. The 



54 THE SIMPLE OSCILLATOR [H.7 

constant K s is called the coupling constant. It is usually very small 
compared with Ki or K 2 . We shall not lose generality by assuming 
that v\ is not less than v 2 . 

The equations of motion of the two masses are 

d 2 xi d 2 x 2 T . , _ 

m l ~^2 = —-^1^1 + K SX2, m 2 -J72 = -K 2 X 2 + K3X1 

If we change the scale of the displacements to allow for the difference 
in mass of the two oscillators, letting x = xi V^i and y = x 2 Vmi, 
then we can write these equations in a simpler form. 



~ + Wv\x = 4ttVV, ^f + ^v\y = 4xV 2 x 



4tt 2 ^ = 






K, \ ( 7 -D 



If one of the oscillators is clamped down, then the other will 
oscillate with simple harmonic motion; but if both oscillators are 
allowed to move, the resulting motion will usually not even be periodic. 
Curves c of Fig. 8 show a typical case. The curves for x and y as 
functions of time are certainly not sinusoidal. This is annoying, for 
we do not like to have to develop and define some new sort of function 
to express the motion. 

Simple Harmonic Motion. — So we first ask whether it is not possi- 
ble to start the two masses in some special way so that the motion 
is simple harmonic, even though the motion in general is not. Of 
course if the motion is to be simple harmonic, both oscillators must 
be vibrating with the same frequency. We therefore try the solution 
x = Ae- 2 * ivt , y = Be- 2 ™* in Eqs. (7.1) and see that it satisfies these 
equations if the coefficients A and B are related in the following 
manner: (v\ - v 2 )A = ^B, (?f - v 2 )B = ^A. 

From these two equations we can find the ratio of A to B and the 
value of the frequency v. s Multiplying one equation by the other gets 
rid of both A and B and leaves an equation for v, v i — (v\ + v \)v 2 + 
v\v\ — n A = 0, having for a solution 

v = [*(»? + 4) +i VW - v\y + 4 M 4 ]* 

± i V(K 1 m 2 - K 2 mxY +.4iqm 1 m s l (7.2) 



H.7] COUPLED OSCILLATIONS 55 

Thus there are two possible frequencies of simple harmonic oscil- 
lations for the combined system, and we shall see that (unlike the 
simple oscillator) which of the frequencies it will vibrate with depends 
on how we start the system into motion (i.e., what the values of A 
and B are). Neither of the allowed frequencies is equal to either of 
the natural frequencies v x or v 2 of the individual oscillators taken 
separately. Suppose that we call the frequency involving the plus 
sign in front of the radical v + and that involving the negative v_. 
The value of v- is smaller than either v\ or v 2 , that of v+ is larger than 
both vx and v 2 . Coupling always spreads apart the natural frequencies. 

Normal Modes of Vibration. — We can say that although the gen- 
eral motion of the system is not periodic, nevertheless if the masses 
are started into motion in just the right way so that the amplitudes 
of motion of m x and m 2 are related by B+ = A + (v\ — v\)/n 2 = 
A + n 2 /{v\ — v\), then, and only then, will the system oscillate with 
simple harmonic motion of frequency v+, these ratios between the 
amplitudes of motion A+ and B + remaining the same throughout 
the motion. Similarly, if the masses are started so that their 
amplitudes are related by the equations #_ = A-(v\ — v 2 _)/y. 2 = 
A-/i 2 /(i>% — vt), then, and only then, will the system oscillate with simple 
harmonic motion of frequency V-. If the motion is started in any other 
way, there will be no permanent ratio between the displacements of the 
two masses, and the motion will not be periodic. These two especially 
simple ways of motion of the system are called its two normal modes 
of vibration. By a little juggling of terms, these motions can be 
represented as follows: If the frequency is v+, 

x = C+ cos a 6-2""+', xi = [a+ cos (%rv + t) 

Vmi 

+ 6+ sin (2irv+t)] cos a 

y = -C+ sin a er 2 ***-**, x 2 = —= [a+ cos (2irv+t). 

V m 2 

+ b+ sin (2irv + t)] sin a 
If the frequency is i>_, \ (7.3) 

x = C_ sin a er 2ri '-*, xi = — j= [a_ cos (2tvJ,) 

VWi 

+ b- sin (2rv-t)] sin a 

y = C_ cos a er™'-*, x 2 = —== [«_ cos (2ncvJL) 

\/m 2 

+ 6_ sin (2irv_t)] cos a 



56 



THE SIMPLE OSCILLATOR 



[H.7 



where the angle a has been introduced to unify and simplify the nota- 
tion. It is related to the v's, etc., as follows: 



tan< 



= {v\ - vj) = _ 



("i - vi) 



tan 2a = 



2m 2 



(v\ - v\) (v\ - v_) 



n - "2 

C+cosa = A+, C+sina = — B + , C-Smet = A-, C-Cosa = J5_ 

One may ask why we are so interested in these normal modes of 
vibration when they are such a specialized way for the system to 
oscillate. The answer is that as soon as these normal modes are 
found the problem of determining the general motion of the system is 
suddenly seen to be quite simple. For it turns out that the general 
motion can always be represented as a combination of both the normal 
modes of vibration. The general solution of Eqs. (6.1) is 



(7.4) 



\ y 
\ 



\ 



\ 



\ 



\ 



a. 



\ 



\ 



x = C+ cos a e-*™* 1 + C- sin a er M '-* 
y = — C+ sin a e- 2wir+t + C_ cos a e' 2 ^"- 1 

■7 

as can be verified by substitution in (011). The displacement of 

each individual mass is a combination 
of two oscillations of different frequen- 
ciesj which results in a nonperiodic mo- 
tion (except in the rare case where the 
ratio of v+ to v_ equals the ratio of two 
integers, when the motion will be peri- 
odic no matter how the system is 
started. Why?). 

We can represent the general mo- 
tion of the system in a still simpler 
manner. Let X = C+er 2 ™* 1 and 

Y = C-e- 2 ™-'. Then x = X cos a + 

Y sin a, y = — X sin a + Y cos a. These 
equations are just the ones used in 

analytic geometry to represent the transformation of the coordinates 
of a point in a plane with respect to the axes x and y to a 
new set of axes X and Y inclined at an angle a to the first set, 
as shown in Fig. 7. Suppose that we represent the position of the 
system at any time by a point on a plane whose abscissa is x, the dis- 
placement (in the proper scale) of the first mass, and whose ordinate 
is y, the displacement of the second. The motion of the system 



\ 



\ 



Fig. 7. — Transformation to nor- 
mal coordinates for two coupled 
oscillators. 



n.7] 



COUPLED OSCILLATIONS 



57 



corresponds to a motion of the point on the plane. What we have 
said above is that as the point moves, corresponding to a general 
sort of oscillation of the system, the projection of the point on the 
rc-axis (the displacement of the first mass) moves back and forth in a 
complicated nonperiodic way, and so does the projection of the point 
on the #-axis (the displacement of the second mass). However, the 




(c) f— " 

p IG g. — Motion of two coupled oscillators. Curves to the right show the dis- 
placements x and y as functions of time; those to the left show the path of the point 
representing the system in the x-y plane (configuration space). Cases (a) and (6) 
show the two normal modes of vibration, when the system point travels along a normal 
coordinate. Case (c) shows the general type of motion. 

projection of the point on the X-axis always moves back and forth 
with simple harmonic motion of frequency v+ (since X = C+er 2 ™* 1 ) 
with an amplitude of motion C+, and the projection on the F-axis 
moves with frequency ?_, with amplitude C_. Only when the system 
is so started that its point moves along the X-axis or along the F-axis 
is its motion periodic (see Fig. 8 for examples). The plane in which 



58 THE SIMPLE OSCILLATOR [H.7 

the point moves is called the configuration plane of the system, and 
the axes X and Y are called the normal coordinate axes for the system. 
The general motion of the system can be written in the form given 
in (7.4) or, less symbolically, as 

si = —7=^ [a+ cos (2tv + {) + b+ sin (2wv + t)] 

sin a , , n 

+ —7= [«- cos (frcvJL) + &_ sm (2jri»_i)] 

VWi 

[A+ cos a cos (27t»' + ^ — f> + ) 




(7.5) 



+ yl_ sin a cos (2x^_/ — $_)] 
a: 2 = _ -~Z. [o+ cos (Sto-v+O + 6 + sin (2irM)l 

COS £tf 

H 7= [0- cos (2x^-0 + 6_ sin (2jrv.J)l 

VW-2 

= — 7=[~~ A + sina cos (2x^+2 — $+) 

+ A_ cos a cos(2rv_^ — $_)] 

The general solution involves four arbitrary constants a+, b+, a_, 
and 6_, or A+, <£> + , ii_, and $_, whose values are fixed by specifying 
the initial displacements and velocities of the two masses. 

Energy Relations.— Another example of the special simplicity 
that is given to the equations of motion when normal coordinates 
are used is the expression for the energy of the system. The kinetic 
energy is, of course, frn^dx^dt)* + im 2 (dx 2 /dt)\ To find the 
potential energy, we find the amount of work necessary to push the 
system from equilibrium to the position where the displacements are 
X! and x 2 . Since friction is negligible, we can do this process in any 
order that we wish and get the same result. We choose to push m x 
out first, pushing against a force —K x xi and so requiring an amount 
of work JKiXt dx x = \K x x\. We next push m 2 out, pushing against 
a force — K 2 x 2 + K 3 X! (by our definition of K z ) and doing an amount 
of work \K 2 x\ - K3X1X2. The total energy is therefore 



mi \w) + m2 \w) + KlX * + K2X > ~ 2K ^^] 
A3f) + (off + w ^ + 47r2 ^ 2 " 8*V*v] 

an expression complicated by the term in xy. 



2 

= 1 
2 



H.7] COUPLED OSCILLATIONS 59 

If we substitute the expressions for x and y in terms of X and Y, 
expressing the functions of a in terms of functions of 2a, and use the 
formula for tan (2a), we find a simple formula for W 



w - "2@ 2 + 3©' + ***+** + *•,._ r- (7.6) 

without any term in XY. If X = A+ cos (2irv + t — $ + ) and Y = 
A- cos {2-rcvJt — <£»_), as indicated in (6.5), then 

W = 2ir 2 (ulA% + vlAl) 

the sum of the energies of vibration along the two normal coordinates. 
The Case of Small Coupling. — Now that we have obtained a 
general solution for the motion of two coupled oscillators, it might be 
well to see how the results correspond to the discussion of forced 
oscillations given in Sec. 5. As we said at the beginning of this sec- 
tion, whenever the coupling is small and the frequencies of the two 
oscillators are not equal, the amplitude of motion of one will be much 
larger than that of the other. The oscillator showing the smaller 
motion can then be considered as the driven oscillator, and its ampli- 
tude of vibration will be given by Eq. (5.3) of Sec. 5. For instance, 
if m x is to be the "driver," its amplitude of motion should be larger 
than that of m 2 . If the coupling is small, a is small, and the case 
we should analyze is that for the frequency v + , where 



(vW cos(a) 

\ m 2 v\ — v\ \ m 2 



Xl ~ I .. rzr ) cos(a)e- 2a " 4 '''+' 
and 



#2 

K z xi 

£ 2 - m 2 (2irv + ) 2 

Since a is small, x 2 will be smaller than xi unless mi is very much 
larger than m 2 . 

From the point of view of m 2 , the quantity K3X1 is just the force 
f(t) applied to m 2 through the coupling spring, due to the motion of 
xi. The driver is oscillating with a frequency v+, so that/ = F^e~^ iv ^ 
— K3X1 where F = (i^ 3 C + /\/wi) cos a. The divisor is proportional 
to the mechanical impedance of m 2 at the frequency v + , since 

— 2nriv+Z 2 — K 2 — m i (2rv + ) 2 



60 THE SIMPLE OSCILLATOR [H.7 

Therefore the expression for the displacement of m 2 when it is being 
"driven" by mi at a frequency v+ is 

— Zrlv + Zi2 

which is to be compared with Eq. (5.3). 

The Case of Resonance. — However, our primary purpose in this 
section is not to check the formulas given in Sec. 5 but to find out 
what happens in the cases where the formulas of that section do not 
hold, where the feedback of energy from m 2 to mi is appreciable. As 
an example of this let us consider the case where the two oscillators 
have the same natural frequency (Vi = j> 2 ) and where the friction is 
negligible. If we try to use the formulas of Sec. 5 in this case, we 
find that they predict that the amplitude of the driven oscillator will 
be infinite. The infinity simply means that the amplitude of motion 
of the driven mass will become large enough to absorb a large fraction 
of the driver's energy, so that the formulas of Sec. 5 cannot be used. 
To solve this problem, we must use the formulas of the present section. 

According to Eq. (7.2), if v x — v i} 



V+ = Vv{ + M 2 = vi + ^ + 



,2 



v- = V4 - M 2 = vi - 2^ + * ' ' » a = 4 

If ju is much smaller than v h only the first two terms in the series 
expansions of the square roots need be considered. Therefore, if 
two similar oscillators, each of natural frequency vx, are coupled 
together, they can no longer oscillate with frequency vi but can 
oscillate with a frequency either (^ 2 /2i'i) larger or this same amoun t 
smaller than v\. If they are started so that x\ = —x% v / m 2 /mi, 
then the system has only the higher frequency; but if they are started 
so that Xi = Xi -\/'mz/m\, then it has only the lower frequency. If 
the system is started in any other way, the motion will be a combina- 
tion of both frequencies. 

If the system is started by pulling mass 1 aside a distance x , 
while keeping aj 2 equal to zero, and letting both masses go at t = 0, 
then the values of the constants to be used in (7.5) are 6+ = 6_ = 0, 
a+cosa/VW = a-sma/s/mi = (x /2). The displacements of the 
two masses will be represented by the equations 



n.7] 



COUPLED OSCILLATIONS 



61 



= X COS I IT — t) COS (2TVit) 



(7.7) 



= VS Xo ^Cy sin(w) 

if the trigonometric equation for the sum or difference of two cosines 
is used. The second form of the equations shows that, if the oscil- 




Fig. 9. — Motion of two coupled oscillators having the same natural frequency. 
Solid curves show the displacements of the two oscillators as a function of time and 
illustrate the alternation in amplitude. 

lators are started in the manner described above and if the coupling 
is weak, the motion is like an oscillation of frequency v\ whose ampli- 
tude of oscillation itself oscillates with the small frequency 0u 2 /2i>i). 
This is illustrated in Fig. 9. Such motion is not simple harmonic 
motion, since the amplitude changes with time. As the first forms 
of Eqs. (7.7) show, it is a combination of two harmonic motions whose 
frequencies differ by a small amount, so started that they at first 
reinforce each other for Xi but after a while get out of phase and cancel 
each other out, and so on. We notice that when the amplitude of 
motion of mi is large, then that of w 2 is small, and vice versa. 

Transfer of Energy. — We can compute the approximate value of 
the free energy of oscillation of m x and m 2 by using formula (4.6), 
W = 2w 2 mv 2 A 2 (t). Of course, it is not strictly correct to speak of 
the energy of only a part of a system, and it is not right to use the 



62 THE SIMPLE OSCILLATOR [II.7 

formula for the energy of a simple oscillator to compute the energy 
of a coupled oscillator; but if the coupling is weak, the formula has 
some meaning. We find that the average "free" energy of mass mi 
is 27r 2 mi^x§ cos 2 (tt/xV i 'i) an d * na * °f m 2 is 2r 2 miv\xl sm 2 (ir(i 2 t/vi). 
It indicates that at t = all the energy of the system resides in mass 
mi but that as time goes on the energy of the first mass diminishes 
while that of the second increases until all the energy is 'transferred 
to m 2 . Then the flow of energy reverses, that of mi building up again, 
and so on. The total energy of the system is 2r 2 mivlxl, a constant, 
as it should be. 

This example illustrates how different from the usual forced oscil- 
lations can be the motion of two coupled oscillators if a large coupling 
or an equality of frequencies of the two vibrators makes possible a 
large transfer of energy from one to the other. It also illustrates an 
obstacle encountered in building a resonating system that will respond 
very strongly to only one frequency. In the last section it was pointed 
out that such critical response required a system with very small 
friction. We see now, however, that if we make a system with 
negligible resistance tuned to a natural frequency v\ and try to drive 
it by a system oscillating also with a natural frequency v\, the result 
of the coupling will be that neither driver nor driven system can oscil- 
late with frequency vi but must oscillate with frequency v\ ± (n 2 /2vi). 
The very fact that the response is so large at vi destroys the possibility 
of vibration at that frequency. 

Forced Vibrations. — We also wish to find the response of the sys- 
tem of two masses to an applied force. Here again the analysis goes 
from the simple harmonic case to the more complicated transient 
forces. We first find the response (i.e., the displacement and velocity 
of mi and m 2 ) to a force F = i'V - "" applied to m x or m 2 . If it is 
applied to m h Eqs. (7.1) become 

^f + o*c - x 2 V = a*r"; -^ + o>\y - x 2 x = (7.8) 

co? = (ifi/rai); cof = (i£ 2 /m 2 ), x 2 = \K z /\/m~m^\ a x = Fo/y/rni 
Solving for the steady-state response consists in substituting x = xtf'™* 
and y = y e~ iut in Eq. (7.8) and solving for x and y . The results, 
when reduced back to expressions for the actual displacements, are 

_ Fo (cof - to 2 )*-*" _ Kitr 1 ** Fp 

•^l ... f O 9 \ / 9 \ 5 ^2 



mi (co 2 - co 2 .)^ 2 - «i)' (co 2 - co2.)(co 2 - coi) mim 2 

(7.9) 



H.7] COUPLED OSCILLATIONS 63 

where co + — 2ttv + , w_ = 2irv-, with the v's given in Eq. (7.2). 

«i = i(" 2 i + «?) + i V(col - col) 2 + 4 X 4 
«L = *(«? + col) - i V(«J - cof) 2 + 4x 4 

If we had applied the force to ra 2 , the corresponding response 
would have been 

*! ™ /..2 .* \/..?> «\> # 2 — 



Wim 2 (to 2 - oj2.)(co 2 - »£.)' " m 2 (co 2 - ^ + )(<o 2 - «i) 

(7.10) 

The symmetry of the expressions is apparent. 

Another way of writing the response is specifically in terms of 
impedances corresponding to the circuit of Fig. 6. We define the 
blocked impedance of Wi as the impedance when m 2 is held fixed at 
equilibrium, Z± = —iwmx + i(K\/(a). The corresponding blocked 
impedance of m 2 is Z 2 = — *com 2 + i(K 2 /(o). The coupling can be 
expressed in terms of a mutual impedance M — i(K z /(S). By differ- 
entiating Eqs. (7.9) with respect to time, and by subsequent algebraic 
juggling, we obtain expressions for the ratios between the simple 
harmonic force applied to m\ and the corresponding velocities of m t 
and m 2 

„ F ZrZ % - M\ F ZrZ 2 - M* ■ 

&\\ = — = ~ , ^12 = — = jj (.7.11) 

V\ Z 2 Vi M 

Z\\ is called the mechanical input impedance of the system at point 1; 

Z 12 is called the mechanical transfer impedance between points 1 and 2. 

If the force is applied to ra 2 , the corresponding impedances are 



z 22 


F ZrZi - M\ 

V2 Z X 


5/21 


_ F _ 

Vi " 


Z X Z 2 - M 2 
M 



Z12 (7.12) 

This brings out the interesting fact that transfer impedances are 
symmetrical. The response at point 2 due to a force at point 1 is the 
same as the response at point 1 due to the same force applied at 
point 2. Sometimes this result is referred to as the principle of 
reciprocity. 

Resonance and Normal Modes. — Each of the impedances given in 
Eqs. (7.11) and (7.12) becomes zero at the two resonance frequencies 
v+ and v-. At these frequencies the displacements become infinite, 
as is shown in Eqs. (7.9) and (7.10). The infinity is due to our neglect 
of frictional forces; if a small amount of friction had been included, 
the displacements would have been large but not infinite. The 
ratio between y and x, when the driving frequency is v + , is x 2 /( w i — o>+) 



64 THE SIMPLE OSCILLATOR [H.7 

= l?/(A - "+), according to Eqs. (7.9). A glance at Eqs. (7.3) 
shows that this is just the ratio maintained when the system is in 
free vibration at frequency v + . 

In other words, when the system is driven by a simple harmonic 
force of frequency equal to one of the frequencies of free vibration of 
the system, its response is large, and the relationship between the 
motions of the parts (i.e., the configuration of the system) is the same 
as if the system were in free vibration at that frequency. One of 
the methods of finding the possible frequencies of free vibration of a 
system and the configuration of the corresponding normal modes is to 
drive the system by a simple harmonic force. The resonance fre- 
quencies are the frequencies of free vibration, and the configurations 
at resonance are those of the normal mode. 

In the present case, there are two natural frequencies of free 
oscillation, so there are two resonance frequencies. For three masses 
connected by springs there would be three normal modes and three 
resonance frequencies, and so on. 

When the present system is driven by a force of frequency consider- 
ably smaller than v-, applied to m x , both masses move in phase with 
the force, the ratio between x 2 and x\ being (K3/K2) = Ks/(K 3 + K'^), 
so that X\, the displacement of the point of application of the force, is 
greater than x 2 by a factor depending on the relative sti^ness of the 
two springs connecting with m 2 . 

According to Eqs. (7.9), as the driving frequency increases, the 
amplitudes increase until the resonance at v_, which is a lower fre- 
quency than either vi or p 2 , the resonance frequencies of each mass 
taken separately. For frequencies just above v-, both displacements 
are opposing the force (180 deg out .of phase). 

At v 2f the displacement of mi, however, becomes zero (if friction is 
not zero, xi is small, but not zero). At this driving frequency the 
input impedance Zw of Eq. (7.11) becomes infinite, since the blocked 
impedance Z 2 of m 2 is zero. What has happened is that at this fre- 
quency m 2 resonates when mi is held rigid. If we attempt to move 
mi at this frequency, m 2 will immediately absorb all the available 
energy with only an infinitesimal motion of mi. This corresponds to 
a parallel resonance in the analogous electrical circuit shown in Fig. 6. 

Above v 2 , X\ is in phase with the force and x 2 is opposed. At 
v = v+, higher than vi or v 2 , the amplitudes are again infinite. For 
higher frequencies, X\ opposes the force and x 2 is in phase with it. 

Transient Response. — A few examples of the application of opera- 
tional-calculus methods to the coupled system will be useful to discuss 



H.7] COUPLED OSCILLATIONS 65 

here, just to begin to show how the method works in more complicated 
cases. Suppose that we compute the response to the impulsive 
force 8(t) applied to mi. From Eqs. (7.9) we see that the quantities 
(1/— icaZ) needed to compute the integral of Eq. (6.16) for the coupled 
systems are 

/ 1 \ = 1 i<4 ~ co 2 ) = 

\ — iwZ u J mi (co 2 — co^Xco 2 -r coi) 

/ 1 \ = 1 X 2 = "2" 

\-ia)Zi 2 J y/mmz (« 2 - co 2 h )(« 2 - coi) -ico 

Since we are not considering friction, all the poles of these quantities 
are along the real axis, being at ± co_ and + co+. 

The residues of (l/2x) (e _iu '/ — icoZn) at w = ±« + are 



— Vji 



47rmico + \co 2 h — coi/ 



+ 6 



The residues at the poles « = ± co_ are obtained by interchanging co + 
and co_ in these expressions. 

A great deal of algebra and utilization of the formulas 

„ (co 2 . - CO 2 ) . , (cof - w 2 ) 

cos 2 a = f-± 14-; sin 2 a = f-f =^ 

(C04. — col) (co^ — coi) 

Y 2 

£ sin (2a;) = sine* cosa = —^ — ^ 



u + 



lead one finally to the expressions for the response of the coupled 
system to an impulsive force applied to mi 



t < 0, xu = x 2 & = 

— icaZj 



t > 0, xi S (t) = ^ J „ d» 



1 [cos 2 ** . , . sin 2 a . , ..1 

= ^T sm («+0 H sin (co_2) 

Wi L co + T co_ J 

Ar J-» — tcoZ 12 I 



mim 2 l 03 + 



r 1 

a COS a — 
Leo- 



sin a cos a — sin (co+tf) 



sin (<a-i) 

J 



These formulas are to be compared with Eqs. (7.5). What the oper- 
ational calculus has done is to fix the constants a+, &+, etc., to corre- 



66 THE SIMPLE OSCILLATOR [II.7 

spond to the initial conditions of m 2 at rest and m x being given a unit 
impulse at t = 0. 

Just after the impulse at t - 0, x°& and v-a are both zero; Xi» is 
zero but v-a starts at a value (1/wi), as it should for a mass ra x which has 
just been given a unit impulse. 

The unit impulsive response when the force is applied to ra 2 is 
obtained by interchanging the subscripts 1 and 2 in Eqs. (7.13). 

The response of the system to any force f(t), applied to mi is, from 
Eqs. (6.17) 

Xl (t) = J*^ f(r)x lS (t - t) dr) x*(t) = f'^ f(r)x»(t - t) dr (7.14) 

This analysis of a simple coupled system has had no important 
practical applications, but it has served to indicate the direction our 
results will take for still more complicated systems. We should 
expect to find complex systems (at least those coupled together by 
springlike forces) to have a number of resonance frequencies. If the 
system is started in just the right way, it will oscillate with simple 
harmonic motion, being damped out if there is friction. There 
should be several ways of starting the system to get simple harmonic 
vibration, each different way corresponding to a different normal 
mode and to a different natural frequency. In general, the system 
will vibrate with nonperiodic motion which is a combination of several 
natural frequencies. 

If the system is driven at one point by a simple harmonic driving 
force, the whole system will respond at that frequency. The steady- 
state response of any part will depend on the frequency and point of 
application of the force, the ratio between force and response being 
given in terms of the input transfer impedances of the system. From 
these impedances, as functions of frequency, can be computed the 
response of the system when subjected to a sudden unit impulse at 
t = 0; and from the impulsive response can be obtained the response 
to any transient force. 

We shall work out these same results for a still more complicated 
system, the simple string under tension, in the next chapter. 

Problems 

1. A vibrator consists of a 100-g weight on the end of a spring. The spring's 
restoring force is proportional to the weight's displacement from equilibrium; 
if the weight is displaced 1 cm, this force is 10,000 dynes. The frictional force 
opposing its motion is proportional to its velocity and is 100 dynes when its velocity 
is 1 cm per sec. What is the modulus of decay of the oscillator? What is its 
decrement? What "frequency" do the vibrations have? What frequency would 



H.7] COUPLED OSCILLATIONS 67 

they have if there were no friction? If the weight were originally at rest and then 
were struck so that its initial velocity was 1 cm per sec, what would be its subse- 
quent motion? What would be its maximum displacement from equilibrium? 

2. The diaphragm of a loud-speaker weighs 1 g, and the displacement of its 
driving rod 1 mm from equilibrium requires a force of 1,000,000 dynes. The 
frictional force opposing motion is proportional to the diaphragm's velocity and is 
300 dynes when the velocity is 1 cm per sec. If it is assumed that the diaphragm 
moves like a simple oscillator, what will be its natural frequency, and what its 
modulus of decay? The driving rod is driven by a force of 100,000 cos (2irvt) dynes. 
Plot a curve of the real and imaginary parts of the mechanical impedance of the 
diaphragm as function of frequency, from v = to v — 1,000 cps. 

3. The diaphragm of Prob. 2 is driven by a force of 100,000 cos (2irvt) dynes. 
Plot a curve of the amplitude of motion of the diaphragm as function of frequency, 
from v = to v — 1,000 cps. Over what frequency range is this loud-speaker mass 
controlled? 

4. What is the mechanical impedance of a mass m without spring or friction? 
What is the impedance of a spring without mass? What will be the angle by 
which the displacement of the mass lags behind an oscillating force? What will 
be the angle of lag of the velocity of the mass behind the force? What are the 
corresponding angles for the spring? 

6. The diaphragm of Prob. 2 is coupled electromagnetically to an electric 
circuit by means of a coil of negligible resistance and inductance, whose coupling 
constant r = 10,000. An alternating-current emf of 10 volts and variable fre- 
quency is applied across the coil. Plot the current through the coil and the ampli- 
tude of oscillation of the diaphragm as a function of frequency, from v = to 
v ~ 1,000 cps. Plot the real and imaginary parts of the motional impedance of 
the coil as function of frequency, from v — to 1,000 cps. 

6. An a-c emf of 10 volts and variable frequency is applied to the coil of 
Prob. 5. Plot the current through the coil and the amplitude of oscillation of the 
diaphragm as a function of frequency, from v = to v = 1,000 cps. 

7. A mass m is attached to the lower end of a spring of stiffness constant K. 
The upper end of the spring is moved up and down with an amplitude Be~ iu>t , and 
the frictional force on the mass is proportional to the relative velocity of the mass 
and the upper end of the spring (dr/dt), where r - x — Be~ iu>t . Show that the 
equation of motion of the mass is 

m (iO + Bm {§) + Kx = {K - *«««.)**-*" 

show that the steady-state motion of the mass is 

R m + i(K/co) _ . , 

x = — — ■ — - — — Be~ iwt 

R m — i[wm — {K/o))] 

and that the phase lag of x behind the displacement of the top of the spring is 

taU_1 l * R m — ~ J + tan_1 ( K / wR ™)- What is the amplitude of motion of xt 
What is the phase and amplitude of x at very low frequencies? At very high 
frequencies? 

8. A loud-speaker diaphragm has mass and stiffness that can be neglected in 
the useful frequency range. Its motion is opposed by a force equal to 400 v, 



68 THE SIMPLE OSCILLATOR HI.? 

where v is the diaphragm's velocity. The diaphragm is coupled to an electrical 
circuit by means of a coil of resistance 25 ohms, inductance 10 mh, and coupling 
constant r = 10,000. An a-c emf of 10 volts, with variable frequency, is applied 
to the coil. Assuming that all the energy lost by the diaphragm through friction 
is transformed into sound waves, plot the sound energy radiated per second (in 
watts) as a function of the frequency of the emf, from v = to v = 1,000. 

9. A loud-speaker is coupled to the electrical circuit by means of a coil of 
resistance 25 ohms, of negligible inductance, and of coupling constant r = 10,000. 
It is found that the additional impedance due to the motion of the diaphragm is 
equivalent to a resistance of 25 ohms, an inductance of 10 mh, and a capacitance 
of 1 juf, all in parallel. What are the mechanical constants of the diaphragm? If 
all the energy dissipated by the diaphragm goes into sound energy, plot the over-all 
efficiency of the loud-speaker-coil system {i.e., the ratio of the power radiated as 
sound to the total power dissipated by loud-speaker and coil) as a function of the 
frequency, from v = to v = 1,000 cps. 

10. The sharpness of resonance of a forced damped oscillator is given by the 
"half-breadth of the resonance peak," the difference between the two frequencies 
for which the amplitude of oscillation is half that at the resonance frequency v . 
Prove that if the natural period of oscillation is negligibly small compared with 
2w times the modulus of decay {i.e., if (fc/47r) is small compared with v ), then this 
half-breadth equals (V3A) times the reciprocal of the modulus of decay of the 
oscillator. What is the half-breadth for the diaphragm of Prob. 2? What would 
the frictional force have to be in order that the half-breadth may be 20 cycles? 

11. Two oscillators, each of mass m and natural frequency v {i.e., if one 
oscillator is held at equilibrium, the other will oscillate with a frequency v ), are 
coupled so that moving one mass 1 cm from equilibrium produces a force on the 
other of C dynes. Show that if C is small compared with £x 2 mvl, and if one 
oscillator is held 1 cm from equilibrium and the other at equilibrium and both are 
released at t = 0, then the subsequent displacements of the masses will be 

* = cos (i^bo) cos (w) ' Vt = sin G^To) sin (27r "°° 

12. Suppose that each oscillator of Prob. 11 is acted on by a frictional force 
equal to R times its velocity. Show that the modulus of decay of the oscillations 
equals (2m /R). 

13. Discuss the forced vibrations of the coupled oscillators described in Prob. 
12. 

14. Three masses, each of m g, are equally spaced along a string of length 4a. 
The string is under a tension T dynes. Show that the three allowed frequencies 
and the corresponding relations between the displacements for the normal modes 
are 

v = v y/l - V|, if yi = ?/3 = 2/2/V2 
v — vo, if y\ = —2/2 and y 2 = 

v = vo \A + Vi, if 1/1 = 1/2= —I/2/V2 

where (77a) = 2^v\m. 

16. An "x-cut" Rochelle salt crystal, mounted as shown in Fig. 5, has the 
dimensions d = 1 cm, 6 = 3 cm, a = 2 cm. The external load is a diaphragm in 



n.7] COUPLED OSCILLATIONS 69 

contact with water, which has an equivalent mass load of 10 g, negligible stiffness, 
and a resistance of 400,000 g per sec (radiation resistance). Plot the real and 
imaginary parts of the electrical impedance of the crystal from v = to v — 10,000 
cps for a temperature of 15°C. 

16. What will be the open-circuit voltage across terminals A and B of the 
crystal of Prob. 15 when an oscillating force of amplitude 10 dynes and frequency 
1,000 cps is acting on its upper surface? What is the voltage if the frequency of 
the driving force is 10,000 cps? Consider the temperature of the crystal to be 
15°C. 

17. A voltage 100 cos (2irvt) volts is impressed across the plates of the crystal of 
Prob. 15. Assuming that all the power dissipated in the resistance R m = 400,000 
is radiated into the water, plot the power radiated as a function of frequency v, 
from v = to v = 10,000 cps. The crystal temperature is held at 15°C. 

18. A simple mechanical system of impedance Z m = R m — t'[«m — (A /«)] has 
applied to it a force 

(0 « < 0) 

F(t) = < sin (at) [0 < t < (mr/a)] 
[ [(nr/o) < t] 

Use the results of Prob. 15, Chap. I and of Eq. (6.7) to compute the displacement 
of the system. 

19. A dynamic speaker of the type discussed in Fig. 4 has negligible coil 
resistance R c and mechanical stiffness K (the other quantities, R m , m and L are not 
negligible). An impulsive voltage E = S(t) is impressed across the terminals of the 
coil. Calculate the current through the coil, using the Fourier transform method 
[see Eq. (6.14)]. Show that the motional emf is 

M 2wLm J- <e (w + wo + ik) (w — « + ik) | j e ~ kt sin (u t) (t > 0) 

where k = (R m /2m) and o>l = (T/Lm) — (R m /2m) 2 . What is the velocity of the 
diaphragm? What are the expressions if (R m /2m) 2 > (T/Lm)? If « = 0? 

20. The speaker of Prob. 19 has constants m = 50, R m = 10,000, T - 10,000, 
L = 0.02. Plot the displacement of the diaphragm as a function of time for an 
impressed emf E = 108(f) across the coil. 

21. Use Eq. (6.13) to compute the velocity of the loud-speaker diaphragm of 
Prob. 19 for an applied voltage. 

!0 (t < 0) 

E (0 < t < T) 
(t- > T) 

Calculate the expressions for the three time ranges, t < 0, < t < T and t > T. 
Plot the result for the case of Prob. 20, for E = 10 volts, T = 0.01 sec. 

22. A crystal of Rochelle salt, mounted as shown in Fig. 5, has a mechanical 
load which is pure resistive, Z m = R m . An impulsive force F e = 8(t) is applied to 
the upper surface of the crystal. Using the formulas related to Eq. (6.14), com- 
pute the current through the equivalent circuit if terminals A, B are open, and 



70 THE SIMPLE OSCILLATOR [II.7 

show that the voltage across A, B is 

ir /•" e- iut dco . ' 

27rR m C c J-» (<*+ i[(C c + C P ) aR m C c C P ] V ° ltS 

(t < 0) 

(l/10 7 ri2 m Cc) exp [-(Cc + C P )t/SR m C c C P ] (t > 0) 



= 1 



23. The crystal of Prob. 22 has the following constants: 

C c = C p = 10- 10 , t = 10-», a = 10, R m = 10« 
The applied force is 

!0 (t < 0) 

10 dynes (0 < < < 0.001) 
it < 0.001) 

Plot the open-circuit voltage across A, B as function of time, from t — —0.001 to 
* = 0.005. 

24. The driving rod of the loud-speaker of Prob. 2 is driven by a force of 
10,000[15 sin (200irf) - 10 sin (600x<) + 3 sin (1000tt<)] dynes. Plot the displace- 
ment of the diaphragm during one cycle, and compare it with the curve for the 
force. 

25. The driving coil of Prob. 5 is actuated by an emf of [15 sin (200n-J) 
— 10 sin (600ir£) + 3 sin (lOQOn-2)] volts. Plot the displacement of the diaphragm 
as a function of t for one cycle, and compare it with the curve for the emf. 

26. The emf of Prob. 25 actuates the driving coil of Prob. 8. Plot the dis- 
placement of the diaphragm for one cycle, and compare it with the curve for the 
emf. 



CHAPTER III 
THE FLEXIBLE STRING 

8. WAVES ON A STRING 

.... So far, we have been considering the vibration of a system whose 
mass is all concentrated near one or two points, so that the motion 
of the system is completely specified by giving the displacement 
from equilibrium of the one or more masses as a function of time. 
The vibrators we use are not usually of so simple a nature. Ordi- 
narily the mass is not concentrated at the end of a relatively weightless 
spring but is spread along the spring. The vibrating string of a 
violin cannot be considered as having all its mass concentrated at 
the center of the string or even concentrated at a finite number of 
points along the string; an essential property of the string is that its 
mass is spread uniformly along its length. Similarly, a loud-speaker 
diaphragm has a good portion of its mass spread out uniformly over 
its extent. In these cases, each portion of the system will vibrate 
with a somewhat different motion from that of any other portion. 
The position of just a few parts of the system will not suffice to describe 
its motion; the position of every point must be specified. 

Presumably, we could attack the problem of the string, for instance, 
by considering the motion of N equally spaced masses on a weightless 
string and then letting N go to infinity. We should then have an 
infinite number of equations of motion, whose solution would give the 
position of every one of the infinity of points on the string as a function 
of time. A solution of these equations can be obtained. The solu- 
tion shows, for one thing, that there are an infinite number of allowed 
frequencies of oscillation of the string, as we could induce from the 
discussions of the previous chapter. But this is a very awkward way 
of solving a problem that is essentially simple. What is needed is a 
new point of view, a new method of attack. 

The new point of view can be summarized as follows: We must 
not concern ourselves with the motion of each of the infinite number 
of points of the string, considered as separate points, but we must 
consider the shape of the string as a whole. At any instant the string 
will have a definite shape, which can be expressed mathematically 
by saying that y, the displacement from equilibrium of that part of 

71 



72 



THE FLEXIBLE STRING 



[m.8 



the string a distance x cm from one end, is a function of x. The motion 
of the string at any instant will depend on the shape of the string, 
and the subsequent shape will depend on the motion; what we must 
do is to find the relation between the shape and the motion. In other 
words, the string's displacement y is a function of both x and t, and 
we must discover the relation between y's dependence on x and its 
dependence on t. The same point of view will be necessary in study- 
ing the vibrations of diaphragms and of air, as we shall see later. 

The Wave Velocity. — Before becoming cluttered up with equa- 
tions, we shall utilize a simple, but rather clever, little trick to help us 
in gaining a picture of how the shape of a string changes. Suppose 
that we have a long flexible string of uniform mass e g per cm, unrolling 

from one reel, threaded through 
the glass tube shown in Fig. 10a, 
and being wound up on another 
reel. Suppose that the reels are 
rotating so that the string is 
traveling with a velocity v cm 
per sec through the tube and so 
that the string is under a tension 
of T dynes. Suppose that we 
pick out a small portion of the 
tube of length As and ask what 
force the string is exerting on 
this part of the inside of the 
tube (we neglect frictional forces). If a small enough length is picked, 
the shape of the part of the tube (and the string in this part) will be 
practically equivalent to the arc of a circle of radius R, where the value 
of R will, of course, be different for different parts of the tube. 

Now, if the tube is not bent too sharply anywhere {i.e., if R is every- 
where larger than the maximum deviation of the tube from the straight 
dotted line in Fig. 10a), the tension on the string will everywhere 
be T dynes. A study of Fig. 10b will show that the net force inward 
on the tube due to the tension of the string is AT = <pT = T As/72. 
In addition to this force is the centrifugal "force" due to the motion 
of each successive portion of the string around the curved arc As. 
This force is pointed outward, and, as we learned in elementary 
physics, its value is the mass of the portion of the string eAs times 
its velocity squared v 2 divided by the radius of curvature R. The 
net inward force on the portion of the tube under consideration is 
then (As/R)(T - ev 2 ). 




Fig. 10. — Forces on a moving string. 




m.8] WAVES ON A STRING 73 

We notice that it is possible to run the string at a particular 
velocity c ?= \/T/e such that there will be no net force on this portion 
of the tube. And then we see that at this velocity c the string exerts 
no net force on any part of the tube! For the force is zero regardless 
of the size of R. If the string is run at the velocity c cm per sec, we 
can carefully break away the tube from around the string and leave 
the string moving with velocity c, still retaining the original form of 
the tube, a wave form standing still in space. 

Of course, running the string with a velocity c and having the 
wave form stand still is the same as having the string stand still and 
the wave form travel with a velocity c. What we have just proved is 
that as long as the string has a uniform density e g per cm, and as long 
as the displacement of the string 
from equilibrium is not too great, 
then a wave will travel along the 
string with a velocity c = y/T/e 
regardless of the form of the wave. 
The last phrase is the important 
part of the statement, for it means 
that, subject to the qualifications „ ., w 

' * ^ Fig. 11. — Wave motion on a 

Stated, a wave travels along a String string. Curve (a) is the shape at 

without any change of form as it * = 0; curve (6) the shape at ' = 1 - 
goes. These two properties of a string, or, rather, two aspects 
of the same property, are peculiar to the uniform flexible string. 
They are not possessed by a string of variable mass density, or a 
string with stiffness, for example. This is just the property that 
makes the string so useful in musical instruments, as we shall see. 

The General Solution for Wave Motion. — The foregoing property 
can be expressed mathematically by stating that the dependence 
of the shape of the string on x and t when a wave is going in the posi- 
tive x-direction must be of the general form F(x — ct). Then at 
t = the shape of the string is given by the function F(x), as is shown 
in Fig. 11a. At t = 1 the wave form F is the same, but it has been 
shifted bodily to the right a distance c cm. The general form for any 
motion of the string must be 

y=F(x-d)+f(x + ct) (8.1) 

representing a wave of the form F going to the right and one of the 
form / going to the left. Every motion of the string can be considered 
as a superposition of two waves, each having a speed c, traveling in 
opposite directions. 



74 THE FLEXIBLE STRING [HI.8 

The symbol F(x — ct) simply means that, no matter what function 
F is, the quantities x and t enter into it in the combination (x — ct). 
Examples are A sm[k(x — ct)], B(x — ct) 3 er a < x - ct >% etc. With our 
convention of taking real values, another example is 

C exp [Hz (x — ct)] = Ce ikx e- ikct . 

A peculiarity of this sort of function of x and t, one that will be of use 
to us in further discussion, is the following: F changes in exactly the 
same way and by the same amount when t changes by an amount a as 
when x changes by an amount — ca. A plot of the shape of the string 
at a given instant of time is completely similar to a plot of the displace- 
ment of a given point on the string as a function of time. To find 
the shape of F we can either run our eye over the whole extent of the 
string at a given instant of time, or we can watch the motion of one 
piece of the string as the wave passes by. This is, of course, still 
another way of saying that the wave progresses along the string with- 
out change of shape. The same statement holds for f(x + ct), except 
that the direction of x is reversed. 

A mathematical way of stating the property of F and / discussed 
above is 

| F(x - ct) = -c A F{x - ct), j t f(x + ct) = c A/(* + ct) (8.2) 

where the symbol (d/dt) means the rate of change with respect to t 
when x is held constant, and (d/dx) means the rate of change with 
respect to x when t is held constant. A change in F due to a small 
change in t is equal to (— c) times the change in F due to a change in 
x of an equal size. 

Initial Conditions. — The particular forms of the functions F and / 
are determined by the initial conditions: the initial form and "velocity 
form" of the string. If the shape of the string at t = is yo(x), and 
its velocity at the point x is v (x), then F(x) + f(x) = y (x) and 
d/dt [F(x — ct) + fix + ct)] = v (x) when t = 0, or, what is the same 
thing, 

«[-?(*) +/(«)] -(i)..(x) 

using Eq. (8.2). Since v (x) must be integrated to solve the last 
equation, we shall define the function 



S(z) = r v (x) dx 



in.8j 



WAVES ON A STRING 



75 



where z = % ± ct. It is not difficult to see that the shape of the string 
that corresponds to the specified initial conditions is represented by 
the function 



- \ \yo(x - ct) + y (x + ct) - \ S(x - ct) + \ S(x + ct) j 



(8.3) 



At t = 0, y = y (x), and (dy/dt) = dS(x)/dx = v (x), as is required. 
The solution is built up of two "partial waves" going in opposite 
directions, which combine to give the required behavior at t = and 
spread apart thereafter. Two examples of this are given in Fig. 12, 



^•-f I 



Fig. 12. — Motions of plucked and struck strings. The solid lines give the shapes 
of the strings at successive times, and the dotted lines give the shapes of the two "partial 
waves" traveling in opposite directions, whose sum is the actual shape of the string. 

one for a string pulled aside and started with zero velocity, and one 
for a string struck by a hammer so that it starts from equilibrium with 
a specified velocity. The spreading apart of the partial waves is 
apparent in the successive drawings. 

Boundary Conditions. — So far, we have been treating the string 
as though it had an infinite extent ; actually, it is fastened down some- 
where, and this fastening affects the motion of the string. The fact 
that the string is fastened to a support is an example of a boundary 
condition. It is a requirement on the string at a given point in space 
which must be true for alL time, as opposed to initial conditions, which 
fix the dependence of y and v on x at a given time. Boundary con- 
ditions are more important in determining the general behavior of the 
string, its allowed frequencies, etc., than initial conditions are. 

If the support is rigid, and the distance along the string is measured 



76 THE FLEXIBLE STRING tDLS 

from it, the boundary condition is that y must be zero when x = 0, 
for all values of the time. If the support is springy, so that it is dis- 
placed sideward a distance CF cm for a sideward force F dynes, then 
the boundary condition is that y must always be equal to C times the 
component of the string's tension perpendicular to the equilibrium line: 
y = CT(dy/dx) at x = 0. Many other sorts of boundary conditions 
are possible. 

Reflection at a Boundary. — Let us take the simple case of the rigid 
support, requiring that y = when x = 0, and see what effect this 
has on the motion of the string. The solutions y = F(x — ct) or 
y = f(x + ct) cannot be used in this case, for they will not always 
be zero when x = 0. However, the solution 

y = -F(x - ct) + F{-x - ct) 

will satisfy the boundary condition. At x = 0, 

y = -F(-ct) + F(-ct) = 0, 

(a) 

^ '-* 

(b) 



^(c) 



' Ky^ 



x\-_. 



Fig. 13. — Reflection of a wave from the end support of a string. The solid lines 
show the shape of the string at successive instants of time; the dotted lines, the imagi- 
nary extension of the wave form beyond the end of the string. 

for all values of t. To see what motion of the string this expression 
corresponds to, let us suppose that F(z) is a function that is large only 
when 2 is zero and drops off to zero on both sides of this maximum. 
Then when t = —10, the function F(—x — ct) will have a peak at 
x = 10c representing a single pulse traveling leftward along the string. 
The function F(x — ct) at this time would have its peak at x = — 10c 
if there were any string to the left of the support, but since no string 



III.8] 



WAVES ON A STRING 



77 



is there, the term —F(x — ct) is not apparent, in the shape of the 
string at t = — 10. It has been represented in Fig. 13a by a (Jotted 
line to the left of x = 0. 

As t increases, the wave in the actual string moves to the left, 
and the wave in the imagined extension of the string moves to the 
right until they begin to coalesce at x = 0. During the coalescing 
the displacement of the point at x = is always zero, for the effects 
of the two waves just cancel each other here. A little later, the waves 
have passed by each other, the wave that had been on the imaginary 
part of the string now being on the actual string, and vice versa. 



i^il_L 



--X^ 




Fig. 14. — Motions of plucked and struck strings fixed at one end. Dotted lines 
show the traveling partial waves; their sum is the solid line, the actual shape of the 
string. 

The succession of events is pictured in Fig. 13. What has happened 
is that the pulse which had originally been traveling leftward is 
reflected at the point of support x = and comes back headed toward 
the right, as a pulse of similar form but of opposite sign. The bound- 
ary condition at x = has required this reflection, and the particularly 
simple sort of condition that we have imposed has required this very 
symmetric sort of reflection. Most other boundary conditions would 
require a greater difference between the original and the reflected 
wave. When a wave strikes it, a rigid support must pull up or down 
on the string by just the right amount to keep y zero; and in doing so 
it "generates" a reflected wave. 



78 THE FLEXIBLE STRING [III.8 

The expression for the motion of a string satisfying the boundary 
condition y = Oat a; = and the initial conditions y = y (x),v = v (x) 
at t = is 

y = ^ I Y(x - ct) + Y(x + ct) -- c H(x- ct) + ~H(x + c<) (S.4) 

where 



Y(z) 
H{z) 



= f y («) (z > 0) 

\-</o(-z) (2 < 0) 

= (S(z) (z > 0) 

\S(-z) (z < 0) 

S(z) = f* v (z) dx 

These definitions of Y and H are necessary because y and y are 
defined only for positive values of x (where the string actually is), 
whereas the form of the partial waves used to build up the subsequent 
forms of the string must be given for all values of z = x + ct. The 
particular forms of Y and H are chosen so that they automatically 
satisfy the boundary conditions at x — for all values of t. Two 
examples of the way in which the motion of the string can be built 
up by the use of these partial waves are given in Fig. 14. 

Strings of Finite Length. — Actual strings are fastened at both ends, 
so that really two boundary conditions are imposed. For instance, 
the string can be fastened to rigid supports a distance I cm apart, so 
that y must always be zero both at x = and at x = I. The most 
important effect of a second boundary condition of this sort is to 
require that the motion of the string be periodic. A pulse started at 
x = travels to the other support at x = I in a time (l/c), is reflected, 
travels back to x = 0, and is" again reflected. If the supports are 
rigid, the shape of the pulse after its second reflection is just the 
same as that of the original pulse, and the motion is periodic with 
a period equal to 21/ 'c. The motion in this case is not, in general, har- 
monic, as we shall see, but it is always periodic. This periodicity 
of all motion of the string depends entirely on the fact that we have 
imposed a particular sort of boundary condition; if other conditions 
are imposed at a; = and x = I (i.e., if the supports are not perfectly 
rigid), then it may not be true that every motion is periodic; in fact it 
may never be periodic. 

The quantitative manner of dealing with the two boundary con- 
ditions is by means of the partial waves F. When the string is only 



IH.8] WAVES ON A STRINO 79 

I cm long, we are free to give any shape to F(z) for z larger than I or 
smaller than zero. "Free" is not the correct word, however, for we 
must choose that form of F which satisfies both boundary conditions. 
If we start out at t = with a pulse of the form F traveling to the 
left, then, as before, we can satisfy the condition y = at x = by 
setting y = -F(x - ct) + F(-x - ct). To have y = at x = I, 
we must arrange the rest of the function F, beyond the limits of the 






Ky= 



V-/^*~T 



is^l 2s/ 



^\^r ^" N 



Fig. 15.— Periodic motion of a string fixed at both ends. Solid lines give the shape 
of the actual string at successive instants; dotted lines show the imaginary extension 
of the wave form beyond the ends of the string. The motion is made up of two partial 
waves going m opposite directions, each being periodic in x with period 21. 

actual string, so that F (I - ct) = F(-l - ct), or, setting z = -I - ct, 
so that F(z) = F(z + 21) for all values of z. This means that the 
function F(z), which must be defined for all values of z, must be 
periodic in z, repeating itself at intervals of 21 all along its length. An 
illustration of how this sort of partial wave can be used to determine 
the motion of the string is given in Fig. 15. 

To satisfy the boundary conditions y = at x = and at x = I 
and the initial conditions y = y (x), v = v (x) at t = 0, we build up 
a combination similar to that given in Eq. (8.4): 

V = \\Y{x - ct) + Y(x + ct) - ±H(x - ct) + \H{x + ct)] (8.5) 



80 THE FLEXIBLE STRING [m.9 

where 



Y(z) = 



-yo(-z) 




(-1 <z < 0) 


2/0(2) 




(0 < z < I) 


-Z/o(2Z - 


z) 


(I <z < 21) 


yo(z - - 


21) 


(21 < z < 31) 


etc. 






S(-z) 




(-1 <z < 0) 


S(z) 




(0 < z < I) 


S(2l - z) 




(I <z < 21) 


S(z - 21) 




(21 < z < SI) 


etc. 







H(z) = 

S(z) = j" v (x) dx 

Two examples of the motion of such strings are given in Fig. 16. 
In Fig. 17 the displacement of a point on the string is plotted as a 

^s S* ^v 

k^' < i > < ^ :>< ^ t" — v — > — % - 



e*^t 



Fig. 16. — Motions of plucked and struck strings fixed at both ends. The solid 
lines show the successive shapes of the string during one half cycle. Shapes for the 
other half cycle are obtained by reversing the sign of the curves. 

function of time, showing that the motion is periodic but not simple 
harmonic. 

9. SIMPLE HARMONIC OSCILLATIONS 

It has been seen in the last section that the imposing of boundary- 
conditions limits the sorts of motion that a string can have and that 
if the boundary conditions correspond to the fixing of both ends of 



m.9] 



SIMPLE HARMONIC OSCILLATIONS 



81 




the string to rigid supports the motion is limited to periodic motion. 
This last result is an unusual one, for we found in the last chapter that 
even as simple a system as a pair of coupled oscillators does not, in 
general, move with periodic motion. It is not unusual for a system 
to oscillate with simple harmonic motion (which is a special type of 
periodic motion) when it is started off properly (we shall see that 
practically every vibrating system can do this); what is unusual in 
the string between rigid supports is that every motion is periodic, no 
matter how it is started. 

Our problem in this section is to find the possible simple harmonic 
oscillations of the string (the normal modes of vibration) and to see 
what the relation is between the frequencies of these vibrations that 
makes the resulting combined motion always periodic. The problem 
of determining the normal modes 
of vibration of a system is not 
just an academic exercise. For 
systems more complicated than 
that of the string between rigid 
supports we have no method of 
graphical analysis similar to that 
of the last section, and the only 
feasible method of discussing the 
motion is to "take it apart" into 
its constituent simple harmonic 
components. There is also a 
physiological reason for studying the problem, for the ear itself analyzes 
a sound into its simple harmonic parts (if there are any). We dis- 
tinguish between a note from a violin and a note from a bell, for instance, 
because of this analysis. If the frequencies present in a sound are all 
integral multiples of a fundamental frequency, as they are in a violin, 
the sound seems more musical than when the frequencies are not so 
simply related, as in the note from a bell. 

The Wave Equation. — To obtain the normal modes of vibration of 
a string, we must take up the problem that we laid aside in the last 
section, that of finding the equation relating the shape of a string and 
its motion. This equation was almost derived in the previous section. 
Looking back at Eqs. (8.2) given there, we find that, by repeating the 
differentiation, both F(x — ct) and /(a; -f ct) satisfy the equation 




Fig. 17. — Displacements of the points 
marked (c) on the strings shown in Fig. 16, 
plotted as functions of the time. 



d 2 y _ 1 d 2 y 



Bx 2 



at 2 



c = \/T/e 



(9.1) 



82 



THE FLEXIBLE STRING 



[III.9 



Since every motion of the string is a combination of the two waves, 
every motion of the string must satisfy this equation. It is called 
the wave equation. We shall meet it often in this book. 

The wave equation corresponds to a number of statements con- 
cerning the motion of a string. We saw in the last section that it 
implies that the wave motion travels with its shape unchanged, at a 
velocity c, independent of this shape. Since the derivative (d 2 y/dx 2 ) is 
proportional to the curvature of the shape of the string at a given 
instant, Eq. (9.1) states that the acceleration of any portion of the 
string is directly ■ proportional to the curvature of that portion. If the 
curvature is downward, the acceleration is downward, and vice versa; 
and the greater the curvature, the faster the velocity changes. 

This second statement corresponds to the usual method of deriving 

the wave equation. The net force 
on a portion of string at any instant 
is proportional to the curvature of 
this portion at that instant. A 
study of Fig. 18 shows us that the 
net force perpendicular to the z-axis 
on a piece of string of length ds is 
T(sm<p 2 — sin<pi). If the displace- 
ment of the string from equilibrium 
is not large (and we have already 
had to assume this), the angles >i and <p 2 will be small, so that 
sin^>i will be practically equal to tan<pi, and similarly for <p 2 . How- 
ever, tan <pi is equal to the slope of the string at the instant in question, 
(dy/dx), at the point x. Similarly tan <p 2 is the slope at the point 
x + dx. Now, from the definition of the symbol (d/dx), the value 
of a function / of x at the point x + dx is equal to the value of / at 
the point x, added to dx times the rate of change of / with respect to x, 




* x+dx 

Fig. 18. — Forces on an elementary 
length of flexible string. 



f(x + dx) = f(x) + dx 



1 
dx 



(9.2) 



so that the net vertical force on the element of string is 

T \(dx\ _ (dy\ ] = Tdx ±(§y\ = Tdx ejy 

l\dy/x+dx \dx/x] dx \dxj dx 2 

The mass of the element of string is e ds; and if the angles <p are 
small, this is practically equal to e dx. The equation of motion of the 
element is therefore 



HI.9] SIMPLE [HARMON IC OSCILLATIONS 83 

edx^=Tdx^, or ^- € ^ 
edx dt2 1 dx dx2 , or ___ — 

which is identical with Eq. (9.1). 

Standing Waves. — But we are still looking for the possible ways 
in which the string can execute simple harmonic vibrations. To exe- 
cute such motion, every portion of the string must oscillate with the 
same frequency. The mathematical counterpart of this requirement 
is that the shape of the string y(x, t) must be equal to the exponential 
e -2*irt multiplied by a function of x alone: y = Y(x)e- 2rirt . If this 
expression is substituted in the wave Eq. (9.1), we find that Y must 
satisfy the equation 

d 2 Y 4tt 2 i/ 2 

^+-^"^ = 0, c*=(T/e) (9.3) 

Solutions of this equation are the exponentials e 2vivx/c and e- 2vivx/c . 
All the simple harmonic motions of the string must therefore conform 
to the expression 

y = Q e (.2rir/c)(x-ct) _J_ Q _ e (2iciv/c){-x-ct) 

= A+ cos -^ (x - ct) - <S> + + A- cos ^ (x + ct) - $_ (9.4) 

representing two sinusoidal waves, of different amplitudes, traveling 
in opposite directions. This result is another illustration of the 
relation between the shape and the motion of the string; when we 
require that the dependence of y on t be sinusoidal, the dependence 
of y on x must also be sinusoidal. 

The wavelength of these waves X = (c/v) is the distance between 
the beginning and end of each cycle of the sinusoidal wave form. 

In the special case when the amplitudes of the two waves are 
equal, the trigonometric formula for the sum of two cosines can be 
used, giving 

A = 2A+ = 2A_ 

y = A cosf — x — Q J cos (2irvt — <f>), = £($_ + $ + ) (9.5) 

$ = i($_ - $ + ) 

Here the traveling waves combine to form standing waves. At points 

on the string where cos ( — a — £2 J = 0, the two traveling waves 

always just cancel each other, and the string never moves. These 



84 THE FLEXIBLE STRING [HI.9 

points are called the nodal points of the wave motion. In the case 
that we are considering, where the density and tension are uniform, 
the nodal points are equally spaced along the string a distance (c/2v) 
apart, two for each wavelength. Halfway between each pair of nodal 
points is the part of the string having the largest amplitude of motion, 
where the two traveling waves always add their effects. This portion 
of the wave is called a loop, or antinode. 

Normal Modes. — So far, we have neglected boundary conditions. 
If we require that y = when x = 0, the general form of (9.4) can 
no longer be used; the number of possible harmonic motions is limited. 
The expression for y that must be used is the standing wave form 
(9.5) with the angle ft so chosen that a nodal point coincides with the 
point of support x = 0: 

y = A sin( x J cos (2-irvt — $) (9.6) 

This agrees with the discussion in the previous section. For the 
simple boundary condition that we have used, the reflected wave has 
the same amplitude as the incident wave; and when the incident one 
is sinusoidal, the result is a set of standing waves. Any frequency 
is allowed, however. 

When the second boundary condition y = at x = I is added, the 
number of possible simple harmonic motions is still more severely 
limited. For now, of all the possible standing waves indicated in 
(9.6), only those which have a nodal point at x = I can be used. Since 
the distance between nodal points depends on the frequency, the 
string fixed at both ends cannot vibrate with simple harmonic motion 
of any frequency; only a discrete set of frequencies is allowed, the set 

that makes sin ( -^ I ) zero. The distance between nodal points 

must be I, or it must be (1/2), or (1/3) . . . etc. The allowed fre- 
quencies are therefore (c/2l), (2c/2l), (3c/2Z) . . . etc., and the differ- 
ent allowed simple harmonic motions are all given by the expression 

y = A„sinfe) cos(™ t - ^ (n = 1, 2, 3, 4 • • •) (9.7) 

The lowest allowed frequency v x = (c/2l) is called the fundamental 
frequency of vibration of the string. It is the frequency of the general 



IH.9] SIMPLE HARMONIC OSCILLATIONS 85 

periodic motion of the string, as we showed in the last section. The 
higher frequencies are called overtones, the first overtone being vz, the 
second v%, and so on. 

The equation for the allowed frequencies given in Eq. (9.7) 
expresses an extremely important property of the uniform flexible 
string stretched between rigid supports. It states that the frequencies 
of all the overtones of such a string are integral multiples of the funda- 
mental frequency. Overtones bearing this simple relation to the 
fundamental are called harmonics, the fundamental frequency being 
called the first harmonic, the first overtone (twice the fundamental) 
being the second harmonic, and so on. 

Very few vibrating systems have harmonic overtones, but these, 
few are the bases of nearly all musical instruments. For when the 
overtones are harmonic, the sound seems particularly satisfying, or 
musical, to the ear. 

Fourier Series.— To recapitulate: The string has an infinite num- 
ber of possible frequencies of vibration; and if the supports are rigid, 
these frequencies have a particularly simple interrelation. If such 
a string is started in just the proper manner, it will vibrate with just 
one of these frequencies, but its general motion will be a combination 
of all of them: 

y = Ai sin( -y J cosf -j t — 3>i J + A 2 sin( — p J cosf — j- t — # 2 ) 

+ . . . 
or, symbolically, 

■ - 2<^x)[ B "°K ! T*) + c ""K s r)] (9 ' 8) 

where the symbol 2 indicates the summation over the number n, 
going from n = 1 to n = «> . The value of A n is called the amplitude 
of the nth harmonic. 

Equation (9.8) is just another way of writing Eq. (8.5). The 
present form, however, shows clearly why all motion of the string 
must be periodic in character. Since all the overtones are harmonic, 
by the time the fundamental has finished one cycle, the second har- 
monic has finished just two cycles, the third harmonic just three 
cycles, and so on, so that during the second cycle of the fundamental 



86 THE FLEXIBLE STRING [HI.0 

the motion is an exact repetition of the first cycle. This is, of course, 
what we mean by periodic motion. 

Equation (9.8) is in many ways more useful for writing the depend- 
ence of y on t and x than is Eq. (8.5). For it gives us a means of finding 
the relative intensities of the different harmonics of the sound given 
out by the string (corresponding to the analysis that the ear makes 
of the sound) and thus gives us a method of correlating the motion 
of the string with the tone quality of the resulting sound. We shall 
have to wait until farther along in the book to discuss the quantitative 
relations between the vibrations of bodies and the intensity of the 
resulting sounds, but it is obvious that the intensity of the nth har- 
.monic in the sound depends on the value of the amplitude A n . Once 
the values of all the A n 's are determined, the future motion of the 
string and the quality of the sound which it will emit will both be 
determined. 

Initial Conditions. — The A n 's and $ n 's, or the B n 's and C n 's, are 
an infinite number of arbitrary constants, whose values are fixed by 
the initial conditions, corresponding to the infinite number of points 
along the string whose positions and velocities must all be specified 
at t = 0. Our analysis will not be complete until we devise a method 
for determining their values when the initial shape and velocity shape 
of the string are given. 

The initial conditions must satisfy the equations, obtained from 
(9.8) by setting t = 0, 

y(x,0) = y (x) = ^ B n sm[~) 

Series like the right-hand sides of these equations are called Fourier 
series. 

Now, the initial shape and velocity shape of the string, the func- 
tions yo(x) and v (x), can be any sort of functions. that go to zero at 
x = and x = I. It therefore must be possible to express any func- 
tion satisfying these boundary conditions in terms of a Fourier series 
of the type S5„ sin (rnx/l). Subtle. mathematical reasoning must be 
used to prove rigorously that this is true (in fact, a completely satis- 
factory proof has not yet been devised), but since we know as physicists 
that the motion of the string is definitely specified by its initial shape 
and velocity shape, we shall assume that it must be true. 



m - 9 ] SIMPLE HARMONIC OSCILLATIONS 87 

The Series Coefficients.— The trick to obtain the values of the 
B n 'a and C n 's consists in multiplying both sides of the preceding 
equations by sin (irmx/l), where m is some integer, and integrating over 
x from x = to x = I. The utility of the trick lies in the fact that 
J o sin {irnx/l) sin (irmx/l) dx is zero unless n equals m, in which case 
it equals (1/2), so that the infinite series of integrals on the right-hand 
side has only one term not zero, involving just one B m or C m . For 
instance, 

X sin (^) y ° (x) dx = % B « £ sin (^) sin (^) ** 

= \2j Bm 

so that 

„ 2 C . (irmx\ . . . \ 

m = T ) sin \T/ y °w ) 

and, similarly, ( (Q Q , 

Cwi = ^ Jo sin V-T^o(x) rfz (m = 1, 2, 3, 4 ■ • • ) ) 

Equations (9.9) provide a means of determining the values of the 
Bn& and C n 's in terms of the initial conditions. 

Plucked String, Struck String.— A few examples will indicate how 
the method works. For instance, if we pull the center of the string 
out h cm and then let it go at t = 0, all the C m 's will be zero, and 

8h . firmS 

= ^ sin V-27 

{0 if m is an even integer 

(_1)(— d/ 2 if w is an odd integer 

Therefore, 

y - ¥ r n Vi7 cos \tJ - 9 sm \ir) cos {-r) +'.■■■ 

• (9-iP) 



88 



THE FLEXIBLE STRING 



[m.9 



Computing this series for y as a function of x and t gives the same 
values for the shape of the string at successive instants as are shown 
in the first sequence of Fig. 16. .Figure 19 shows how the correct 
form is approached closer and closer the more terms of the series are 
used. 

At first sight, the foregoing series appears to be simply a more 
awkward way of finding the shape of the string than the method used 
in the previous section. However, the series can tell us more about 
the string's motion than the results of the last section can. It tells 




/\ First Te 




First Four* Terms 

Fig. 19. — Fourier series representations of the initial form of the string given in 
Eq. (9.10) and the initial velocity form given in Eq. (9.11). Successive solid curves 
show the effect of adding successive terms of the series; dotted curves show the actual 
form, given by the entire series. 



us, for instance, that the second, fourth, sixth, etc., harmonics will 
be absent from the sound given out by the string, for they are not 
present in the motion. It tells us that if, for example, the intensity 
of the sound emitted is proportional to the square of the amplitude 
of motion of the string, then the fundamental frequency will be 81 times 
more intense than the third harmonic, 625 times more intense than 
the fifth harmonic, etc. 

The absent harmonics correspond to standing waves that have a 
nodal point at the center, the point pulled aside. This is an example 
of the general rule (which can be proved by computing the # required 
integrals for B m ) that in the motion of any plucked string all those 
harmonics are absent which have a node at the point pulled aside. 



IH.9] SIMPLE HARMONIC OSCILLATIONS 89 

If the string is struck, so that y = and 

((*) ( 0< *<d 
(o Q<»<») 

then all the B m 's are zero, and 

iron ~ 2 f P z/4 A/xx\ . /xmx\ , 

so that 

. 1 + V2 . /3tx\ . (Zicct\ 

+ — 27- sm VT-J sin V~ry 

"m^—jsm^—J- • • • J (9.11) 



^ 7T°C 



125 



We note that the fourth, eighth, etc., harmonics, those having nodes 
at x = (1/4:), are absent in this case. 

Energy of Vibration. — There is a general analogy between the 
amplitudes A n of the various harmonics and the amplitudes of motion 
along the normal coordinates X and F discussed in Sec. 6. This 
can be illustrated by computing the energy of vibration of the string. 
The kinetic energy of the string is the integral of the kinetic energy 

of each element of length: ie I i-^Jdx. Its potential energy is 

equal to the amount of work necessary to move the string into its 
instantaneous form from the equilibrium form y = 0. Suppose that 
its form at a given time tis y, a function of x. Then we can imagine 
changing the string from equilibrium form to final form by making 
its intermediate form be ky, where k changes from zero to unity. The 

force on any element of string of the form ky is T f — 2 ky) dx, and the 



90 



THE FLEXIBLE STRINO 



[III.9 



force that we must use to oppose this is equal to this value and oppo- 
site in sign. As we displace the string from equilibrium by changing 
k, the element of displacement is y dk, and so the work required to 
bring this element of string into place is 



J>(s) 



^)kdxydk = -Ty[^ 2 



(5)*J>--«>(S) 



dx 



The potential energy of the whole string, the work required to bring 
the string from equilibrium into the form represented by the function 
V, is 

if we integrate by parts. Since y = at x = and x = I, the first 
term is zero, and the potential energy is just (T/2)j(dy/dx) 2 dx. The 
total energy of the string is therefore 

When series (9.8) is substituted into (9.12), the expression for the 
energy becomes 

12 



-*(")' 



-I- 4AM2 s i n (-/ 



^ nA n cosf^y- J cos(^ 3>„ J \ dx 

A\ sin 2 (?f - <h) £ sin 2 (y) dx + • • • 

$i ) sin Cr - * 2 ) Jo sin Vr) sin (nr) dx 

+ • • • 

+ A\ cos 2 fe* - * x ) Pcos 2 fe)^ + • ■ • } 

All the integrals of the sort / sin (ttx/T) sin (2rx/l) dx are zero, whereas 
integrals of the type of / sin 2 (rx/l) dx and / cos 2 (irx/l) dx are equal 
to {1/2), This simplifies the expression enormously,- and after adding 
sin 2 to cos 2 , term by term, we can write 

w = w.) 2 (™J A i = 2 2t ° (I) vlAl ' " -■■ (l) (9 - 13) 



111.10] FORCED VIBRATIONS 91 

The energy is therefore a series of terms, each term depending on 
just one of the harmonics, an expression similar to that given in 
Eq. (6.6). The different harmonics are the different normal modes 
of vibration of the string, and the quantity A n is the amplitude of 
vibration along the nth normal coordinate of the system. We can 
say that the energy of a vibrating string is equal to the total energy 
of an infinite number of equivalent harmonic oscillators, each having 
a mass equal to half the total mass of the string {It/ 2), one having 
frequency vi and amplitude A i, another having frequency v 2 and 
amplitude A 2 , and so on. 

10. FORCED VIBRATIONS 

So far we have studied the particularly simple case of the free 
vibrations of an idealized string. The analysis was useful, for it 
gave us an insight into the general properties of the motion without 
entangling us in the algebraic complications that crop up as soon as 
some of the idealism is relaxed. 

Before we study the effects of motion of the end supports, friction, 
and nonuniformity on the behavior of the string, let us study the 
response of the idealized string to a driving force. Here again we 
find that a neglect of complications, at first, enables us to bring out 
the salient behavior with the least amount of mathematical camouflage. 

Wave Impedance and Admittance. — The simplest case is that of a 
string of infinite length stretched at a tension T between supports 
at x = and x = °o. The mass of the string per unit length is e, 
so that t he v elocity of wave motion 
is c = -y/T/e. The support is rigid in 
the x-direction, so as to sustain the 
tension of the string; but it is hinged 
so that it can move in the ^-direction, 
transverse to the string. A transverse 
force applied to the support will move 
the end of the string as well as the 
support, so that the mechanical im- 
pedance offered to the force is the sum FlG - 20 -- Forces on a flexible su PP° r *- 
of the transverse mechanical impedance of the support and the 
mechanical impedance of the end of the string to transverse motion. 

The impedance of the support is of the general type discussed in 
Chap. II, and need not concern us here. What is now of interest is 
the wave impedance of the string, the ratio of the transverse force 
applied to the end of the string to the transverse velocity of the end 




92 THE FLEXIBLE STRING [111.10 

of the string, when the driving force is simple harmonic and the trans- 
verse impedance of the end support is neglected. 

The displacement of the string at a distance x from the end and 
at a time t is y(x,t). The angle that the end of the string makes with 
the equilibrium line is 9 = tan _1 [(dy/dx)_o]. The longitudinal force 
on the end of the string is — T cos 0. Since we are assuming, every- 
where in this chapter, that (dy/dx) is small compared with unity, we 
can say that, to the first approximation, the longitudinal force is T, 
the constant tension, and the transverse force exerted by the support 
on the end of the string is 



■ - r (2) 



-rt».--r.^-/«> (io.i) 

If the driving force is simple harmonic, / = Foe - *"', the shape of the 
string must be sinusoidal, as is indicated in Eq. (9.3). In the present 
case the wave must be going in the positive ^-direction, since it is 
being generated at x = 0. Therefore the space factor will be Ae+ ikx , 
where 



— A oihx — iut 



Incidentally, because we have chosen the negative exponential 
e -u>t f or our convention, it turns out that the sign of the exponential 
e ikx indicates the direction of motion of the wave. A wave in the 
negative ^-direction would be e - i( ~ kx+a>t) , with a negative sign in the 
a>part. 

The value of A is determined from the expression for the driving 
force in Eq. (10.1) 



Foe-™' = 

so that 

V = 



\dx) x= o 



1 ** ° e ik{x-ct) 



— l(j) eC 



V ~ dt ec 6 



(10.2) 



The input impedance of the string, the ratio between the driving force 
and the transverse velocity of the string at x = 0, is 



111.10] FORCED VIBRATIONS 93 

ec = (±\ = y/Te (10.3) 

This quantity, the input impedance of an infinite string, is sometimes 
called the wave or characteristic impedance of the string. We see it 
is real, being a pure resistance for a simple string. This implies that 
energy is being continuously fed into the string, energy of wave 
motion that never returns, since the string is infinite in extent. When 
we come to consider strings of finite length, we shall see that the input 
impedance differs from the characteristic impedance ec and is not 
purely resistive. 

The power input to the string is the average value of Fv at x = 

p = i &!! = i €C | y p (io.4) 

In many cases we wish to know the velocity of some other part 
of the string when the force is applied at x = 0. This is obtained in 
terms of the transfer impedance, already discussed in Sec. 7. It is 
usually easier to deal with the reciprocal quantity, the transfer admit- 
tance, the ratio between the transverse velocity of the string at point x 
and the simple harmonic driving force (in this case at x = 0). For 
the simple case of the infinite string, 

^=r„(0,*;«)= jgl^i;^ (10.5) 

General Driving Force. — When the transverse force on the end of 
the string is more complicated than a simple harmonic force, the 
response of the string can be obtained by the methods of Sec. 6. We 
find the Fourier transform of the force function 



(W) = ^J- 



Then, in accordance with the analysis leading to Eqs. (6.15), we 
obtain the equation for the velocity shape of the infinite string when 
the end at re = is acted on by a transverse force f{t), 

v(x,t) = f Y m {0,x;o>)e-^ da> I ^ J ^ f(j)e™ dr J 

.. /» 00 (* 00 . 

= %T~ \ e (Wc) - i < tf ' du I /(r)e iwT dr 



94 THE FLEXIBLE STRING [HI.10 

Using the general formulas for Fourier transform given by Eqs. (2.19) 
and (2.20), we find that the velocity shape of the string is 

^-sK'-O-M'-?) (ia6) 

This is a very interesting and simple result, one that we. should 
have been able to obtain without recourse to the machinery of Fourier 
transforms. It restates the simple fact that was discussed in Sec. 8, 
that the shape of the string, as function of x for a constant time, is 
simply related to the motion of one part of the string, as a function 
of time. Since the input impedance of the infinite string is constant 
and resistive, the velocity of the string at the point of application of 
the force (x = 0) is proportional to the force no matter what form f(t) 
has. Therefore for x = 0, v(0,t) = (l/ec)f(t), where ec is the input 
resistance. And since the motion of the string causes waves in the 
positive x-direction, the expression for v(x,t) must be that given in 
Eq. (10.6), with the characteristic quantity (x — ct) appearing. 

The corresponding expression for the displacement is obtained 
by integrating /, 

y(x,t) = \- c Q (t - fj; Q(Z) = j* f(t) dt (10.7) 

We can now verify our calculations still further. From Eq. (10.1) 
we have that 

. *>--*&L- -s[=«H)L 

which checks, if we use Eq. (10.7) for Q and the relation c 2 = (T/e). 

The reason that there is such a simple relationship between / and 
y in this case is that the input impedance is a constant, independent 
of «. There are no resonant frequencies, where the impedance is 
zero and the admittance has a pole, for the infinite string; since there 
is no reflection of waves from the far end, there is no periodicity of the 
wave motion and, therefore, no natural frequencies of free vibration. 

String of Finite Length. — When we come to consider the simple 
string of finite length with a rigid support at x = I, we find that the 
input impedance at x = is considerably more complicated. The 
string now has resonance frequencies, so that the impedance goes to 
zero for certain values of <a. The waves are reflected from the rigid 
support at the far end (x ~ I), so that no energy is lost. Therefore 



m.10] FORCED VIBRATIONS 95 

if the support at x = I is rigid, and if there is no resistance to the 
string's motion, then the input impedance is purely reactive, with 
no resistive term. 

We find the displacement of the string from Eq. (9.3). The two 
simple harmonic waves of frequency (w/2tt) = v are combined to form 
a standing wave of zero amplitude at the rigid support x = I, 

y = A sin[k(l — xjje-^, k = (w/c) 

The amplitude o. the wave, A, is adjusted to fit the amplitude of the 

force at x = 0, 

, (dy\ . To, /<al\ . , 

1-^1 — A — cos I — )e-" at 
\dx/ x= o c \ c ) 

Therefore the expressions for the displacement and velocity of a string 
of length I driven from one end are 



Foe-* * = 



El sin[(co/c)(S ~ x)] e _^ t 
ecu cos (col/c) 

.F 8w[(a/c)(l — x)] 
ec cos (ool/c) 



(10.8) 



and the expression for the input admittance and the transfer admit- 
tance (reciprocals of the impedances) are 



r(0,0; w )=-(i)tan(^) 

--©MtM?)--"©] 



(10.9) 



Y(0,x;co) = 



Foe-™ 1 

This case, therefore, gives definite resonances; the amplitude of 
motion becomes infinite (since we are neglecting friction) whenever 
tan (o)l/c) becomes infinite. This occurs at the frequencies for which 
M/c) = (tt/2), (&r/2), (&r/2) • • • ; or 

Vn = (^) (2n + 1} (n = °' 1 ' 2 ' 3 ' ' ' > 

io n = 2irv n = fe)(2n + 1) 

For these frequencies the input admittance is infinite (the input imped- 
ance is zero). 

For the frequencies v = (c/2l), (2c/2Z), (3c/ 21) • • • the input 
admittance is zero, the input impedance is infinite (for the zero 
friction case). This is analogous to the parallel resonance in electrical 
circuits. At these antiresonance frequencies the motion of the end 



\k*~ 



JJ^~ 






06 THE FLEXIBLE STRING [HI.10 

point x = is infinitesimally small, though the rest of the string is in 
motion. 

For very low frequencies, the input impedance 1/F(0,0;«) has 
the limiting value i(ec 2 /lu) = i(T/fa>), a " capacitative " reactance, 
with an effective stiffness constant (T/l) and "capacitance" (l/T). 

The next step in this analysis would be to compute the response 
of this system to impulsive forces. This case will be left for a prob- 
lem, and we shall go on to a somewhat more useful example. 

Driving Force Applied Anywhere. — Often the driving force is not 
applied at one end of the string, but elsewhere along the string. Some- 
times, indeed, the force is distributed along the string, instead of being 
concentrated at a point. A solution of this general case is best 
obtained in terms of solutions for concentrated forces, however; so 
the next problem we shall study is that of the string of length I, held 
between two rigid supports, and acted on by a force F(£)e -iwt applied 
at the point x = £. 

When we obtain the steady-state response for a simple harmonic 
force applied at a point, we can use the operational calculus to obtain 
the response to an impulsive force applied to a point. Once this is 
obtained, the response of the string to a force that is any function 
of time and that is distributed in any manner along the string is com- 
puted in terms of integrations over | (the point of application) and r 
(the instant of impulse), analogous to the integrals of Eqs. (7.14). 

If a simple harmonic force of frequency (w/27r) is applied to the 
point x = £, the part of the string f or x < £ should be a part of a 
standing wave that is zero at x — 0, such as a sin (cax/c). The part 
for x > £ should be a standing wave that goes to zero at x = I, such 
as b sin[(<o/c)(£ — x)]. There will be a sudden change of slope at 
x = £, and — T times this change in slope must equal the applied 
force. Calculations similar to those made in the previous example 
show that the correct solution is 



>-- s \* \ eco sin(«Z/c) 



ecu sin {ool/c) \ c / 



F(£)e-™ 1 sin (w|/c) . 



{-:>-*>] 



(10.10) 



(x> 



The transfer admittance (v/f) for this case is 

i_ sm[(o)/c)(l - £)] sin (a>x/c) . < ^ 

_ i_ sin ((o^/c) sin[(co/c)(f - x)\ . > . 

ec sin (wZ/c) 



m.10] FORCED VIBRATIONS 97 

The input admittance is obtained by setting x = £ in the expression 
for Fax;co). 

Alternative Series Form. — Equation (10.10) is a closed form for 
the shape of the string driven by a simple harmonic foree applied to 
x = £. Sometimes it is useful to express this function in terms of a 
Fourier series of the type given in Eq. (9.8). As shown there, if 
we set y = XB n sin (rnx/l) the integrals for B n can be worked out 



?« = j I sin {jyJ v(x,t) dx 



where y is given in Eq. (10.10). After a great deal of algebraic 
manipulation, we obtain the final series 

n = 1 

which is equal to the expression given in Eq. (10.10). While it is an 
infinite series, it is sometimes easier to use it for computation than is 
the closed form of Eq. (10.10). 

As a matter of fact we could have obtained this series in a much 
more straightforward manner by going back to the equation giving 
the force on each portion of the string. According to the discussion 
following Eq. (9.2), the force on the element of string between x and 
x + dx is T dx (d 2 y/dx 2 ), due to the tension. This is the only force 
on most of the string in the present case; except for the point where the 
driving force is applied, x = £. Here the applied force per unit length 
of string is very large, so that the integral over x of the applied force 
near x = £ is equal to .Foer**'. The applied force per unit length is 
therefore given in terms of the delta function discussed on page 48. 

The equation of motion of the string then becomes 

If we are interested in the steady-state response, we consider that 
the transient oscillations have died out, and all that is left is a shape 
that oscillates with frequency («/2x). In brief, we set y{x,t) = 
Y^e-^K The resulting equation for Y is 



d?Y 
dx 2 



-® r -[¥\«?-o < io - is > 



98 THE FLEXIBLE STRING [111.10 

where c 2 =. (T/e). Now if we can find a Fourier series expansion for 
8(x — £), we shall be able to find a Fourier series expansion for F. 
A series expansion for the delta function can be found by the 
methods of Eq. (9.9). The important property of the delta function 
K x ~ £) i s that for any function f(x) 

£f(x)8(x- Z)dx=f(® (10.14) 

Therefore if 8 can be expressed as a Fourier series SZ)„ sin {irnx/l), the 
series coefficients are 



D.-f 



X*-e-»(T)*-?W ! f). 

Formally, therefore, the series representing the delta function is 

«(* - e - r 2 ""Or) sin (t 1 ) (10 - 15) 

The only awkward part is that the series does not converge. This 
should not be surprising, however, for one would expect a peculiar 
sort of series to represent as peculiar a function as 8(x — £). It need 
not disturb us particularly, however, for we do not intend to use series 
(10.15) to compute 8(x — Q; we shall use it only as a means to com- 
pute the series for y. As long as we use the series for 5 only as a short- 
cut means of obtaining other series which do converge, we can perhaps 
justify our use of a nonconvergent series. Naturally we must use the 
short cut with care and check our results by other methods whenever 
we can. 

Going back to Eq. (10.13), we assume that Y = SA n sin (irnx/l) 
and substitute this and the series for 8(x — |) into the equation. The 
result is 

Equating coefficients of sin (mx/l) on both sides of the equality sign 
gives equations for A n , and the resulting series for y = Ye*™ 1 is just 
that given in Eq. (10,12). This justifies the use of the series for 



111.10] FORCED VIBRATIONS 99 

b{x — £), since the series for y converges, and also indicates how much 
shorter is the short cut than the earlier method. 

The series of Eq. (10.12) shows very clearly the phenomenon of 
resonance and the resonant properties of systems, which we began to 
discuss in Sec. 7 on coupled systems. The steady-state shape of the 
string is usually a combination of all the shapes of all the normal modes 
of free vibration of the string, sin (irnx/l). When the driving frequency 
(oj/2t) approaches one of the natural frequencies (nc/2Z), the contribu- 
tion of the corresponding mode to the shape of the string becomes 
larger than all the others. In other words, as the driving frequency 
approaches one of the natural frequencies, the amplitude of motion 
increases without limit and the shape of the string approaches that of 
the corresponding normal mode. 

The series also indicates the effect of the point of application of the 
force. If the force is applied at a node of one of the normal modes, 
the corresponding factor sin (irn^/l) is zero and that normal mode and 
resonance is absent from the motion. 

Distributed Driving Force. — To find the response of the string to 
a simple harmonic driving force that is distributed along the string, 
we need only integrate the expressions given in Eqs. (10.10) and 
(10.12) over the point of applications | of the force. If the previous 
expressions give the response to a delta functio nTiorce at £, it is not 
difficult to see that the response of the string to a force ^(De - ^ dynes 
per cm length distributed along the string is given by the formulas 






sin (tux/1) 
(im/iy - (a>/c) 2 _ 



e-**' (10.16) 



One example of the use of this equation should probably be given. 
When a plane wave of sound of frequency a)/2ir passes over a string, 
it produces a force on it which is in phase over the whole length of the 
string if the direction of the wave is perpendicular to the string, but 
which varies in phase from point to point if the wave front is at an 
angle. In general the force per unit length at the point £ cm from one 
end can be given by the general expression 

F(f)er-*"« = /Toe*"* - *"', a = (2t/X s ) COS^ 

where X s is the wavelength of the sound wave in air and # is the angle 
the direction of the wave makes with the string. The relation between 



100 THE FLEXIBLE STRING [HI.10 

F and the intensity of the sound will be discussed later in this book 
[see Eq. (29.6)]. 

Taking the simple case of normal incidence first, where the force 
Foe - "*' is uniform and in phase over all the string, we use the closed 
form of Eq. (10.10) in Eq. (10.16) to obtain 



■fc(-!)] 



J? -lot ) C0S I 

= b^_ < ,_„ N -/-■ _ ! i (10 17) 

9 €0) 2 \ COS(cdZ/2c) ) 

This has resonances at every other harmonic (v = co/2tt = nc/2l; 
n = 1,3,5,7 • • • ), the symmetry of the normal modes for the even 
harmonics precluding their excitation. The shapes exhibited by the 
string for different driving frequencies are shown in Fig. 21. 

For the more general driving force Foe*"* - ™', it is easier to use Eq., 
(10.12) for y{£,x,t), and the final result can be expressed in the series 



V °\ T J ^™* (xw) 2 - (aiy 

(10.18) 



(ttw) 2 - (a>Z/c) 5 

This series is equal to the closed form of Eq. (10.17) for a = 0. 

Transient Driving Force. — To calculate the response of the string 
for a transient force, we use the operational-calculus methods again. 
We begin by computing the response of the string to an impulsive 
force at t = applied at x = £; f(t) = 5(x — £)8(t). Referring to 
Eq. (6.16), we see that the proper expression is given by the integral 

Using Eq. (10.11), this becomes 



, x . t) = _L f" *"*" s(»,z) 

" ' 2irec J - «, o) sin (coZ/c) 



where 

(sin(^)sin[(f) (*-*)] (*>{) 

The poles of the integrand are atu= (rnrc/l) where n is any integer, 
positive or negative (these correspond to the natural frequencies). 
Near one of the poles for an even n, the quantity sin (coZ/c) approaches 



111.10] 



FORCED VIBRATIONS 



101 




v = 4v, £ 



43 

» s io»i 



Fig. 21. — Shapes of steady-state motion of an undamped string between rigid sup- 
ports, driven by a uniform force of frequency v. The fundamental frequency of the 
string is v\. Those parts of the string above the dashed equilibrium line are in phase 
with the force; those below the line are 180 deg out of phase. 

the value (l/c)[w — (ntrc/l)]. In other words, for <a very near (nirc/l) 
(n even) the integrand becomes 



1 Q—inrct/l 



sin I 



* - T ) sm V" ) 

2rel (rnrc/l) w — (nwc/l) 



(*< Q 



The residue of this expression, its limiting value when multiplied by 
[w — (nirc/Vj\, as this factor approaches zero, turns out to be 



-1 
2ir 2 ncc 



o — invct/l 



sin 



-r) sm KT) 



(0 < x < I) 



102 THE FLEXIBLE STRING [HI.10 

For n odd, the factor sin (cd/c) approaches the quantity 



fc\ ( rarc\ 



but the factor sin(nir — mr^/l) turns out to be — sm(nir£/l), so the 
two minus signs cancel, and the result is the same as before. The 
final value for the integral, — 2wi times the sum of the residues on or 
below the real axis of <a, gives the value f or t > 

/ (t < 0) 

Vitefi = ]zl ^ I ^(^) ^J^A Pr ^,i {t > o) 
l iwec ±^ n n \ I J \ I / 

Utilizing the equation sinz = {l/2i)(e iz — e~ iz ), we have 

/ (t < 0) 

y s a,x;t) = ] _2_ ^ 1 S in (™*) sin (™) sin (^l) (t > 0) 

1 n = 1 

(10.19) 

Incidentally, this equation could also have been obtained by substi- 
tuting series (10.12) in the contour integral. 

The series for y& does not converge well, any more than the series 
for the delta function does. However, as with the delta function, we 
are not interested in computing y s , the behavior of the string after 
being hit by an idealized impulsive force at a mathematical point on 
the string. We only intend to use the series as an easy means of com- 
puting the behavior of the string when acted on by more realistic 
forces, distributed along the length of the string and spread out in time. 

For the most general type of force /(£,£), a function of time and of 
position, the response of the string is 

dr I d£ f(£,T)ys(£,x,t - t) 

-=3i|jJ>[T (, -' ) ]*-. (1 °- 20) 

This series does converge for reasonable forms of /((•,<)• 

The Piano String. — A reasonable approximation to the force of a 
piano hammer on a string is 



m.10] FORCED VIBRATIONS 103 

/0 (-icr > *> fr; . a* - if > $ > a* + fr) 
*U0 = A cos [(^ - * )] cos(^) (10.21) 

\ (-iff < * < i<r; z - if < £ < x + if) 

The quantity a is the time duration of the application of force, and f 
is the length of the portion of string acted on by the force. The force 
starts acting at t = —io-, rises to maximum at t = 0, and goes again 
to zero at t = io-. The distribution of the force along the string is also 
like the positive half of the cosine curve, with the center of force, where 
it is greatest, falling at £ = x . 

Substituting this into Eq. (10.20) and carrying out the usual 
accompaniment of trigonometry and algebra, we can work out the 
expressions for the shape of the string. The integration over £ is 
from x Q — if to x + if, but the integration over t is a bit more 
elusive. When t is less than - i<r, y s (£,x; t - r) is zero for all values 
of r for which F(£,r) differs from zero, so the integral is zero, as it 
should be (since the string has not been hit yet). For t > io- the 
range of integration over t is from - io- to + io-; but f or - io- < t < io- 
the only range over which Fy s is not zero is between r = - io-, where 
F goes to. zero, and t = t, where y»(£,x; t - t) goes to zero. Conse- 
quently, we have three expressions for the resulting shape y(x,t) 
Fort<- io-, y(x,t) = 

For - io- < i < i<r, 

For t > io-, 

. U ( X t) = *M. ^S i l" cog(™rr/2Q l ["cos (n7rco-/2Z)l 
x^ec ^ n Ll - (nf/0 2 J |_1 - (WO 2 J 
D ;^ (TrnxA . (t71x\ . • (irnci\ 

8m \—) sm \-r) sm \-r) 

This formula appears quite formidable, but it can be computed if need 
be. Certainly it is not very difficult to obtain from it the relative 
magnitude of the various harmonics in the free vibration after the 
end of the blow (t > io-). 

Incidentally, this example is a good one to show the power of 



104 THE FLEXIBLE STRING [111.10 

the operational-calculus methods. The algebraic gymnastics neces- 
sary to obtain Eq. (10.22) from the combination of Eqs. (10.20) and 
(10.21) are not particularly easy for one mathematically muscle-bound. 
Nevertheless, the calculations are only laborious, not subtle. On the 
other hand, to obtain the final formula for y(x,t) by any other method 
would involve still more labor, and more mathematical subtleties than 
we care to include in this volume. Our applications of operational 
calculus in Chap. II may have seemed rather like using a sledge ham- 
mer to drive a tack. We see now, however, that the problem of the 
simple string already provides a spike worthy of the sledge. 

To be honest, the resulting series for y(x,t) is not too good an 
approximation for the actual motion of an actual piano string when it 
is struck, partly because the actual piano string is not a perfect string 
but has stiffness. We shall indicate how to correct for this in the 
next chapter. 

The Effect of Friction. — In the foregoing analysis we have neglected 
friction, although it is present in every vibrating string. To complete 
our discussion we should show, as with the simple oscillator, that the 
effect of friction is to damp out the free vibrations and to change 
slightly the allowed frequencies. To show that this is so is not diffi- 
cult by the use of operational calculus, although it would be difficult 
by any other method. 

The difficulty lies with the nature of the frictional term. The 
resistive force per unit length opposing the string's motion is due to 
the medium surrounding the string, the medium gaining the energy 
that the string loses. Part of the energy goes into heating the medium, 
the amount depending on the viscosity of the medium; and part goes 
into outgoing sound waves in the medium, the amount depending on 
the radiation resistance of the medium. The medium also adds an 
effective mass per unit length, which may not be negligible if the 
medium is a liquid. The important point, however, the one that is 
responsible for our difficulties, is that the effective resistance due to 
the medium (and also its added effective mass) depends on the fre- 
quency of the string's motion. 

The equation Of motion for the string when friction is included is 

d 2 y_ T d 2 y MsdV 
e dF- T dx-*- R{03) M 

where R is the effective frictional resistance per unit length of string. 
To find the "normal modes" of the string involves a sort of circular 
process, since we cannot solve for the natural frequencies until we 



IH.10] FORCED VIBRATIONS 105 

know the value of R, and we cannot obtain the value of R unless we 
know the frequency of motion. In this case it is actually easier to 
start with steady-state driven motion, for then the frequency is known 
and the value of R is definite. 

The equation for the string acted on by a simple harmonic force 
of frequency (a)/2ir) exerted on the point x = £ is 

*4 + 2 *(co) |f - c* g « m 8( x - fr** (10.23) 

c 2 = (f/e), *(«) = R(<a)/2e 

We are assuming that the added mass due to the reaction of the medium 
is negligible compared with the weight of the string. The more 
general case, where we must assume that e is also a function of <a, 
will be discussed later. 

As has been done previously, we shall assume that the steady- 
state motion of the string can be expressed in terms of a Fourier series 

V = ^j °» sin ( nr) e_i "' 

Substituting this in Eq. (10.23), multiplying both sides by sin (ttwz/Z) 
and integrating over x from to I, gives an equation for a m , from which 
one eventually obtains a series for y 

le ~{ C" + ** ~~ «*»)(« + tk + tow) 
where 

-w = [(T) 2 -^ 

We note that both w n and k are functions of w, the driving frequency. 
Characteristic Impedances and Admittances. — An interesting 
alternative method of writing this equation is in terms of transfer 
admittances. The ratio between the string's velocity v = — iuy and 
the driving force F{£)er iMt is 

Y m (S,x;o>) = 2 \~^ , 

7T X \-Zm (£,x;a},n) 

- s \-zr) J csc v— ) csc \— ) 



,106 THE FLEXIBLE STRING [111.10 

The input admittance is obtained By setting x = f. Considering v 
to be analogous to a current and F analogous to a voltage, the reaction 
of the string (at x = £) is analogous to that of an electric circuit of 
an infinite number of parallel branches, the nth branch consisting of an 
inductance (le/2) csc 2 (rn^/l), a resistance of (IR/2) csc 2 (tw^/0> and a 
capacitance of (2l/ir 2 n 2 T) sm 2 (irn^/l), all three in series. 

The response of the string to a unit impulsive force concentrated 
at x = £ is obtained by computing the contour integral 

mU:X . t) .if { 2 < ?T* , \f* ( H!l ^ e ~n d » 

y vs ' ' ' irk J- „ ( ^-J (w + ik — w n ){w + ik + w n ) ) 

One pole for the nth term occurs when a; + ik(o>) — w n (u>) is zero. 
This may be difficult to solve algebraically if k(w) is a complicated 
function of «. However, it can usually be solved graphically or by 
successive approximations. We can write the solution symbolically as 

<a = io n — ikn', k n = k(w n — ik n ) \ HO 26") 

an = w n (<a n - ik n ) = [{nrnc/iy - A:*]* j 

It turns out that the other pole of the nth term is at co = — u> n — ik n . 
Taking residues at all the poles we finally obtain 

/ (* < 0) 



.J ^sin(^f)sin(^)sin M (t > 0) 



y»(Z,x$) = 2\ e-*»< ^ (Tn%\ ^ (irnx\ ^ / A (t ^ ft ^ (10.27) 



which is to be compared with Eq. (10.19). This expression can be 
used in Eq. (10.20) to obtain the response of the string, with friction, 
to any transient force. 

The expression for ys gives the free vibrations of a string when 
started with an impulsive blow. It shows that the effect of friction 
is to introduce a damping term e~ knt into each of the component vibra- 
tions. The frequencies of free vibration (co„/27r) do not greatly differ 
from the harmonics for the undamped motion, (nc/2/), if the frictional 
constant k n is small. In a good many cases k n increases as n increases, 
so that the higher harmonics damp out more rapidly than the lower. 
In such a case the sound emitted by the string will be harsh just after 
the start of the motion, owing to the initial intensity of the higher 
harmonics, becoming "smoother" as the motion damps out. 

The amount of energy radiated by a string directly into the air is 
quite small compared with that which can be radiated by a sounding 
board attached to the string supports, as may be determined by com- 



in.il] STRINGS OF VARIABLE DENSITY AND TENSION 107 

paring the intensity of sound radiated by a violin with and without 
backboard. The effect of the transmission of energy from a string to 
a sounding board, via the motion of the end support, will be discussed 
in Sec. 13. 

11. STRINGS OF VARIABLE DENSITY AND TENSION 

The Fourier-series method discussed above is but one special case 
of a method of dealing with vibrating bodies which we shall use in all 
our subsequent work. We first find the shapes of the possible modes 
of simple harmonic motion of the system, the modes that satisfy the 
boundary conditions. In the case discussed above, these shapes were 
given by the functions 



A n sin 



( irnx \ 
I ) 



but for other boundary conditions or for other systems {e.g., for a 
string with nonuniform mass distribution or for a stiff bar) the func- 
tions of x will be different ones, which we can represent by the symbols 
$n(x), A set of such functions, all satisfying the same equation and the 
same boundary conditions, is called a set of characteristic functions, 
and the corresponding allowed values of the frequency are called char- 
acteristic values. 

General Equation of Motion.— The most general equation for the 
string will involve a density e(x) and a tension T(x), both of which 
vary with x. The net force on an element dx of string can be found 
as before 

K2)L-[K2)1 

The equation of motion of the string is finally 

where f(x,t) is the applied force per unit length. 

To study the normal modes of oscillation and to compute the char- 
acteristic functions and values, we consider the cases where f(x,t) = 0. 
For simple harmonic motion we set y(x,t) = Y{x)e~^ ivt . The equation 



dx 



108 THE FLEXIBLE STRING [111.11 

for Y is 

T{x) ^1 + (2tv)Hx)Y = (11.2) 

The solutions of this equation which satisfy the boundary conditions 
at x = and x = I are the characteristic functions \f/ n , and the cor- 
responding allowed values of the frequency are the characteristic 
values v n = (w«/2tt). Examples of these solutions will be given later 
in this section. 

The general free vibration of the system is then given by the series 

y = % A ^(x)e-^ (H.3) 

71 = 1 

where A n = B n + iC n = \A n \e^\ The determination of the con- 
stants B n and C n from the initial conditions is effected by the same 
methods that obtained Eqs. (9.9). The characteristic functions turn 
out to be orthogonal; i.e., 

J^(»)*.(*M«) «*«-&. (*-") (1L4) 

where the constant M n is called the normalization constant, or the 
effective mass, for the nth normal mode. We note that for the uniform 
string of Sec. 9, M n = iZe. 

Therefore if we are to represent the initial shape of the string, y , 

by the series 

00 

yo(x) = X B ^{x) 

n = X 

we obtain the values of B n by multiplying both sides by ^ m (x)e{x) and 
integrating over x from to I. The result is 

B m = Q-) f yo(x)*~(x)e(x) dx (11.5) 

which is analogous to Eqs. (9.9). 

Orthogonality of Characteristic Functions.— The property 
expressed by Eq. (11.4) is a general property of the characteristic 
functions discussed here. This can be shown in the following maimer : 
We multiply the equation for ^ n by \l/ m 



*4( r t) --»*■* 



m.ll] STRINGS OF VARIABLE DENSITY AND TENSION 109 

Next we multiply the equation for i// m by \(/ n and subtract the two 
equations. The left side of the equation is 

Both sides are now integrated over #, giving 

[rw (*. f - *.*£)][ = (Bt - «a JJ *** * 

In all the cases we are to consider, the quantity in brackets on the 
left side of the equation is zero at both ends of the string, either because 
the ^'s or their derivatives are zero, or because T is zero there. There- 
fore the right-hand side of the equation is zero, and therefore the 
integral must be zero unless o»2, = <a\, i.e., unless m = n. This is just 
the orthogonality property given in Eq. (11.4). 

Driven Motion. — If the string is driven by a simple harmonic force 
of frequency (w/2r) which is concentrated at the point x = £, so that 
f(x,t) = F(£)8(x — %)e- iai , we find the steady-state motion by using 
a series of characteristic functions, just as we did in the previous sec- 
tion. We expand the delta function for the force into the series 

F(&S(x - = F(Q 2 (Jfc) <*)Mk)Mx) 

as can be proved by multiplying both sides by f m (x) and integrating 
over x. This series, of course, does not converge, but we are using it 
as a short-cut means of getting y. 

We also set y(t,x;o),t) = ^a n ^ n (x)e- iat and insert both series into 
Eq. (11.1), which gives 

2 M*)(«i - «»)*•(*) = F(0 2 (^) <x)M&Mx) 

Equating coefficients of $ n (x), we obtain an expression for a n , such that 
the series for y is 

^-J M n (« — o>«)(ci> -f «») v ' 

This is to be compared with Eq. (10.12) for the simple case. 

The corresponding transfer admittance (l/F)(dy/dt) is given by 
the series ■ - , 



110 THE FLEXIBLE STRING [111.11 

t j£1 M n (CO — «„)(« + <ti n ) 



n = l 



It is purely imaginary, indicating that thelmpedance (1/F) is purely 
reactive. At low frequencies (w < «i), y is in phase with the force, 
and the impedance is stiffness controlled (see page 33). 

The next stage is to use contour integration to obtain an expression 
for the response of the string to an impulsive force at t = 0, localized 
at x = £, symbolized by the expression S(t)8(x — £). The integral 
for this is obtained from Eq. (6.16) and is 

Each term in the series for Y m has two simple poles, atw = + o>„. 
Adding the residues for each term, we finally obtain the impulsive 
response 

i0 (t < 0) 

IS (-V) M&Ux) sinW) (t > 0) (11.8) 

This series does not always converge, any more than the series of Eq. 
(10.19) did. The final result, however, will converge. 

The desired final result is the shape of the string, y(x,t), when acted 
on by a general force per unit length /(£,£), distributed along the string 
in an arbitrary manner, and varying in an arbitrary manner with 
time 

y(x,t) = J dr J J^ y, (Z,x;t - t)/($,t) d$J 

- % tM {£ . * ™ M - ^ ] [ J>*-«> d "]} (lL9) 

We notice that the motion may involve components with each of the 
natural frequencies (co„/2tt). The relative magnitude of these com- 
ponents depends on the shape of /(£,£)• 

This general series is the formal solution for the motion of the string 
of variable mass and tension when acted upon by a transient force. 
In the special case of the uniform string, it reduces to the series of 
Eq. (10.20). For other cases, it will not be possible to go further 
until we have worked out the specific form of the characteristic func- 



III.11] STRINGS OF VARIABLE DENSITY AND TENSION 111 

tions \p n for the case we happen to be interested in. We shall work out 
two cases in detail to show how these functions" may be computed 
and then will indicate briefly how friction can be dealt with. 

Nonuniform Mass. — Any variation in the distribution of mass 
along the string also disturbs the harmonic relation between the over- 
> tones. As an example of this we shall consider a string of length I, 
stretched between rigid supports, which is heavier at the center than 
at the ends. Taking the point x = at the center of the string, we 
specify that its mass per centimeter be e(x) = e [l — (x 2 /a 2 )], where a 
must be larger than (1/2) . (Why ?) The wave equation then becomes 



dx 2 \tJ\ a 2 ) dt 2 



Waves do not travel with a constant speed along this sort of string, 
nor is the shape of the wave unaltered during its travel. This means 
that the graphical methods discussed in Sec. 8 cannot be used, and we 
must analyze the motion into its simple harmonic modes of vibration. 
If we set y = if/(x)e~ 2 * irt : , the wave equation reduces to 

dfy . 471-VeoA x 2 \ , 

We must find the solution of this equation and then pick out the 
values of v that allow the solution to be zero at x = ±(l/2). Equa- 
tion (11.10) was considered in Prob. 4 of Chap. I. The solution was 
shown to be 

z = 2ttv Veo/Tx 
and where 

D.(k,z) = oos(2) + i s « - ggg*' + ■ • ' 

Bo(M) = sin(*)+±*.-J| 2 T+... 

The function D e is even with respect to z = 0[i.e.,D e (k, —z) = D e (k,z)], 
and Do is odd [i.e., D (k, —z) = —D (k, z)]. 

Since we have picked the origin at the Center of the string, we must 
use either D e or D a as a possible solution to satisfy the boundary condi- 
tions. Any other combination will not have its nodal points placed 
symmetrically about x = 0. The shape of the first mode of vibration, 
which will vibrate with the fundamental frequency, will be given by 



112 THE FLEXIBLE STRING [m.ll 

D e for the value of v that makes the first nodal point of D e fall at x = 
(1/2). Since D e is symmetric about x = 0, it will also have a node at 
x — — (1/2) and will thus satisfy both boundary conditions. The shape 
of the next mode, for the first overtone, will be given by D for the 
value of v that makes the first nodal point of D (the first one, aside 
from the one at x = 0) fall at x = (1/2). This function will also have » 
a node at x = —(1/2). Going back to D e , if we increase v until the 
second nodal point falls at x = (1/2), we obtain the third mode, or the 
second overtone, and so on. 

The Sequence of Characteristic Functions. — This illustrates a 
general property of modes of vibration of strings, no matter what their 
mass distribution or boundary conditions; if we arrange the charac- 
teristic functions giving the shapes of the different modes of vibration 
in a sequence of increasing frequency of vibration, then each function 
has one more nodal point between the points of support than the 
preceding one, the curve for the fundamental having none, that for 
the first overtone having one, that for the second overtone two, and 
so on. 

These functions can be labeled in order: fa, fa, fa ■ • • , a sequence 
alternating between D e and D for successively larger values of v = 
(u/2ir). The corresponding characteristic values or frequencies are 
v\, v 2 . . . , or a>i, o>2 .- • 

The functions are orthogonal, as we have shown on page 108. 
From them we can compute the motion of the string when acted on 
by a driving force, by substituting in Eq. (11.9). 

The general form of the string of variable mass which corresponds 
to the initial conditions that y = y (x) and (dy/dt) = v (x) at t = 
is given by the series 

oo 

V = % fa(x)[B n cos(2irv n t) + C n sm(2Trv n t) 

n = l 

where 

B n = Ij^J J yofae(x) dx 

Cn = ( o 1 \ T ) I v fae(x) dx, and N n = #€ dx 

\27rv„iV w / Jo Jo 

This can be shown by the method given in the first part of Sec. 10, 
now that we have proved that the functions fa are mutually orthogonal. 
The Allowed Frequencies. — The calculation of the allowed fre- 
quencies for the string corresponding to Eq. (11.10) is a somewhat 



m.ll] STRINGS OF VARIABLE DENSITY AND TENSION 113 

tedious task, but the results will be given here. In Table 2 are given 
the values of the ratio between the frequency v n and the fundamental 
frequency of a string of uniform mass e , for different values of l/2a 
and of n. It should be noticed that the frequencies of the overtones 
are not integral multiples of the fundamental, so that the general 
motion of the string is not periodic. 



n= I 



n=2 



n = 3 




n = 4 



Fig. 22.— Normal modes of a string of nonuniform density. Solid lines show the 
shape of the characteristic functions for a string of variable density eo[l - (x/a) 2 ] 
where (l/2a) = 0.8. Dotted lines show the corresponding sinusoidal functions for a 
string of uniform density eo. 

The shapes of the first four modes of vibration are given in Fig. 22 
and are compared with the sinusoidal shape of the uniform string. 
It is seen that the change in shape due to nonuniformity is not great. 
It consists in a pushing out of those parts of the wave where the string 
is heaviest (i.e., near the center) and a consequent pulling of the nodes 
toward the center. 



114 



THE FLEXIBLE STRING 
Table 2. — The String of Nonuniform Density 



[in.li 





0.2 


0.5 


0.8 


1 


1.003 


1.016 


1.042 


2 


2.012 


2.071 


2.181 


3 


3.019 


3.117 


3.299 


4 


4.026 


4.161 


4.411 


5 


5.033 


5.203 


5.521 


6 


6.039 


6.246 


6.630 


7 


7.046 


7.289 


7.738 


8 


8.053 


8.331 


8.846 



Frequencies of the string of variable density e [l — (x 2 /a 2 )], stretched between 
rigid supports a distance I apart. The table gives values of j3 n for different values 
of n and of l/2a. The allowed frequency v n is equal to (1/2Z) -y/T/eo multiplied 
by/3„. 

Vibrations of a Whirling String. — There are cases where gravi- 
tational, or "centrifugal/' forces act directly on the various portions 
of the string, so that the tension varies from point to point along the 
string. In these cases the wave equation for the motion of the string 
takes on a different form. Returning to the discussion following 
Eq. (9.2), we see that when the tension is a function of x the net 
vertical force on an element of string is 

\ dx/ x+ d x \ dx/ x ~ dx\ dx) 



The resulting wave equation is 



d_ 
dx 



( r S) 



d 2 y 
dt 2 



(11.12) 



which reduces to the usual wave equation when T is independent of x. 
As an example of such motion, let us consider the vibrations of a 
string of length I pivoted at one end to a rigid support and whirling 
about this support. We shall neglect gravity, so that the string's 
motion will be in a plane. If we neglect the drag of the air, the position 
of equilibrium will be a straight line rotating in a plane about the 
central support with a uniform angular velocity co„ radians per sec. 
The string can oscillate about this position of equilibrium if it is dis- 
turbed, as long as its total angular momentum remains constant. 
In studying this oscillation we can neglect the uniform motion of the 
equilibrium line and deal only with the string's displacement from this 



IIL11] STRINGS OF VARIABLE DENSITY AND TENSION 115 

line, #. The displacement is a function of t and of x, the distance 
from the support. At first we take y perpendicular to the plane of 
rotation. 

To find the expressions for velocity and acceleration in terms of 
these whirling coordinates, we study the behavior of a vector of vari- 
able length z, rotating with constant angular velocity co„ about its 
end, as shown in Fig. 23. The velocity of the outer end of the vector 
has two components: one along the vector, of magnitude (dz/dt), and 
one perpendicular to it, of magnitude co^. There are two components 
of acceleration parallel to z of magnitudes (d 2 z/dt 2 ) and - w *z and a 
component perpendicular to z of magnitude <a a (dz/dt). 






co a (az/at) 




c) 2 z/dt* 



Accelerations Displacements for the 

Whirling String 
Fig 23.— Velocity and acceleration components for a vector g rotating about one 
SJ. Jit t^ f, , Veloci . t y "«•.,.!» the ca «e of the whirling string, vector x (pointing 
along the line of dynamic equilibrium) rotates but does not change in length; vector y 
the displacement from equilibrium, changes length and rotates. 

In the case of the whirling string, we are to find the acceleration 
of a point that is represented by the sum of two vectors, one of con- 
stant length x and the other, perpendicular to x, of variable length y 
Both are rotating with angular velocity co a . If y is parallel to the axis 
of rotation (perpendicular to the plane of rotation), then the accelera- 
tion of the point is -calx along x (x is constant, so d 2 x/dt 2 = 0) and 
d*y/dt* along y. But if y is in the plane of rotation, the acceleration 
is -<a a x - ca a (dy/dt) along x and (d 2 y/dt 2 ) - a>ly along y. 

Since dy/dt is assumed smaU compared with w a x, in both cases the 
acceleration along x is -««*, which acceleration must be due to the 
force of the tension in the string. The force on an element dx of the 
string at a point x cm from the center will therefore be e dx x<S where 



116 THE FLEXIBLE STRING [HI.11 

e is the density of the string in grams per centimeter (which is supposed 
to be uniform). The tension at the point x will be the sum of the 
forces on all the elements of string beyond the point x out to the outer 
tip of the string 



T(x) = £ eulx dx = (?f\ (P - 



The wave equation for vibrations perpendicular to the plane of rota- 
tion will then be 



@*[<r -*>$-% 



To study the normal modes of vibration of the string, we set y = 
^/{x)e~ 2vivt ' } and to simplify the equation, we change the scale of length, 
making x = h, so that z = is the center, and z = 1 is the outer end 
of the string. The resulting equation for yp is 

We can see without any further analysis that the frequencies of 
vibration will depend oh the angular velocity co but will not depend 
on the string's length or its density (as long as the density is uniform;. 
For increasing the length or density increases the mass, which tends 
to diminish the frequency; it also increases the tension, which tends 
to increase the frequency; and the two tendencies just cancel. For 
vibrations in the plane of rotation (coplanar) the additional term 
— uly comes into the equation of motion, making the equation for $ 
become 

The analysis is the same in both cases, since we are determining the 
value of v in either case. If we determine the frequencies v p and the 
modes of oscillation for the vibrations perpendicular to the plane of 
rotation (transverse) the shapes of the modes will be the same for 
the coplanar vibrations; but the allowed frequencies v a for the coplanar 
case will be related to the transverse ones, v p , by the relation 



-4-fey 



m.ll] STRINGS OF VARIABLE DENSITY AND TENSION 117 

We shall first find the frequencies for the transverse vibrations, per- 
pendicular to the plane of rotation. 

The Allowed Frequencies. — To find the allowed frequencies of 
oscillation, we must obtain the solutions of Eq. (11.13) which are 
zero at z = 0. The easiest method of solving the equation is to assume 
that \f/ is given by a power-series expansion 

\f/ = a\Z + a 2 z 2 + azz z + • • • 

where we omit the constant term a in order to satisfy the boundary 
condition at z = 0. If we set a = (Sir 2 v 2 / col), the equation becomes 



I (1 - z 2 ) ^j = 2a 2 + 2(3a 3 - a x )z + 3(4a 4 - 2a 2 )z 2 + 



dz _ 

= — oaf/ = ■— adiZ — aa 2 z 2 



Equating powers of z, we obtain a 2 = a 4 = a 6 = • • • = 0; a 3 = 

2 — a 12 — a 

— g — ai; a 5 = — on — az ' ' ' • ^ ne solution of Eq. (11.13) which is 

zero at z = is therefore the series 

,-4 + ii«, + p--)ff-.o , 

(2 - q)(12 - q)(30 - «) _ , 1 

" 1 " 5,040 z -t- • • -J 

Since this infinite series satisfies the one boundary condition no matter 
what value a has, it would seem that all values of the frequency would 
be allowed, since v is proportional to the square root of a. On the 
other hand, when we pick some value of a and compute the values of 
\f/ as a function of z, we find that the series becomes infinite at z = 1. 
Since we cannot allow an infinite displacement of the tip of the string, 
this would seem to indicate that no value of the frequency is allowed! 
The dilemma is resolved when we notice that if a equals 2, or 12, or 
30, etc., the series is not an infinite one but breaks off after the first or 
the second or the third, etc., term; in these cases ^ is not infinite at 
2=1. The applying of the two boundary conditions, that ^ = at 
z = and that rp be not infinite at z = 1, rules out all but a discrete 
set of values of a and therefore allows only a discrete set of 
frequencies, corresponding to the set of values a = 2, 12, 30 • • • 
2n(2n — 1) • • • . If we always choose a\ so that if/ = 1 at z = 1, then 
the allowed frequencies of vibration and the corresponding character- 
istic functions are 



118 THE FLEXIBLE STRING [III.ll 

v 2 = 2.4495 f^\ fc(z) = U5z 3 - 3z) 

v z = 3.8730 (g^, * 8 (s) = |(63z 5 - 702 3 + 15s) / (U ' 14) 

v* = 5.2915 ( ( ^\, M*) = inr(429z 7 - 693s 6 



+ 315s 3 - 35s 



*-fe) 



Vw(2n - 1) 



The functions ^ n are equal to the Legendre functions P m (z) [denned 
in (Eq. 27.6)] of odd order (m = 2n - 1). 

The fundamental vibration goes through one cycle every time the 
equilibrium line rotates through 360 deg. The overtone frequencies, 
however, are not integral multiples of the fundamental and therefore 
bear no simple relation to the frequency of uniform rotation of the 
equilibrium line. The forms of the string for the first four normal 



Fig. 24. — Forms of the normal modes of vibration for the whirled string. The 
displacements are from the dotted equilibrium line, which is supposed to be rotating 
in a plane about its left-hand end with a constant angular velocity a> a . 

modes are shown in Fig. 24. It is to be noticed that each function in 
the sequence has one more node than the one preceding, also that these 
nodes tend to concentrate near the outer end of the string, where the 
tension is least. 

The Shape of the String.— It can be verified that these functions 
are mutually orthogonal; that 

( (n ^ m) 

i *•(«)*«(«) dz = { 1 (H.15) 

{ (4n — 1) v ' 



ni.ll] STRINGS OF VARIABLE DENSITY AND TENSION 119 

We can show, by the methods discussed earlier, that if the string is 
started into oscillation by making its initial displacement from equilib- 
rium equal to y (x) and initial velocity (over and above the uniform 
velocity of the equilibrium line) equal to v (x), the subsequent motion 
will be given by the series 



y(x 
where 



A n — 



00 

»0 = ^| #» (j) l A n cos(2jr* n + B n sin {2wvJ)] (11.16) 

n = l 



(4n - 1) 



Since the overtones are not harmonics, this motion will not be periodic 
unless it is started in one of the normal modes, so that all the A's and 
B's are zero except one pair. It should be noticed that the funda- 
mental mode of this transverse motion corresponds simply to a shifting 
of the string's plane of rotation. 

For the case of vibration in the plane of rotation, we mentioned 
earlier that if a were made (8irV/coi[) + 2, instead of (8irV/«2), the 
discussions on pages 117 and 118 would still be valid. This means 
that the shape of the normal modes of vibration in the plane have 
the same shape as the transverse modes, so that Fig. 24 is good for 
both types of vibration. However, the allowed frequencies for the 
two types differ; instead of those given in Eq. (11.14), the modes in 
the plane have the frequencies 



V! .= 0, V2 = 2.2361 



fe) '■ = fe) V(« - D(2n + 1) 



The equations for the ^'s are the same as given in Eq. (11.14). We 
note that the lowest "frequency" is zero, indicating that the funda- 
mental mode is not a vibration but a steady motion, corresponding to 
a change in angular velocity of the string. 

Only the lowest modes for coplanar and transverse motions repre- 
sent a change in the string's total angular momentum, the coplanar 
mode corresponding to a change in magnitude, and the transverse 
mode corresponding to a change of direction. The higher modes 
represent vibrations that do not affect this total angular momentum; 
nor are these higher modes periodic with respect to the rotation of the 
string about its support. 

Driven Motion for the Whirling String.— The general formulas 
worked out at the beginning of this section «an be applied to the prob- 



120 THE FLEXIBLE STRING pi.ll 

lem of the transient or steady-state motion of the string about its 
dynamic equilibrium. We shall see in Sec. 27 that the characteristic 
functions that we have derived for the whirling string are proportional 
to the Legendre functions $ n {z) = P^n-iix/l). Equations (27.7) give 
some of the mathematical properties of the functions. One other 
property is of use here 

P (0) - ( 1V . 1 • 3 • 5 • • • (2n - 1) J2n)^ 

Equation (11.6) indicates that, if a whirling string is subjected to a 
simple harmonic force F(x)e~ i<ot dynes per cm distributed along the 
length of the string, the steady-state displacement of the string from 
its dynamical equilibrium will be 

n = l L 

where 

*•(*) = P ^-i(j\ M n = ^^rY «i = «J(» - D(2n + 1) 

when the force applied is in the plane of rotation and F(x)e^ iat is the 
component perpendicular to the equilibrium line of the whirling string. 

A simple example of the use of this formula is the problem of the 
effect of gravity on a string whirling in a vertical plane. The com- 
ponent of force per unit length perpendicular to the string, when the 
string is at an angle 6 with the horizontal, is eg cos 0, uniform along the 
string (g is the acceleration due to gravity). ■ Since the angular velocity 
of the string is «„, 6 = co a t, so that the effective driving force is the 
real part of ege-^K The formula above can therefore be used, with 
F(£) = eg. The driving force in this case has a frequency co equal to 
the frequency co of rotation of the string; but since w does not equal 
any of the natural frequencies w n of the whirling string, no resonance 
occurs, so that the motion of the string away from equilibrium due to 
the effect of gravity is not very great unless co is small. 

To obtain numerical results we must compute the integral fegif/ n d£. 
Using Eqs. (27.7) and the results given above, 

f *.(*) # = I f P*n-i(z) dz = -^zn l p ^~2(0) - P 2 „(0)] 

_ , _°n*-i Z 1-1-3 • • • (2n-3) 
" ^ L) l 2 • 4 • 6 • • • 2n 

2n(2n-2)l 
~ y L ) l (2»n!) 2 



STRINGS OF VARIABLE DENSITY AND TENSION 



121 



Putting this all together into Eq. (11.17), we finally obtain a series 
for the displacement of the string from dynamical equilibrium y, or 
better the dimensionless quantity y/l, analogous to z = (x/l), 



(j)-m^-^ 1 



1-3 



(2n - 3) 



2-4-6 



• 2n 

4:71—1 



2n 2 - n - 2 



*-©} 




This gives the steady-state effect of gravity on a string whirling 
in a vertical plane. We see that the relative magnitude of the effect 
depends on (g/lafi), the ratio be- 
tween the acceleration of gravity 
and the acceleration of the tip of the 
string in its circular path (as should 
have been expected). 

Figure 25 shows the shape of the 
string at the eight parts of the cycle 
t = 0, (««/8), (« a /4), (3«„/8) • • • 
(7««/8), for the ratio (g/la>l) = (i). 
The dashed lines show the equilib- 
rium positions of the string if grav- 
ity were not acting. We note that 
on the average the displacement is 
out of phase with the force, being 
above the equilibrium lines on both 
sides, ahead of them when it is going 
up, and behind them when it is going 
down. 

Physically, this means that the string goes slower at the top of its 
swing than at the bottom, which corresponds to the well-known rela- 
tion between kinetic and potential energy. Mathematically, this 
comes about because the "driving frequency" (o) a /2ir) is above the 
natural frequency v x = of the first normal node, so that this mode 
(which is the largest term in the summation) is out of phase with the 
force. The natural frequencies of all the other modes are higher than 
(w /27t), so they are in phase with the force. The presence of the higher 
modes accounts for the curved shape of the string. Physically, it is 
due to the fact that there is a greater variation in the speeds of the 
string, between the top and bottom of its swing, for the outer portion 
of the string than for the portion near the axis. 



Fig. 25. — Shapes of the whirling 
string acted on by gravity, at eight 
equally spaced parts of the cycle. Ratio 
(ff/wa 2 Q has been chosen to be 0.20, 
large enough to exaggerate the effect. 



122 THE FLEXIBLE STRING [111.12 

12. PERTURBATION CALCULATIONS 

The motions of some strings, with simple distributions of mass and 
tension, can be computed by the methods discussed in the previous 
section. An exact solution of Eq. (11.2) can sometimes be obtained 
without excessive labor, and the integrations of Eq. (11.9) can then 
be worked out for the general motion. However, in many cases an 
exact solution of Eq. (11.2) is not easy to obtain, so it will be useful 
to work out an alternative, approximation method. This approxima- 
tion technique turns out to be most effective when the variations in 
density and tension do not change greatly along the total length of 
the string, so that the string does not differ much from a uniform 
string. The deviations from uniformity are called perturbations, 
and the technique of calculations is often called the perturbation 
method. 

The Equation of Motion. — As long as we are at it, let us make the 
formulas as general as possible. Suppose that the mass of the string 
per unit length at a distance x from one end is e [l + b(x)], where 6 
is small compared with unity over the whole range of x from zero to 
I. Suppose also that the tension is ^[l + h(x)] and that the trans- 
verse frictional resistance per unit length is V^Vo r (%), where both 
h and r are everywhere small compared with unity. We assume, 
however (until the next section), that the string supports are 
rigid. 

In one of the cases considered in the last section, b was equal to 
-(l/a) 2 [x — (1/2)] 2 and h and g were zero. When a is large com- 
pared with (1/2), b is everywhere small compared with unity. The 
case of the whirling string, also considered in the previous section, 
cannot be dealt with by perturbation methods, since the tension varies 
from zero to a maximum value, and the corresponding h would not 
be small compared with unity. The case of a string hanging vertically 
between two rigid supports would be a workable example, however. 
The tension at the bottom (x = 0) would be T Q , that at the top 
(x = I) would be To + (e lg) (where g is the acceleration of gravity), 
and the tension would vary linearly in between, so that h would equal 
(e g/T )x. If the tension at the bottom, T , is larger than the weight 
of the string, (e lg), then the perturbation method can be used. Many 
other examples, some of practical utility, can be thought of. 

The quantities b, h, and r are the perturbations. If t hey ar e all 
zero, the string is a uniform one, with wave velocity c = y/To/eo, and 
with normal modes and frequencies 



IH.12] PERTURBATION CALCULATIONS 123 



, » . (irnx\ 

yl(x) = sin I — \ 



(12.1) 



If they are not zero, but are small, we can expect that the shape of the 
nth. mode will be yl(x) plus a small correction and that the correspond- 
ing natural frequency will be vl plus a small correction. We wish to 
find how to compute these corrections. 

The equation of motion of the string, with no driving force applied, 
is ' 

*U + «*)] + VT^, r(x) | = T ± J [1 + «,)] I j (12.2) 

To find the correction terms for the free vibrations, we assume that 
the corrected normal mode can be expressed in terms of a series 



^ m (x)e- ia - f = \y° m (x) + 2) a n yl{x)\ er*""* 
c& = (2x^(1 + Vm ) = (x) 2 (5) (1 + Vm) 



(12.3) 



where the coefficients a n and the correction to the frequency, t] m , are 
supposed to be small compared with unity. The summation indi- 
cated by the 2 sign is over all values of n except m (i.e., the symbol 
5* means "not equal to"). 

We insert Eq. (12.3) into Eq. (12.2) and drop out "second-order" 
terms (i.e., bt], ah, etc.) and finally obtain 



First-order Corrections. — The values of the coefficient a n can be 
obtained by the device of multiplying all terms by sin (rsx/l) (where s 
is some integer) and integrating over x from to I. The orthogonal 
properties of this function make a large number of integrals vanish; 
in fact all terms of the summation vanish except that for which n = s. 
The resulting equation is 



124 THE FLEXIBLE STRING [m.12 

nm — (n/m)h nm + i(l/irm)r nm 



a n <^ 



1 — (n/m)' 



where 



b nm = (j\ J o sin^j sin (^r) b W dx 
^ = (?)^in(^)sin(^)Kx)^ 

where we have changed back from s to n after performing the integra- 
tions. 

The value of the correction to the frequency is obtained by multi- 
plying Eq. (12.4) by sin (irmx/l) and integrating 

r\ m ^ —b m m + h mm ~ 1 \~ZZ J Tmm J 

a m = o) m - ik m ; ^ — ( 2) Tmm ( ( 12 - 6 ) 

C0 m CH ( — ; — 1 Vl + Jl mm — bmm J 

Since b mm is proportional to the average value of the mass perturbation, 
averaged over the wth mode, and since h mm is proportional to the simi- 
lar average of the tension perturbation, we can say that if the mass 
perturbation tends to increase the mass the value of b mm will be posi- 
tive and the natural frequency («,»/2w) will be thereby decreased. 
Conversely, if the tension perturbation is positive, the natural fre- 
quency will be correspondingly increased. (This is not surprising.) 
The free vibrations are damped, due to the resistive term r, the expo- 
nential time factor for the wth mode being e-*-*-*— «. 

The coefficients a n representing the difference (to the first approxi- 
mation) between the shape of the perturbed mode of vibration and the 
mode for the unperturbed (uniform) string are proportional to the 
"transfer" integrals b nm , etc., which measure the amount by which 
the perturbations " couple" the various/modes together. If the only 
perturbation is a f rictional resistance r independent of x, then all 
these transfer integrals are zero, the a's are zero, the perturbed modes 
have the same shape as the unperturbed, and the sole effect of the 



m.12] PERTURBATION CALCULATIONS 125 

friction is to introduce damping into the amplitude of free vibration, 
via the factor e~ kmt . Other examples are not as simple. 

Examples of the Method. — We take up again the case of non- 
uniform density discussed in the previous section, in order to show 
how easily (?) the calculations go. The perturbation is 

and the other terms h and r are zero. The necessary integrals are 

bmm = ~ \iw) l 1 ~ v^yj 

b — mnl2 n + (—i) m + n ] 

° nm ir 2 (m 2 - n*)*a* L + { } J 

and the corresponding natural frequencies and shapes of the normal 
modes are, therefore, 

, , N . Ux\ %l 2 . (3irx\ 

Mx) c* sm^J - ^-^ sm^—j 

(5ttx\ 



_ 5Z 2 

34567r 2 a 2Sm| 
, . , . (2tx\ 2l 2 . /Wv V /io^ 

Mx) ^ sm \^)-2^ sm \~r) ) (12 - 7) 

_ SI 2 . (Qttx\ _ 
512x 2 a 2Sin \ I ) 
. , , . (3irx\ . 271 2 . (tx\ 

_ 135Z 2 . (5tx\ _ 

l,024x 2 a 2Sm \ I ) 

Since the perturbation is symmetric about x = (1/2), the series for the 
normal modes contains either all even modes or all odd modes. (Why 
is this?) We note again that the overtones are not strictly harmonic 
in this case. 

Incidentally, these results check fairly well with the values given 
on page 114 and with the curves of Fig. 22. The amount of work 
required for the present calculations is much less than for the exact 
solution. The approximate formulas are no longer satisfactorily 



126 THE FLEXIBLE STRING [111.12 

accurate, however, when a becomes smaller than about (3Z/4), and 
the exact solution is then the only satisfactory one. 

The effect of other types of mass loading can be computed by 
this method. A few examples will be given as problems. 

The other perturbation considered here in detail will be that of 
the string hanging vertically between rigid supports, where the tension 
is (To + € gx), with x measured from the bottom end. The perturba- 
tion here is h(x) = (gx/c 2 ), where c 2 = (T /e ), and the required 
integrals are 

, _ feY h _ _ 2^o m 2 + n 2 _ ( _ 1)m+n] 
n mm - \2ToJ' * 2 T Q (m 2 - n 2 ) 2 L V J J 



so that 



_ 2lge mn(m 2 + n 2 ) , _ ,_.s m+n] 
an ~ t 2 To (m 2 - n 2 Y L V J-; J 



Therefore the natural frequencies and the first two modes are 



n [To L , (lge \ 

(tx\ . (2lge \\l0 . faA , 68 . (4rx\ 

-»(t s )+"-] 



sin 



sin I 



,290 
+ 91261 Sm| 
etc. 

Here the overtones are still harmonic (at least to this order of approxi- 
mation), and the even modes are modified by additional terms of odd 
modes, and vice versa. The result is to move all the nodes toward 
the bottom end, as will be seen if one plots out the i^'s. 

Characteristic Impedances. — It is now time to expand on the 
analogy, touched upon at the end of Sec. 10, which is suggested by 
writing the reaction of each mode of the string as an impedance 
or admittance. In Eq. (10.25) we denned a transfer impedance 
Z m {Z,x)w)ri) for the nth mode, relating the motion at point x due to 
force at point £. Now that we have seen that modes can be coupled 
together by perturbations, we see that there must be a transfer imped- 
ance corresponding to this coupling, which can be labeled Z m (£,x;a>;n,m). 



ni.12] PERTURBATION CALCULATIONS 127 

This symbol may seem overmuch endowed with labels and subscripts, 
but we shall see that its use "saves considerable time and space and 
suggests several useful analogies. (We note here that the subscript 
m denotes "mechanical" and the m in the brackets denotes an integer. 
We shall omit the subscript in this section to reduce complexities.) 
Before we come to the impedances, let us define an average mass, 
resistance, and stiffness of the string, averaged over the nth normal 
mode in a manner analogous to that of Eq. (10.25): 

M{x,Z;n) = [yl(x)yl(k)]- 1 j Q [^)J 2 e [l + b(u)}du 

= (|J (1 + U CSclyJcSclyj 

R(x,!t;n) = faftaOriK*)]- 1 J Q [yl(u)Y VnTo r(u) du 

= (f Z ) r nn csc(^) csc(^) }> (12.9) 

K(x,i-;n) = -te(x)M&\- 1 £ V°M £{w + *(«)] 



iy°M 



du 



(12.10) 



= y^r) (1 + hnn) csc \t) csg vi) 

One can also write a corresponding "transfer" mass, etc., as follows 
M(x,£;n,m) = I , f° ^J I Hu)yl(u)y° m (u) du 

= (*-Ab esc (—} esc (^) 

D/ . /e cA (irnx\ /VraA 

R{x,^;n,m) — I -^ \r nm cscl -y- ) cscl — j- J 

trr y \ /V 2 ranT \ , (irnx\ /VraA 

K{x,^;n,m) = I — ^7 — J h ™ csc l ~T~ ) CSC \~T~ ) 

If this analogous symbolization is to have any meaning at all, 
it should serve to express natural frequencies, etc., by formulas analo- 
gous to those for a simple oscillator. For example, from Eq. (4.1), we 
see that the expression for the damping constant of the simple oscil- 
lator is k = (R/2m). The corresponding expression for the nth mode 
of the perturbed string is, from Eqs. (12.9) and (12.6), 



128 THE FLEXIBLE STRING [111.12 

I" R(x,$;n) 1 /e\r r nn 1 ,M h 

l2M{x,Z;n) J \2j |_(1 + b nn ) J ~ W *** ~ " 

if we neglect all but first-order terms. Similarly, the expression for 
the natural frequency of the simple oscillator is co = (2irv ) = y/K/m. 
In the present case, we have 

\ K (*,M "1* _ (*nc\ f (l + U 1* 






Vl + Ann - K 



again neglecting second-order terms. 

Therefore the analogy is a fruitful one. Each normal mode is 
analogous to a simple oscillator, or analogous to a circuit of inductance 
M(x,£;n), resistance R(x,£;ri), and capacitance l/K(x,l-;ri) in series. 
The characteristic impedance of the nth mode of vibration of the string 
is .. 

Z(x,Swn) = -ia>M(x,Z;n) + R(x&n) + (m K(x,£;n) (12.11) 

This impedance is a function of the frequency (co/2x), not only because 
of the factors (— iw) and (i/<a), but also because the quantities 6, r, 
and h which enter into M, R, and X may be functions of the frequency. 
For instance, if the string radiates sound, then both r(x) and b(x) 
will have radiation components that depend on frequency, as we shall 
see in Chap. VII. 

There is also a characteristic transfer impedance between the nth 
and mth modes 

Z(x,£;o>;n,m) = —i(j}M(x,£;n,m) + R(x,£yn,m) 

+ (Mi£(z,S;n,m) (12.12) 

which suggests that it. is a sort of mutual impedance, coupling the 
nth and mth modes, as indeed it turns out to be. We notice that if 
the string is truly uniform (6 = h = 0, r independent of x) then all 
the transfer impedances are zero and there is no coupling between 
modes, which is as it should be. 

Finally, the expression for the perturbed normal mode has a par- 
ticularly simple form when written in terms of these impedances. For 
instance, the transfer impedance between m and n at the frequency 
<a m /2T of the mth mode [see (Eq. 12.6)] is, to the first order, 



m.12] PERTURBATION CALCULATIONS 129 

Z(x,£;a) m ;m,ri) m\—i ( ~yj [%) & ™ + \1T/ Tnm 

Similarly, the impedance for the nth mode at the frequency of the 
rath mode is, to the zeroth order, 

A comparison of these formulas with Eq. (12.5) shows that an 
alternative fashion of writing the series for the perturbed normal mode 
is 

This is interesting in that it shows that the amount by which the nth 
mode modifies the mth mode is proportional to the transfer impedance 
Z(x,£;o) m ;m,ri), which is in turn dependent on the perturbation integrals 

Omnj l^mn) and. T mn . 

Forced Oscillation. — The full usefulness of the characteristic 
impedances defined in Eqs. (12.11) and (12.12) becomes apparent when 
we take up the forced motion of the string. Suppose that we apply 
a concentrated force at the point x — £. The equation of motion is 
then 

*(i + 6) g* + V^To r I - T 1 [(1 + h) I] = F{&B(x - 0e- 

(12.14) 
Setting 

into this equation, multiplying it by sin (irmx/l), integrating over x 
from zero to I, and finally dividing by sin (irm^/l), we obtain the funda- 
mental equations for the forced motion of the string 



130 THE FLEXIBLE STRING [m.12 

U m Z(x,Z;a>;m) + X U n Z(x,£wn,m) = F(Q - (12.15) 



where 

U m = —io)A m sin 



( Tmx\ 



The quantity A n sin (irnx /I) is that component of the amplitude of 
motion of the point x on the string which corresponds to the nth 
mode, for a driving force F(£) applied at the point £. Similarly, the 
quantities U n are the corresponding modal components of velocities. 
The electrical analogue is therefore a network of infinitely many paths 
in parallel, the self-impedance of the rath path being Z(x,£;o);m), with 
each path coupled to every other by mutual impedances Z(x,£;o);m,n). 
The analogue to the applied force is a voltage F(g) applied across the 
network ; and the U n 's correspond to the currents in the parallel paths. 
The frequencies of free vibration correspond to the allowed frequencies 
of vibration of the network when the generator is short-circuited. 

In order to calculate the values of the A' a or U's by successive 
approximations, we first neglect the transfer impedances Z(x,£;a);m,ri) 
which are supposed to be considerably smaller than the impedances 
Z(x,%;o};m). (Why?) Therefore, to the "zeroth*' approximation, 

A G in 6^— F ^ 

This can then be inserted back in the series in Eq. (12.15) and the 
equation solved again to get the first approximation 

a • f Tmx \ ~ to |\ - "5! ?&jwmo1 

AmSin \ l ) ~~ -iuZixfr^m) l l £d Zfofcwjn) J 

— io)Z(x,£',ai;m) sin(xraa;/7) 

if we utilize (and generalize) the definition of Eq. (12.13). Second and 
higher approximations could be obtained by repeating this process; 
but we shall let well enough alone. 
Therefore the expression 

y(x,^,t) c* 2i -i.Z(x^;m) [' ~ Jj ~Z^^rJ (12 ' 16) 

is an expression for the steady-state shape of the string when driven 
by the force F(£)5(£ — x)e~ ia \ which is correct to the first order in the 
small quantities [Z(x,%;a>;m,n)/Z(x,£;o>;ri)]. 



111.12] 



PERTURBATION CALCULATIONS 



131 



When the string is uniform (6 = h = 0, r independent of x), then 
the quantities Z(x,%;<ti)m,ri) are all zero, and the expression becomes 
an exact one. 





Fig. 26. — Sequences showing the successive shapes of a damped string, driven by 
a force of frequency v, during a half-cycle of the steady-state motion. The force is 
concentrated at the point marked by the circle, and the arrows show its successive 
magnitudes and directions. The three sequences are for the driving frequency v, equal 
to f , two and three times the fundamental frequency of the corresponding undamped 
string. 

Uniform String, Steady State. 

y(x,£ya,t) = > }V m 

<** —uaZ(x,S;u;n) 

Z(x,$;a,;n) = [ -ico (^) + (^f) } (12.17) 

+ ©(^)M~M?) 



132 THE FLEXIBLE STRING [111.12 

which is identical with Eq. (10.25). Figure 26 shows the shapes of 
such a string when driven by a force applied at £ = (1/3). The center 
sequence shows a case of resonance, with large amplitude (not infinite 
because the resistance is not zero). The left-hand sequence shows a 
case that should be another resonance, were it not for the fact that the 
force is applied at a node of the corresponding free vibration (in which 
case sin (rn^/l) is zero and the corresponding Z is infinite) so only 
nonresonant modes appear. 

As noted in the discussion of Eq. (10.25), the analogy of the input 
admittance to that of a parallel network of impedances Z(£,£;co;n) is 
obvious. In the present case when the perturbations are not uniform, 
the individual paths are coupled by the impedances Z(£,£;co;ra,?i.) = 
Z(£,£;co;w,ra). 

Transient Motion. — The calculation of the transient motion of the 
perturbed string follows the same procedure as has been outlined in 
Sec. 11. We set up the steady-state response of the string to a con- 
centrated force, which has been done in Eq. (12.16). We then express 
the Z's in terms of co, b nm , r nm , etc., and perform the contour integra- 
tions necessary to obtain the response to a unit impulse at £ 

y*M;t) = ^ J_ m [f^] vOM;»,0 ^ ( 12 - 18 ) 

The details of computing the residues about all the poles of this 
function are quite wearisome but involve nothing particularly different 
from the cases discussed before. Each of the first terms in the series 
for y(x,£;o),t) has two poles at co = ±co TO — ik m , with values that have 
been given in Eq. (12.6). The second terms, which are summed over 
both m and n, have four poles: two at co = ±co m — ik m , and two at 
co = +co„ — ik n . When the residues have been computed and the 
results expressed again in terms of the impedances, we have the 
formula 

t < 0, y»(x,$;0 = 

t > 0, yb(x,%;t) ~ real part of 

Op—kmt—iumt 



{'2 



^m(03m,^m(0} m ,X) 



(xmeoc) Vl + h nn + b r , 

{ — - o — kmt — iifmt 

^Mix^nOKix^m) (12 " 19) 

TO 

f ^ Z(x£',o) m ;m,n) _ ^ Z(x,%;o) m ;n,m) T\ 
L Si Z M'>">»ri Si Z(x^ m ;n) J/ 



in.13] EFFECT OF MOTION OF THE END SUPPORTS 133 

which is correct to the first order of the small quantities b, h, and r 
(but is not correct to the second order). Here we refer to Eq. (12.13) 
for the formula for the perturbed characteristic functions, to Eqs. 
(12.9) through (12.12) for definitions of the impedances, and we again 
note that 



k m c^ 



[ R(x,Z;m) 1 _ [ K(x,!-;m) ~\ 

l2M(x,^m)]' " m -lM(x,Z;m)] 



Thus we have arrived again, by the route of the operational calculus, 
to the result that ^ ro (w m ,x), as defined in Eq. (12.13), gives the shape 
of the wth mode of free vibration of the perturbed string; and that 
k m and w m , as defined in Eq. (12.6), are the damping constant and fre- 
quency of this vibration; all within the range of validity of the first- 
order approximation considered here. The formula of Eq. (12.19) 
corresponds to that of Eq. (11.8). The integral for the general motion 
due to a general force similarly corresponds to that of Eq. (11.9). 

13. EFFECT OF MOTION OF THE END SUPPORTS 

At the end of Sec. 10 we mentioned that strings are poor radiators 
of energy by themselves. In fact the most effective method of radi- 
ating the energy of vibration of a string is to attach its supports to a 
sounding board which will do the radiating. Before we can get very 
far with a study of the effect of the sounding board, we must discuss 
the effect on the motion of the string of the motion of the end supports. 
Such an investigation will be doubly useful, for we shall find that the 
methods developed here will also be necessary to the study of sound 
waves in tubes and in rooms. We shall consider here only strings of 
uniform density and tension: further complications of nonuniformity 
can be treated, if need be, by the methods of the previous section. 

Impedance of the Support. — Presumably the supports are capable 
of bearing the string tension in the direction of the equilibrium line 
of the string. Any force on the support at right angles to this line, 
however, will cause some sidewise displacement of the support, small 
or large depending on the transverse mechanical impedance of the sup- 
port. This impedance, the ratio between the sidewise force on the 
support and the transverse velocity ( — iu times the transverse displace- 
ment) will be called Z for the support at x = and Zi for that at 
x = I. The real part of Zq is a resistance R , and the imaginary part 
is a reactance X (Z = Ro — iXo = \Z Q \e- iv o) ) the phase angle being 
<po = tan -1 (X /JSo). 



134 THE FLEXIBLE STRING [111.13 

When the string is in equilibrium, the supports are in equilibrium, 
and their sideward displacements y(0) and y(l) are zero. When the 
string is displaced, however, there is a transverse component of the 
tension T, which will tend to pull the supports sideward, as shown in 
Fig. 20 (on page 91). This transverse force on the support at x = 
is T y = T sin 0, and when is small (as it usually is) it is almost 
equal to T tan 0, which' is equal to T times the slope of the string, 
(dy/dx), at x = 0. The new boundary condition is therefore that 
T^dy/dx), at x = 0, is equal to the transverse impedance of the sup- 
port, Z , times the string velocity {—io>y), at x = 0. At x = I the 
force is minus T times the slope at x = I. The two conditions may 
therefore be summarized as follows, for strings oscillating with simple 
harmonic motion of frequency (co/2tt) : 

.,«*-«> "-ty®! (I8J) 

<*-» »- + (£)(£)) 

We see that when the supports have infinite impedance the boundary 
conditions reduce to the usual ones requiring y to be zero at x = and I. 

Reflection of Waves. — A wave coming from the right and meeting 
the support at x = would not undergo perfect reflection with change 
of sign. Therefore we must first reexamine the reflection of simple 
harmonic waves under these boundary conditions, in order to see 
what the mathematical expression should be for the shape of the stand- 
ing wave. We consider first the support at x = 0, The incident 
wave, coming from the right and traveling in the negative x-direction, 
is Ate-W") (x+ct) , where c 2 ={T/e) and a = 2tv. The reflected 
wave must also be simple harmonic, but its amplitude A r will differ 
in magnitude and phase from Ai. To emphase this we can set A r = 
A ie -2Ta +2Tip 0} w here the value of <x determines the reduction in ampli- 
tude of the reflected wave and the value of jS determines the phase 
change on reflection. 

The total displacement of the string is therefore 

y — (Aie-^ ivx/c) + A r e +(2Tivx/c) )e- 2 * i ' ,t cm 2) 

where X = (c/v) = (2tc/o)) is the wavelength of the incident (and 
reflected) wave. The values of «o and /3 must be adjusted so that 
the boundary condition at x = is satisfied. The slope of the string 
is 



m.i3] 



EFFECT OF MOTION OF THE END SUPPORTS 135 

(— ) = — [— ^^(e-CaTisA) — ^-2*(« -i/j -fa/X)) e -&»* 

The boundary condition of Eq. (13.1) for x = therefore requires that 

* - a - * - (t) 



-5 



e -2T(a -i^ ) = 



+ <z 
i -r 
i + r 



5 j-0 j-iK>d |io9y = e 
I- Z- 2- tr- 



-4 



(13.3) 




-3-2-1 1 1 

a = Real Part of q. 

Fig. 27. — Conformal transformation of the q = o — ib plane onto the f = — i\ plane, 
and vice versa (when the drawing is turned upside down). 

These equations display a very interesting reciprocal relationship 
between q, the ratio between the amplitudes of the reflected and inci- 
dent wave, and f , the ratio between the transverse impedance of the 
support causing the reflection and the wave impedance of the string 
defined in Eq. (10.3). Relationships between complex quantities 



136 THE FLEXIBLE STRING [111.13 

expressable by equations relating them can be represented graphically 
by what are called conformal transformations, from which approximate 
values of the relationships can be read off. For instance, in this case 
the line a = — i is a straight line on the g-plane, parallel to the 6-axis, 
one-half unit to the left of the origin; but on the f -plane it is represented 
by a circle of radius two units, with center at f = 2, as shown in Fig. 
27. For the particular transformation symbolized by Eq. (13.3), 
circles on the f -plane go into circles on the <?-plane, and vice versa. 
A straight line, being a limiting form of a circle, also goes into a circle. 

The coordinate lines of the g-plane are drawn on the f-plane in 
Fig. 27 and constitute a representation of the transformation. From 
it we can read off the values of and x corresponding to values of a 
and 6. For instance, corresponding to q = — 0.5 + 0.'5iis{" = 1 — 2i, 
and corresponding to q = — i is f = i, and so on. Since the reverse 
transformation (f to q) has the same form, Fig. 27 need only be turned 
upside down to have the reciprocal representation. 

Several interesting things are apparent on studying Fig. 27. In 
the first place, as long as is positive (and it must be in any actual 
case, for the real part of an impedance must be positive) the magnitude 
of q will be less than unity. This means that in any actual case the 
reflected wave will never be larger than the incident wave (which 
seems sensible). In the second place, when f = 1 (Z = ec) there is 
no reflected wave at all (q = 0); when f = °o, q = —1; and when 
f = o, q = 1. When |f | = 1, the real part of q is zero, so that the 
phase change on reflection is ± 90 deg. However, when the real part 
of f is zero, \q\ = 1, i.e. , the reflected wave is as large as the incident one. 

Hyperbolic Functions. — The relations displayed by Eq. (13.3) are 
interesting enough, but a modified form of the transformation will 
turn out to be much more useful in further calculations. The equa- 
tions will utilize hyperbolic functions, so that it will be well to review 
their properties. The fundamental definitions are parallel to those 
of Eq. (2.7), which defined the trigonometric functions in terms of the 
exponential function, 



cosh (x) = \{e x + e~ x ) ; e x = cosh (x) + sinh (x) 
sinh (x) = \{e x — e~ x ) ; e-* = cosh (x) — sinh (x) 

-j- cosh (x) = sinh (x) ; ' -r- sinh (x) = cosh (x) 



(13.4) 



The relationship between these functions and the trigonometric 
functions are given by the equations 



in.13] 



EFFECT OF MOTION OF THE END SUPPORTS 



137 



cosh (x + iy) = cosh (x) cos (y) + i sinh (x) sin (y) 

sinh (x + ty) = sinh (x) cos (?/) + i cosh (x) sin (y) 

cos ix + tV) = cos (x) cosh (y) — i sin (x) sinh (?/) 

sin ( x + *2/) = sin (x) cosh (?/) + i cos (z) sinh (?/) 



(13.5) 



The amplitude and phase angle of cosh (a; + iy) and of sinh(x + iy) is 
given in Plate I at the end of the book. 

We also define a hyperbolic tangent and cotangent in an analogous 



manner 



tanh (x) = 



sinh (x) _ 1 — e~ 
cosh (x) 1 + e~ 



coth (x) 



tanh (x) 



(13.6) 



Tables of the hyperbolic sine and cosine are given on page 438. 
More interesting to us at present are the values of the hyperbolic 
functions of complex quantities. The values can be obtained by using 
Eqs. (13.5), which result in the useful relations given in Eqs. (13.7). 

cosh[ir(o! .— iff)] = |cosh[7r(a — #)]|er*o 
|cosh[x(a — i0)]\ = \/cosh 2 (7rQ:) — sin 2 (xj8); 

tan £2 = tanh(ira) tan(x)3) 



[*« -if- 0] 



cosh[ir(a — ifi)] = i sinh | ir(a — ifi 

= — cosh[7r(a — i(3 — i)] 
tanh[ir(a - i&)\ = 6 - i X = f = |f \<T* 
sinh(27ra) _ _ sin(2x/3) 



e = 



x = 



cosh (2ira) + cos (2*0) ' A cosh (2™) + cos (2irj8) 
2 + [x + cot(2x/3)] 2 = csc 2 (2tt|8) 
[6 - coth(27m)] 2 + x 2 = csch 2 (2W) 
tanh [ir(a — ip)] = tanh [ir(a — i@ + in)] 

1 



(13.7) 



tanh [ir(a — ifi + in + i*')] 

(n = 0, ±1, ±2 



) 






cosh (2x0:) — cos(27T|8) _ _ T [* sin (2tt/8) T 
cosh (2™) + cos (2x0) '^ ~ tan [sinn ( 27ra ) J 
(1 + , )2 + x2]= ^ tanh _ i 



4x L(l - 0) 2 + X 2 . 



l_sinh (2xa) 

r 2 " i. 



>-£*""' [l^l?]/ 



From these equations we see that the line on the f = 6 — ix plane 
which corresponds to ft = constant is a circle of radius esc (2x/3), with 



138 THE FLEXIBLE STRING [111.13 

center at f = i cot (2tt/3) ; and that the line corresponding to a-con- 
stant is also a circle, of radius csch(2xa) and with center at f = 
coth (2ra). This transformation is plotted in Plates I and II on pages 
453 and 454. Plate III gives the magnitude and phase angle of the 
hyperbolic sine and cosine. The transformation for the hyperbolic 
tangent, shown in Plate I, is sometimes called the bipolar transforma- 
tion. - Tables III and IV, on pages 440 and 443, also give values for 
the transformation, if greater accuracy is needed than can be obtained 
from the plates. 

( Both tables and plates are given only for values of /3 between 
and 1, and only for positive values of a and 0. Only positive values 
of and a are included, because for systems encountered in "real 
life " the resistive term is never negative. The range of fi is restricted 
because the whole transformation is a periodic function of 0, and 
values for other ranges of /? can be obtained from the ones given. 
Both and x can be considered as functions of a and /5; 0(a,/3), x(a,(i). 
Values of and x for /3 increased (or decreased) by a whole integer are 
equal to the original values: 

0(a,/3 + n) = 0(«,/3); X (<*,P + n) = xK/3); n = 0,±l,+2 • • • 

We can also see from Eqs. (13.7) that changing from fi to n — 
changes the sign of x but leaves unchanged : 

6(a,n - 0) = 0(a,P); x(<*,n - 0) = -x(«,/3) 

Therefore we see that 6(a,0. 2) = 0(a,1.2) = 0(a,27.2) = 0(a,-6.8) 
= 0(a,6.8) = 0(a,O.8) = 0(a,-3.2), etc.; and x(«,0.2) = x(M-2) = 
x(«,27.2) .'= x(«,-6.8) = -x(«,6.8) = -x(«,0.8) = - x ( a ,-3.2), etc. 

String Driven from One End. — We must now go back to our physi- 
cal problem to try to show why it is necessary to bring in such a lot 
of new mathematical machinery. Suppose that the string is fastened, 
at x = t, to a support having transverse mechanical impedance 
Zi = ec£i and that a sideward force is impressed on_ the end x = 0. 
We wish to compute the transverse impedance of the string at x = 
and its shape and motion when acted on by the force. The impedance 
of the support at x = Z will modify the reflected wave returning from 
that end of the string and will, therefore, modify the whole reaction 
of the string by an amount that we must now compute. 

If the driving force is simple harmonic, with frequency (co/2x), 
the motion of the string must be some combination of the two exponen- 
tials e {iax/6) and e~ (iax/c) , multiplied by the time term e~ iwt . Examination 
of Eq; (13.4) shows that this combination can be expressed as 



111.13] EFFECT OF MOTION OF THE END SUPPORTS 139 

y(x,t) = A cosher a + j(y) - ifa fe-™ 1 

where X = (2irc/co) is the wavelength of waves on the string. The 
constants a and j8 are to be determined by the value of the impedance 
of the support at x = I, by use of Eq. (13.1) 

A cosh[7r(aj — ifa)] = A ( - — =- ) sinh[7r(aj — ifa)] 



\ia)cZiJ 



or 



-(f) 



tanh[x(ai - ifa)] = ( — ) = ft = 61 - ix t (13.8) 

where the constants for the end x = I are related to those for x — 
by the equations 



m = a ; 



-(f> *-* + (?) 



Therefore the calculation of the wave impedance of the string involves 
a knowledge of the hyperbolic tangent of complex quantities and is 
the excuse for the previous several pages of formulas. 
The wave impedance of the string at x = is 

Z(0,0;o>) = (J-)(^) = ectanh[7r(«o - *j8 )] 

= ec tanh<7raj — iir fa + ( y ) ( 

To go from the impedance of the support at x = I to the impedance of 
string plus support, we find the values of a and P, on Plate I, corre- 
sponding to an equal value of a but a value of j3 increased by the 
length of the string in half wavelengths, (21/X). We follow the circle 
corresponding to the fixed value of a, from fa to fa -f- (2Z/X), and the 
result is the transverse impedance of the string in units of (cc). 

Shape of the String. — If the transverse driving force at x = is 
Foe"^ 1 , the shape of the driven string is 

^o cosh {ir[a + i(2x/\) - ifa ]} e _ iat 
— t'coec sinh[7r(a — iPo)] 

which corresponds to Eqs. (10.8) for the case of the rigid support. 
By using some of Eqs. (13.7) we can express this in terms of amplitude 
and phase angle: 



140 



THE FLEXIBLE STRING 



[111.13 





c 






& m 








ft .2 


> 

i 






3 g 


<3" 






-dTS 


.C 






H ft 


"« 






§ a 


> 








IS 






5& 














£ ft 








O a] 








^2 


o 


E 

3 


3 


+3 © 


Ui 


E 




+3 






03 ® 


- c - 
o 


c 


E 


gS 


a. 

v .• 


2 


< 


o* o 






^_ Ets 


o ^ 


-O 3 3 




u>E.^ 


ft 03 




C x Q. 


o |E 


-o d 


a.2<£ 


d ® 




v 3 









111.13] EFFECT OF MOTION OF THE END SUPPORTS 141 

= El / cosh 2 Qr«o) ~ sin 2 (fcc - TftO V 
y uec \ cosh 2 (xa ) — cos 2 (71-ft) ) 

exp <t tan _1 [tanh(xo:o) tan (Ave — irfio)] 

— t tan -1 [tanh (ira ) cot (fl-ft)] — iwf + ( - )> 

where k = (2tt/X) = (w/c) and ft = ft + (2Z/X). Figure 28 shows 
a time sequence of such wave motion for several different values of 
ai and ft. The one for ai = is for no energy loss to the support at 
x = I and is a true standing wave, with true nodal points spaced a 
half wavelength apart. When ai is not zero, energy is lost to the 
support, the reflected wave is not so large as the incident wave, and 
at no point on the string is the amplitude exactly zero. 

The amplitude of motion of the string, as a function of x, does 
show maxima and minima, however, spaced a half wavelength apart. 
The dependence of this amplitude on x is through the factor 

-\/cosh 2 (7rao) — sin 2 (fcc — xft) 

= VcoshVaj) - sin 2 [(27r/X)(a; - I) - rft]. 

The maxima come when (2ir/\)(x — I) = 7rft + irn, (n = 0, ±1,±2 
• • • ), and the maximum nearest the support at x — I is a distance 
d = (I — x) = (X/2)(l — ft) from this support (we assume that the 
value of ft lies between and 1, as it can always be made to do). The 
minima come when (2ir/X)(x — I) = irfii + ir(n + i), and the mini- 
mum nearest the support comes at d = (I — x) = (X/2)(£ — ft) if 
< ft < £, or (X/2)(f - ft) if 1 > ft > i. Therefore the value of 
ft for the support at x = I can be determined by measuring what part 
of a half wavelength lies between the support and the nearest point of 
maximum amplitude and subtracting this fraction from 1. 

At the maxima, the factor in the amplitude depending on x has 
the value cosh(xaz) ; at the minima this factor has the value sinh(iraj). 
Consequently, the ratio between the minimum amplitude of motion 
of the string and the maximum is tanh(7raj), from which we can deter- 
mine ai. Therefore we can measure the transverse impedance of the 
support at x = I by observing the driven motion of the string attached 
to it. The ratio of minimum to maximum amplitude gives ai, the 
distance of the maximum from the support gives ft, and the impedance 
Zi can be obtained from Eq. (13.8) and the plates or tables at the back 
of the book. 



142 THE FLEXIBLE STRING [111.13 

This same sort of analysis can be made for sound waves in a tube 
closed at x = I by a diaphragm having mechanical impedance Z h as 
will be shown in Chap. VI. 

Standing Wave Ratio and Position of Minima. — An alternative 
technique is to use Eqs. (13.3) and Fig. 27 to correlate measurements 
of standing waves with the impedance of the driven support (the 
load) or the impedance that the string exhibits to a transverse driving 
force (the generator). The magnitude of q, \q\, the ratio between 
the amplitudes of reflected and incident waves, is called the standing 
wave ratio. At the points where the two waves are in phase, the 
amplitude of motion Fmax is largest and is proportional to (1 + \q\); 
at the points where the waves are out of phase, the amplitude 7^ 
is least and is proportional to (1 — |g|). Consequently, a measure of 
the ratios between minimum and maximum amplitudes serves to 
determine the standing wave ratio, or vice versa 

P" = TT^ ! I?! - y- I y* = «-" (13.9) 

-t max -L I j C^ | -l max r -1 min 

The phase angle 2ir/3 of q = \q\e 2Ti ? gives the phase shift on reflec- 
tion. If it is zero, the amplitude at that point is maximum, and the 
point of minimum amplitude is i wavelength away; if /? is ■£- (phase 
angle 180 deg) the amplitude is there a minimum. Consequently, 
the number of half -wavelengths measured from the load to the nearest 
minimum is equal to i - p lo&d ; or the number of half wavelengths 
measured from the generator to the nearest minimum is equal to 
i + iSgen (which is a restatement of the discussion on page 141). 

This suggests using the chart of Plate IV, which is the portion of 
Fig. 27 inside the circle = 0. The contours for the resistance and 
reactance and % are plotted. The standing wave ratio \q\ is then 
simply the radial distance from the center (0 = 1, % = 0) to the point 
(0,x). The value of /3 is the angle of this radius vector. 

If the value of the load impedance is known, the corresponding 
point can be found on the chart and the standing wave ratio deter- 
mined by measuring the distance from the center to this point. The 
impedance of the string for the generator can be found by traveling 
along a circle with center (0 = 1, x = 0), starting from the point 
(0ioad, Xioad), and traversing the circle through an arc in degrees equal 
to 360 deg times the number of half wavelengths from load to gener- 
ator. We note that we must traverse the circle in a clockwise direc- 
tion in this case (increasing /3) . 



111.13] EFFECT OF MOTION OF THE END SUPPORTS 143 

If the value of the string impedance for the generator is known, 
we find the load impedance by going from the point (0 gen , Xsen) along 
a circle in the counterclockwise direction (decreasing /3) through an 
arc equal to 360 deg times the number of half wavelengths from load 
to generator. 

If the ratio (Fmm/Fmax) is known and the position of one mini- 
mum (or maximum) is known, either the impedance of the load 
or the string driving impedance can be computed by drawing a circle 
of center (6 = 1, x = 0) and radius equal to \q\ = [1 — (Y^/Ym^)]/ 
[1 + Yrwn/Yma*)]. Going around this circle, from the point /? = i, in 
the counterclockwise direction through an arc equal to 360 deg times 
the number of half wavelengths from the minimum to the load, ends 
at a point corresponding to the load impedance. Going around the 
circle clockwise through an arc equal to 360 deg X (number of half 
wavelengths between the minimum and the generator) reaches the 
point corresponding to the driving-point impedance of the string. 

Both types of charts (for a, j3 and for \q\, /3) are given at the back 
of the book. One is more useful for some calculations, the other 
more useful for others. It is well to get used to both. 

Characteristic Functions. — To discuss the driven motion of the 
string further we shall have to obtain the characteristic functions that 
satisfy the boundary conditions at a given frequency. To make 
the problem as simple as possible initially, without leaving out the 
essential parts, we shall assume that the transverse impedance of the 
supports at x = and x = I are large compared witTi ec, the wave 
impedance of the string. In this case approximate methods can be 
applied, and the technique is not too different from that employed 
in Sec. 12. We consider that the string itself is uniform, with uniform 
tension and zero distributed resistance. (These additional complica- 
tions can be brought in later; at present the only perturbations con- 
sidered are those due to the motion of the supports.) 

We assume that the nth characteristic function is 



^n(x,ca) = sin f — ) (x — a n ) 



(n = 1,2,3 • • • ) 



where w n and a n are to be determined by the boundary conditions. 
Setting this in Eqs. (13.1) and assuming that (T/uZ) is small enough 
to neglect quantities higher than the first order, we finally arrive at 
the approximate formulas 



144 THE FLEXIBLE STRING [111.13 

a "^{ ! f)[ 1 + (dk) + (sass)] 

where T is the tension (T = ec 2 ). 

Therefore the characteristic function is 

\l/n(x,w) ~ sin< (ira) 1 - ( -^J (<r + tJto) - ( ^J (<ri + «i) 

[(f) + (a) <•* + *•>]} 

s^ ( — 7- J (o- + Wo) (at a; = 0) 

~ ( ~~ 1)n ~ l fe) (<Tl + ^ (at X = l) 

where (ec/Z) = (1/r) = k - *V; k = ed?/(fl 2 + x 2 ) = 6/(6* + x 2 ); 

<r = -ecX/(R* + x 2 ) = -x/(6 2 + x 2 ) 

The quantity (Z/ec) is the specific impedance of the end support, the 
impedance in units of the wave impedance of the string. The recip- 
rocal of this is the specific admittance of the support, here taken to be 
a small quantity. The real part of this, k, can be called the specific 
conductance and the imaginary part a, the specific susceptance of the 
support. The* limiting values of the function yf/ n correspond to the 
motions of the end supports produced by the motion of the string. 
The characteristic function ip n is a complex quantity and is a function 
of the driving frequency co, both explicitly and also through the depend- 
ence of the admittances k and a on co. 

Following the procedure leading to Eq. (11.6), we see that the 
shape of the string, when acted on by a simple harmonic driving force 
concentrated at x = £, is 



^-J M n (co — w n 



F(j) M MMx t u) ^ 



)(« + 0) n ) 



where 



Mn ~ \2 ) l 1 \io>lZ ) [iuizj] — \2coJ 



Transient Response. — The calculation of the response of the string 
due to an impulsive force at t = involves the calculation of the resi- 



m.13] EFFECT OF MOTION OF THE END SUPPORTS 145 

dues of y at its poles in the w-plane. This is a little tedious, since co„ 
depends on w; but it can be carried out to the first order in the small 
quantities k and <r. The most difficult point comes when we set the 
approximate values w = + (rnc/l) into u n to calculate the first order 
correction to the position of the poles. It is a general property of 
impedances to change the sign of their imaginary parts (but not of 
the real parts) when the real part of co changes sign. Consequently, 
if 

. «„(«) = y hrn + \^~1 ) ( KOn + Kln ~ t(r0n — *°"* n ) 

the two roots for « are, to the first order, 



c , . V 

J {irn — 0-Qn — (Tin — 1*0n — Win) 



t) 

(—Tnc\ c , . . .v 

— 7 — J ^ j {—irn + a 0n + <n n — *«0n — lain) 

where we understand that the admittances in the nth term, <r» and 
R n , are computed for the frequency (irnc/l). This behavior must also 
be taken into account in evaluating $ n at the poles. The final result 
is 

!0 (t < 0) 

Jj\imc) (13.10) 

n = 1 N ' 

[ypn{%, - )^n(x, — )e i( - e/l) c™-«>»-«»)* 
-^(^,+)^„(a:,+)e- i ^^-^-«»^] (t > 0) 
where 

&»(z,+) = sin<[7rn — (<r 0n + zVc 0n ) — («ri* + im*)] 

[(i) + (£) ('-+ *->]} 

and 

^ n (a?,-) C~ Sinf ~) + COsf ^y- J ( 1 — J J ((Ton — tKOn) 



— J ((Tin — IK-In) 



I 



146 THE FLEXIBLE STRING [111.13 

The series in (13.10) is correct only to the first order and, there- 
fore, does not give exactly the correct answer near t = 0; but the gen- 
eral behavior is a close enough approximation to see what is happening. 
A string supported between nonrigid supports, started into motion 
by any arbitrary force, vibrates with a complex of damped, harmonic 
motions. The damping constant for the nth mode is (c/l) times the 
sum of the specific mechanical conductances (/c 0w + ni n ) for transverse 
motion of the two supports at the nth natural frequency. If the 
supports are rigid or if their transverse impedances are purely reactive, 
then these conductances will be zero and the string will not be damped 
owing to support motion; though it may be damped owing to the 
reaction of the air (which we have treated in Sec. 12, but are neglecting 
here). But if the supports are attached to sounding boards that 
radiate the energy of vibration, then the conductances will not be 
zero, and the motion will be damped. We note that, because of the 
factor (c/l), the longer the string is, the less will it be damped. It is 
easy to see why this is so : for the same support conductance, a longer 
string will feed a smaller fraction of its total energy to the supports in 
any second. 

We note also that the overtones of the string are not harmonic, 
for the nth natural frequency is (cn/2l) — (c/2wl) (cr 0n + <xi n ), which 
is only approximately equal to n times the fundamental frequency 
(c/2l) - (c/2irl)(aoi + an). 

If the supports are stiffness controlled their reactances are nega- 
tive, their susceptances <r = —ecX/(R 2 + X 2 ) are positive, and all 
the natural frequencies are lowered. This is because the supports 
will move in phase with the part of the string nearest them, and the 
outer nodes, which are at the supports, if they are rigid, will be some- 
what outside the supports (virtual nodes, not on the actual string). 
This means that the wavelength has been increased a bit, so the 
frequency is lowered. 

If the transverse impedance of the supports is mass controlled, 
the susceptances are negative and all the natural frequencies are 
raised. There is a node in the string a short distance in from each 
support so that the support can move out of phase with the transverse 
force from the vibrating string. If the conductances are not zero, 
none of the nodes will be perfect; but the amplitude of motion of the 
string will be a minimum at these points, and the phase lag of the 
supports' displacement behind the string will be less than 180 deg. 

This analysis will be taken up again when we study the behavior of 
sound waves in tubes, and more accurate solutions will be derived. 



111.13] EFFECT OF MOTION OF THE END SUPPORTS 147 

Recapitulation. — We have gone into considerable detail in studying 
the motions of the string, perhaps more detail than seems necessary. 
This has been done because the string is the simplest case of a system 
with an infinite number of allowed frequencies, and it is best to discuss 
some of the properties common to all such systems for as simple a 
system as we can find, lest the mathematical complications completely 
obscure the physical ideas. The effect of friction, both on the system 
and through the supports, and the phenomenon of multiple resonance 
are both properties that are true of systems more complicated than 
the string. The damping effect of the air's reaction will be more 
important than the effect of the supports in systems more extended 
than the string, but the general effect will be the same as that dis- 
cussed above. 

We also have been developing methods of handling vibration prob- 
lems which will be exceedingly useful in our later work, developing 
them on problems where the general method is not too much obscured 
by details. In particular, we have been giving example after example 
of the utility of the study of the normal modes of vibration of a sys- 
tem. Once the normal frequencies and corresponding characteristic 
functions have been worked out for a system with a given set of 
boundary conditions, we can determine its motion for any set of 
initial conditions and for any sort of applied force. We can also 
discuss, by methods similar to those developed in Sec. 12, the effect 
on the motion of the system of slight changes in the system's properties 
(such as its distribution of mass or its distribution of tension). And, 
by expressing the applied force in terms of the characteristic functions, 
we can work out the forced motion. We can show, for instance, that 
when the driving frequency is equal to one of the allowed frequencies 
of the system, then the system takes on the shape of the corresponding 
characteristic function with an amplitude that is infinite if there is no 
frictional damping force. (Compare this with the discussion in the 
last paragraphs of Chap. II.) 

All the methods discussed above will be used again later in more 
complicated problems, either in the text or in the problems. 

Problems 

1. A string, clamped at one end, is struck at a point a distance D from the 
clamp, by a hammer of width (D/4). The head of the hammer is shaped so that 
the initial velocity given the string is maximum at the center of the head and is 
zero at the edge, the initial "velocity shape" of this portion being like an inverted 
V. Plot the shapes of the string at the times t = 0, (D/2c), (D/c)', (3D /2c), 



148 THE FLEXIBLE STRING [111.13 

(2D/c). Draw a curve showing the vertical component of the force on the clamp 
as a function of time. 

2. A harp string is plucked so that its initial velocity is zero and its initial 
shape is 

(i*r) x i° < x < 20J 

KT)\2- X ) \& <x< 2o) 

S20h\ , n an . . \ 
KW) {x ' l) V20 < ' < l ) 

Plot the successive shapes of the string during one cycle of the motion. Draw a 
curve showing the vertical component of the force on one of the supports as a 
function of time. 

3. What are the total energies of the two strings shown in Fig. 16? 

4. Show that if a string, having a wave velocity c, is plucked at a point whose 
distance from one end is 1/3 the equation for its subsequent shape is 

y = w L sm \t) cos vt) + 4 sin \rr) cos \tt) - tV sm vt) cos \rr) 

, . (5ttX\ /5wct\ , . . /7rx\ /7irct\ . 1 

- A sm {-j- j cos {— j + A ^ {—) cos ^ j + • • J 

The string has a mass of 0.01 g per cm, is 25 cm long, and is under a tension of 
1,000,000 dynes. Find the energy of vibration of the first four normal modes when 
the string is pulled aside 1 cm (h = 1). 

5. A uniform string with no friction is stretched between rigid supports a 
distance I apart. It is driven by a force F e~ iut concentrated at its mid-point. 
Show that the amplitude of motion of the mid-point is (F /2ecu) tan (ul/2c) . What 
is the amplitude of motion of the point x = (1/4) ? 

6. A uniform string of small electrical resistance is stretched between rigid 
supports a distance I apart, in a uniform magnetic field of B gauss perpendicular 
to the string. A current Iae~ i03t amp is sent through the string; what is the force 
on the string per unit length? Show that the velocity shape of the string (assume 
zero friction) is 

BJoe-*" {cos[(»/c)(a; - $Z)] ,) , 

v = ^-. < / 7/0 ^ ~ 1 C cm /sec 

lOzwe I cos (wZ/2c) ) 

Use the formula E = 10~ 8 j Bv dx volts to compute the motional emf induced 
in the string by the motion and, therefore, the motional impedance. 

7. A condenser is discharged through the string of Prob. 6, producing a current 
S(t) amp. Compute the subsequent shape of the string. 

8. The string of Prob. 5 is acted on by a force F(£) u(t) concentrated at the 
point x — I [see Eq. (2.6) for a definition of u, and Eq. (6.10) for its use]. Show 
that the shape of the string after t = is 



HI.13] EFFECT OF MOTION OF THE END SUPPORTS 149 

From this formula compute the shape of the string when it has been subjected to 
the constant (independent of time) force F(£) up to t = and then released to 
vibrate freely. 

9. A string of length I and mass e g per cm is hung from one end, so that 
gravitational forces are the only ones acting. Show that if the free end of the string 
is taken as origin, the normal modes of vibration have the form 



y - Jo^.^) 



where v n = (|8„/4) y/gjl) the function J (z) is given by the series (see Prob. 3 of 
Chap. I) 

z 2 z 4 

■W-^-iT + ei 

and the constants /3„ are the solutions of the equation Jo(irp n ) = [see Eq. (19.6)]. 
Is the motion periodic in general? What are the ratios of the lowest three allowed 
frequencies to the frequency of oscillation that the string would have if all its mass 
were concentrated at its lower end? 

10. Utilizing the general formulas of Eqs. (11.6) and (11.8), obtain specific 
formulas for the shape of the string of Prob. 9 when driven by a periodic force and 
when struck by an impulsive force. 

11. Choosing suitable values for the properties of the string of Prob. 9, plot the 
shape of the string when it is vibrating at its lowest three allowed frequencies. 
Plot its shape when driven at its free end at a frequency 1.5 times its fundamental. 

12. The tip of the whirling string is struck an impulsive blow 5(0 in a direction 
perpendicular to the equilibrium plane of motion. Calculate the series repre- 
senting the subsequent displacement. What is the amplitude of motion of the 
lowest mode? 

13. A string of infinite length is acted on by a force F(t) concentrated at the 
point x = 0, where 

(t < 0) 

F(t) = { F (0 < t < to) 
(t > U) 

Plot the shape of the string at the times t = 0, (to/2), U, (32 /2), 2t . What is 
the total energy given to the string? 

14. A string of steel (density 7.7 g per cc) is stretched between rigid supports. 
Its fundamental frequency of vibration is 500 cps. Sound of unit intensity and 
frequency v falls on the string, normal to its axis. Compute the amplitude of 
motion of the mid-point of the string for the frequencies v = 50, 100, 300, 450, 499, 
550, 600, 1,000. Note. — The force per unit length on a string of cross-sectional 
radius a, due to an incident sound wave of intensity T and frequency v 2 , is 0.0074Ka 2 
\/T e~ 2rin dynes per cm length. 

15. Plot the shapes of the first three modes of the stretched hanging string, 
described in Eqs. (12.8), for (lge /To) = 0.25. Compute the series for shape of the 
string when driven by a periodic force concentrated at x = £. 



150 THE FLEXIBLE STRING [111.13 

16. What are the values of tanh[7r(a — i(5)] when a ■= 0, /S = 1.75; when 
a = 0.2, j8 = -0.6; a = 0.1, /3 = 0.45; a = 0, '/3 = 0.5? What are the values of 
cosh [ir (a — ip)] for these same values of a and /3? 

17. A uniform string of length I and tension T is fastened at x = £ to a support 
having transverse mechanical resistance R = ec tanh (ira — ^r) and zero reac- 
tance. The string is originally at equilibrium, and the end at x = is suddenly 
acted on by a transverse impulsive force 8(t). Utilize Eqs. (13.10) and (6.14) to 
obtain the formula for the shape of the string: 

00 

1 ^C^ p-(ira/l)(.x+ct) I r^ "1 

»<*» " k 2 (. + «■ + * I <" + §> sin [r <» + »<* + C,) J 

+ a'cos ^ (n + %)(x + ct)\l 

- s; 2 (»Tw-T- 1 (w + j) 8in B ( " + t)(x - c<) ] 

n = 

+ « cos y (n + |) (x — cf) [ -\ tanh (ira) 

18. A string of density 0.1 g per cm is stretched with a tension of 10 s dynes 
from a support at one end to a device for producing transverse periodic oscillations 
at the other end. When the driving frequency has a given value, it is noted that 
the points of minimum amplitude are 10 cm apart, that the amplitude of motion of 
the minimum is 0.557 times the amplitude at the maximum, and that the nearest 
maximum is 6 cm from the support. What is the driving frequency, what are the 
values of on and ft, and what is the value of the transverse impedance of the 
support? 

19. A string is stretched between two supports having transverse mechanical 
resistance R large compared with ec, and zero reactance. The string is driven by a 
periodic force concentrated at a; = £ . What are the amplitudes of motion of the 
end supports, and by how much do they lag behind the driving force? 

20. The string of Prob. 19 is struck an impulsive blow at the point x =' £. 
Compute the subsequent motion of the two end supports. 



CHAPTER IV 

THE VIBRATION OF BARS 

14. THE EQUATION OF MOTION 

It must have been rather obvious in the previous chapter that 
we were analyzing the motions of a somewhat idealized string. In 
the first place, we assumed that the string was perfectly flexible, that 
the only restoring force was due to the tension. Secondly, we made 
no mention of the possibility of longitudinal motion of alternate 
compression and tension, which can be set up in any actual string as 
well as in any other piece of solid material. This longitudinal wave 
motion will be disregarded for a while longer; we shall spend the 
whole of the last three chapters discussing it. However, we can no 
longer put off studying the effect of stiffness on the string's motion. 
And we shall begin the study by discussing the transverse vibrations 
of bars. 

There is no sharp distinction between what we mean by a bar 
and what we mean by a string. In general, tension is more important, 
as a restoring force, than stiffness for a string, and stiffness is more 
important for a bar; but there is a complete sequence of intermediate 
cases, from stiff strings to bars under tension. The perfectly flexible 
string is one limiting case, where the restoring force due to stiffness 
is negligible compared with that due to the tension. The rod or bar 
under no tension is the other limiting case, the restoring force being 
entirely due to stiffness. The first limiting case was studied in the 
previous chapter. The second case, the bar under no tension, will be 
studied in the first part of this chapter, and the intermediate cases 
will be dealt with in a later part. 

Stresses in a Bar.— To start with, we shall study the bending of a 
straight bar, with uniform cross section, symmetrical about a central 
plane. The motion of the bar is supposed to be perpendicular to this 
plane, and we shall call the displacement from equilibrium of the 
plane y. 

When the bar is bent, its lower -half is compressed and its upper 
half stretched (or vice versa). This bending requires a moment M, 
whose relation to the amount of bending we must find. To compress 

151 



152 



THE VIBRATION OF BARS 



[IV.14 



a rod of cross-sectional area £ and length I by an amount dl requires 
a force QS(dl/l), where Q is a constant, called Young's modulus. The 
values of this constant are given in Table 3, for some of the more 

common materials. Now, imagine the bar 
to be a bundle of fibers of cross-sectional 
area dS, all running parallel to the center 
plane of the bar. If the bar is bent by an 
angle $ in a length dx, then the fibers which 
are a distance z down from the center sur- 
face (it is no longer a plane now that the 
bar is bent) will be compressed by a length 
2$, the force required to compress each 
fiber will be Q dS(z$/dx), and the moment 
of this force about the center line of the 
bar's cross section will be (Q$/dx)z 2 dS. The total moment of these 
forces required to compress and to stretch all the fibers in the bar will be 




Fig. 29. — Moment acting on a 
bent element of a bar. 



M = 



\dx) J 



dS 



(14.1) 



where the integration is over the whole area of the cross section. 
Table 3. — Elastic Constants of Materials 



Material 



Brass, cold rolled 

Bronze, phosphor 

Copper, hard drawn. . 

German silver 

Glass 

Iron, cast 

wrought 

Iron-cobalt (70 % Fe) . 

Nickel 

Nickel-iron (5 % Ni) . . 
Silver, hard drawn. . . 
Steel, annealed 

invar 

Tungsten, drawn 



Q 



9 X 10 11 
12 X 10" 

10 X 10 11 

11 X 10 u 
6 X 10 11 
9 X 10 11 

19 X 10 11 
21 X 10 11 
21 X 10 11 
21 X 10 11 
8 X 10 11 
19 X 10 u 
14 X 10 11 
35 X 10 11 



8.6 
8.8 
8.9 
8.4 
2.6 
7.1 
7.6 
8.0 
8.7 
.7.8 
10.6 
7.7 
8.0 
19.0 



Values of Young's modulus Q in dynes per square centimeter, and of density p, 
in grams per cubic centimeter, for various materials. 

We define a constant k, such that k 2 = (l/S)jz 2 dS, where S is the 
area of the cross section. This constant is called the radius of gyra- 



IV.14] 



THE EQUATION OF MOTION 



153 



lion of the cross section, by analogy with the radius of gyration of 
solids. Its values for some of the simpler cross-sectional shapes are 
as follows: 

Rectangle, length parallel to center line 6, width perpendicular 
to center line a: 



Circle, of radius a: 



VV12) 

-GO 



Circular ring, outer radius a, inner radius b: 
k = £ Va 2 + 6 2 

Bending Moments and Shearing Forces. — Equation (14.1), giving 
the moment required to bend a length dx of rod by an angle $, is then 



M = 



Q$Sk* 
dx 



(14.2) 



If the rod does not bend much, we can say that $ is practically equal 
to the difference between the slopes of the axial line of the rod at the 
two ends of the element dx: 



$ 



\dx/ x+dx \dx/ 



+ ( ^ = -dx 




Therefore the bending moment is 



(14.3) 




M+dM 



F+dF 



Fig. 30. — Bending moments and 
shearing forces to balance. 



This bending moment is not the same 
for every part of the rod; it is a function 
of x, the distance from one end of the rod. 
In order to keep the element of bar in 
equilibrium, we must have the difference 
in the moments acting on the two ends of the element balanced by a 
shearing force represented by F in Fig. 30 (moment and shear are, 
of course, two different aspects of the single stress which is acting on 
the bar). The moment of the shearing force is F dx and this must 
equal dM for equihbrium, which means that 



dx dx 3 



(14.4) 



154 THE VIBRATION OF BARS [IV.14 

This equation is not exactly true when the bar is vibrating (since a 
certain part of the moment must be used in getting the element of bar 
to turn as it bends), but it is very nearly correct when the amplitude 
of vibration is not large compared with the length of bar. 

Properties of the Motion of the Bar. — The shearing force F is also a 
function of x and may be different for different ends of the element of 
bar. This leaves a net force dF = (dF/dx) dx acting on the element, 
perpendicular to the bar's axis; and this force must equal the element's 
acceleration times its mass pS dx where p is the density of the mate- 
rial of the rod. Therefore the equation of motion of the bar is 
dx (dF/dx) = P S dx (d 2 y/dt 2 ), or 

a^ _p_av (145) 

dx* Qk 2 dt 2 K } 

This equation differs from the wave equation in that it has a fourth 
derivative with respect to x instead of a second derivative. The 
general function F(x + ct) is not a solution, so that a bar, satisfying 
Eq. (14.5), cannot have waves traveling along it with constant velocity 
and unchanged shape. In fact, the term "wave velocity" has no 
general meaning in this case, although it can be given certain special 
meanings. For instance, a simple harmonic solution of Eq. (14.5) is 

y = (7 e 2m(Ma:-,« = ^ COS [2tt(^ ~ Vt) — $] (14.6) 

where 



"-(ass?)' '- 2 " A >/? 



This represents a sinusoidal wave traveling in the positive direction, 
of just the sort one finds on strings. There is an important difference 
between the two waves, however; for in the case of the bar the velocity 
of the wave u = (v/n) = (4 7 r 2 Q/c 2 /p) i V~ v depends on the frequency of 
oscillation of the wave, whereas in the case of the string it does not. 
The velocity of propagation of a simple harmonic wave is called the 
phase velocity and is one of the special kinds of "velocity" that have 
meaning for a bar. In the case of the string the phase velocity is 
independent of v and is equal to the velocity of all waves c. For the 
bar the phase velocity depends on v, and there is no general velocity 
for all waves. This is analogous to the case of the transmission of 
light through glass, where light of different frequencies (i.e., of differ- 
ent colors) travels with different velocities and dispersion results. A 
bar is sometimes said to be a dispersing medium for waves of bending. 



IV.14] THE EQUATION OF MOTION 155 

Wave Motion of an Infinite Bar.— Although the phase velocity is 
not a constant, nevertheless we should be able to build up any sort of 
wave out of a suitably chosen combination of sinusoidal waves of 
different frequencies, in a manner analogous to the formation of a 
Fourier series. At present, we are considering the bar to be infinite 
in length, so that all frequencies are allowed, and the sum is an integral 

y = .Co ^"W/O cos (7 VO + C(n) sin (7 VOJ dfi (14.7) 

where y 2 = 4tt 2 k VQ/p, and where the functions B(n) and C(jx), corre- 
sponding to the coefficients B n and C n of a Fourier series, are deter- 
mined by the initial conditions. This integral is analogous to the 
Fourier integral of Eq. (2.19). If the initial shape and velocity shape 
of the bar are y (x) and t> (aO, a derivation similar to that given for 
Eq. (9.9), for the Fourier series, shows that 

-B(m) = ( 4—3 J I y (x)e- 2 ™» x dx \ 

) 1 \"f ( 14 -8) 

° (m) = V&W/ J_ m voWe- 2 *^ dx J 

as was shown in Eq. (2.20). 

This may seem a rather roundabout way of obtaining solutions 
of Eq. (14.5) which satisfy given initial conditions; but since the 
useful functions F(x - ct) and f(x + ct) are not solutions of (14.5), 
there is no other feasible method. 

The case y = e -* 2 / ia \ v Q = is one that can be integrated and 
can be used to show the change in shape of the wave as it traverses 
the bar. Solving Eqs. (14.8) and (14.7), we obtain 

We can now see the utility of Eq. (14.7), for while we can show that 
(14.9) is a solution of the equation of motion (14.5) by direct substi- 
tution, we can also see that it would be exceedingly difficult to obtain 
(14.9) from (14.5) directly. 

The shape of this function, at successive instants of time, is shown 
by the solid fines in Fig. 31. For comparison, the sequence of dotted 
lines shows the variation in shape of a flexible string with equal initial 
conditions. In the case of the string, the two partial waves move out- 
ward with unchanged shape. The shape of each "partial wave" for 



V 



156 



THE VIBRATION OF BARS 



[IV.15 



the bar, however, changes continuously as the wave travels outward. 
In particular, notice the formation of subsidiary "ripples" ahead of 
the principal "crest" of each wave. This is due to the fact, which 
we have already noted, that the high-frequency, short wavelength parts 
of the wave spread outward faster than the rest. 





■p 1G . 31. — Comparison between the motion of waves on a bar (solid line) and on a 
string (dotted line), both of infinite length. Each sequence shows the shapes at suc- 
cessive instants, after starting from rest in the form given at the top of the sequence. 
The constant y was chosen to make the average "velocity" of the waves on the bar 
approximately equal to c, the velocity of the waves on the string. 

15. SIMPLE HARMONIC MOTION 

We cannot pursue our study of the motions of the bar much further 
without examining its normal modes of vibration. As with the 
string, we must ask if there are any ways in which the bar can vibrate 
with simple harmonic motion. We try setting y = Y{x)e- 2vivt and 
find that Y must satisfy the equation 



d 4 Y 
dx* 



= 16.Vr, ,-_£-,- 4^ (£) (15.1) 



IV.16] SIMPLE HARMONIC MOTION 157 

The general solution of this is 

y = de 2 *"* + Cze- 2 *** + Cge 2 ""* + Cie- 2 ™"* 
= a cosh (2irfix) + b sinh (2x,uz) + c cos (2x^r) + d sin (2tthx) 

(15.2) 

For a discussion of the functions cosh and sinh, see Eqs. (13.4), and 
see Tables I and II at the back of the book. 

This general solution satisfies Eq. (15.1) for any value of the fre- 
quency v. It is, of course, the boundary conditions that pick out the 
set of allowed frequencies. 

Bar Clamped at One End. — For example, if we have a bar of length I 
clamped at one end x = 0, the boundary conditions at this end are 
that both y and its slope (dy/dx) must be zero at x — 0. The par- 
ticular combination of the general solution (15.2) that satisfies these 
two conditions is the one with c = —a and d = — b 

Y = a[cosh (2ttmz) - cos (2w/ia;)] + 6[sinh (2t{jlx) - sin (2jr/is)] (15.3) 

If the other end is free, y and its slope will not be zero, but the 
bending moment M = QSn 2 (d 2 Y/dx 2 ) and the shearing force F = 
—QSK 2 (d s Y/dx 3 ) must both be zero, since there is no bar beyond 
x = I to cause a moment or a shearing stress. We see that two con- 
ditions must be specified for each end instead of just one, as in the 
string. This is due to the fact that the- equation for Y is a fourth- 
order differential equation, and its solution involves four arbitrary 
constants whose relations must be fixed, instead of two for the string. 
It corresponds to the physical fact that while the only internal stress 
in the string is tension the bar has two: bending moment and shearing 
force, each depending in a different way on the deformation of the bar. 

The two boundary conditions at x - I can be rewritten as 
(l/4ir 2 n 2 )(d 2 Y/dx 2 ) = and (l/8ir*n 3 )(d*Y/dx*) = at x = I. Sub- 
stituting expression (15.3) in these, we obtain two equations that fix 
the relationship between a and 6 and between n and /: 

a[cosh (%rnl) + cos (2r/*Z)] + 6[sinh (2^0 + sin (2thI)] = 
afsinh (2jr/d) - sin (2ir/d)] + 6[cosh (2rtf) + cos (2r/*J)] = 
or 
b = a sin (2Tfxl) - sinh (2thI) = _ cos (2^1) + cosh (2t(iI) 

cos (2ir(d) + cosh (2rfxl) a sin (2^0 + sinh (2jr/rf) ( ^ 

By dividing out a and multiplying across, we obtain an equation for /x: 

[cosh(27TMZ) + cos(2ttmZ)] 2 = sinh 2 (2rAiO - sm 2 (2rnl) 



158 THE VIBRATION OF BARS [IV.16 

Utilizing some trigonometric relationships, this last equation can be 
reduced to two simpler forms: 

cosh (2ir/j.l) cos (2v/d) = — 1, or coth 2 (7ryuZ) = tan 2 (x/iZ) (15.5) 

where coth(z) = cosh(20/sinh(z). 

The Allowed Frequencies. — We shall label the solutions of this 
equation in order of increasing value. They are (2irfj.il) = 1.8751, 
(2*7*20 = 4.6941, (2*7*80 = 7.8548, etc. To simplify the notation, 
we let (1/tt) times the numbers given above have the labels jS„, so that 



(I) 



(15.6) 



where 0i = 0.597, /3 2 = 1.494, /3 3 = 2.500, etc. 

It turns out that j8„ is practically equal to (n — i) when n is larger 

than two. 

By fixing /*, we fix the allowed values of the frequency. Using 
Eq. (15.1), we have 



(15.7) 



or 

0.55966 

fl = l¥~ 



j> 2 = 6.267 vi 

v z = 17.548vi 
Vi = 34.387^1 
etc. 

Notice that the allowed frequencies depend on the inverse square 
of the length of the bar, whereas the allowed frequencies of the string 
depend on the inverse first power. 

Equation (15.7) shows how far from harmonics are the overtones 
for a vibrating bar. The first overtone has a higher frequency than 
the sixth harmonic of a string of equal fundamental. If the bar 
were struck so that its motion contained a number of overtones with 
appreciable amplitude, it would give out a shrill and nonmusical sound. 
But since these high-frequency overtones are damped out rapidly, 
the harsh initial sound would quickly change to a pure tone, almost 
entirely due to the fundamental. A tuning fork can be considered 
to be two vibrating bars, both clamped at their lower ends. The fork 
exhibits the preceding behavior, the initial metallic "ping" rapidly 
dying out and leaving an almost pure tone. 



IV.16] SIMPLE HARMONIC MOTION 159 

The Characteristic Functions. — The characteristic function cor- 
responding to the allowed frequency v n is given by the equation 

+ n = a n [cosh(^) - cos(^)] 

6„[sinh(^)-sin(^) 



+ 
where 



(15.8) 



_k _ cosh (xpn) + cos (xj8„) _ sJnhQrftn) — sin(ir|8 w ) 
n sinh (t0 u ) + sin (*r/3„) ~~ Un cosh (wj8„) + cos (w0«) 

We shall choose the value of a n so that j ty\ dx = (Z/2), by analogy 

with the sine functions for the string. The resulting values for 
a n and b n are ai = 0.707, &i = -0.518, a 2 = 0.707, 6 2 = -0.721, 
a 3 = 0.707, & 3 = —0.707, etc. For n larger than 2, both a n and b n 
are practically equal to (I/a/2)- Some of the properties of these 
functions that will be of use are 

J,, C (m ?± n) 

o M*)*.(x) dx = j/j\ (m = n) * n 0) = (-l)-i V2 

(^«(-l)-V5(^) and 0. ~ „-*(«> 2) ^ (15 - 9) 



it n ~: — -= [ff-rPnX/l _|_ (_Dn-l e i/J B (ir-0/fl 

V2 



V ^ 4/ 



+ sin(^- -^ (n> 2) 

The shapes of the first five characteristic functions are shown in 
Fig. 32. Note that for the higher overtones most of the length of 
the bar has the sinusoidal shape of the corresponding normal mode 
of the string, with the nodes displaced toward the free end. In terms 
of the approximate form given above for \f/ n , the sine function is sym- 
metrical about the center of the bar; the first exponential alters the 
sinusoidal shape near x = enough to make 4/ n have zero value and 
slope at this point; and the second exponential adds enough near 
x = I to make the second and third derivatives vanish. Note, also, 
that the number of nodal points in yp n is equal to n — 1, as it is for 
the string. 



160 



TH~E VIBRATION OF BARS 



[IV.15 



In accordance with the earlier discussion of series of characteristic 
functions, we can show that a bar started with the initial conditions, 

at t — 0, of y = y (x) and (dy/dt) = 
v (x) will have a subsequent shape 
n=( given by the series 




n=2 



n»3 



y = 2) ^n{x)[B n co^{2irv n t) 

+ C n sin (2rr n t)] (15.10) 



n = l 



where 
B„ = 



yo(x)$n(x) dx 



Cm — 



\irv n lj Jo 



o(x)^„(x) dx 



Fig. 32. — Shapes of the first five 
characteristic functions for a vibrating 
bar clamped at one end and free at the 
other. 



Plucked and Struck Bar. — Two 

examples of such calculations will 
be given. The first example is that 
of a rod suddenly released from an 
initial shape y$ = (hx/l), an undesir- 
able case in actual practice (for it 
bends the bar quite severely at 
x = 0) but one that can be easily 



solved. The values of the coefficients B n turn out to be 



£* = ^- 



JXTp n (x) dx 
o 

2h ( 
-g-p \a n [2 + ^(sinnx/^ — sinTrjSn) — cosh^n — cosx/3 w ] 

+ &„[7rj3 n (cosh'7r/3 n + cos7Tj8„) — sinh-7r/3„ — sinir/Sn]? 



4h 



On 



when the ratios between a n and 6„, given in Eqs. (15.8), are used to 
simplify the expression. The successive shapes taken on by the bar 
when started in this manner are shown in the first sequence of Fig. 33. 
Since the overtones are not harmonics, the motion is not periodic. 

The other case to be considered is that of a rod struck at its free 
end in such a manner that its initial velocity is zero everywhere except 
at x = I, where it is very large, large enough so that jv dx = U. 
This case corresponds to that of a tuning fork struck at one end. If 



IV.16] 



SIMPLE HARMONIC MOTION 



161 



the total impulse given to the end of the fork is P, then U = P/pS. 
The coefficients B n are zero and 

tM Jo 7ry„Z 7r 2 ^ \ Qk 2 




B- 



H- 






Fig. 33.^ — Motions of a bar clamped at one end. Left-hand sequence shows the 
successive shapes of a bar started from rest in the shape given in the topmost curve. 
Eight-hand sequence shows the motion of a bar struck so that its outer tip starts 
upward at t =0. Since the overtones are not harmonics, the motion is not periodic. 

when the value given for ip n (l) in Eqs. (15.9) is used. The shapes 
of such a bar at successive instants of time are shown in the second 
sequence of Fig. 33. This motion is also nonperiodic, as are all 
motions of the bar that correspond to more than one normal mode of 
vibration. 

Clamped-clamped and Free-free Bars. — Other boundary condi- 
tions will give rise to other characteristic functions and frequencies. 



162 THE VIBRATION OF BARS [IV.16 

The bar may be clamped at both ends, in which case expression (15.3) 
must again be used for Y, but instead of Eq. (15.4) we must have 

h — sin (2tt/xQ + sinh (2ir ixl) _ _ cos (2^) — cosh (2x^0 
~ cos {2irid) — cosh (2irnl) si n (2rnl) — sinh (2rnl) 

and instead of Eq. (15.5) Ave have 

cosh(27TM0 cos {2tt id) = 1, or tanh 2 (717*0 = tan 2 (717*0 

The allowed frequencies can be obtained from the formula 

j8i = 1.5056 

Vn==J W = w4^7^ ft = 2.4997 (15.11) 

/3 n ~ n + i (n > 2) 

Still another set of boundary conditions are the ones for a com- 
pletely free bar, where the second and third derivatives must be zero 
at both ends. The characteristic functions (which we can call if/") 
for this case can be obtained from the ones for the bar clamped at 
both ends (which we can call \[/ n , to distinguish them from #,') by 
simply differentiating twice with respect to x, $" = (d 2 if/ n /dx 2 ). By 
using Eq. (15.1) we can show that (d 2 $"/dx 2 ) = (47r 2 ^p/Q/c 2 )^ n , 
so that if ip n and (dif/ n /dx) are both zero at x — and x = I, then 
(dhl/'Jdx 2 ) and (dhp'^/dx 3 ) are both zero at the end points. The 
functions ^" therefore satisfy the boundary conditions for a com- 
pletely free bar. It. is not difficult to see that $„ is a solution of the 
equation of motion, corresponding to the same allowed frequency as 
does the function ^„. The allowed values are given in Eq. (15.11). 

We thus obtain the rather surprising result that the allowed fre- 
quencies for a free bar are the same as those for a similar bar clamped 
at both ends and that the corresponding characteristic functions are 
related by simple differentiation, although their shapes are quite 
different. 

Energy of Vibration. — It is sometimes useful to know the energy 
of vibration of a bar. This energy can be computed by using the 
expressions obtained in the previous section. We saw there [Eqs. 
(14.2) and (14.3)1 that the moment required to bend an element dx 
of the bar by an angle 3? is M = QSn 2 ($/dx). The amount of work 
required to bend it from equilibrium to this angle is 

J>— * ^-^(3)'* 



IV.16] SIMPLE HARMONIC MOTION 163 

Therefore, the potential energy of the bar, the work required to bend 
the bar into its final instantaneous shape y(x,t), is 

The kinetic energy is, of course, 

so that the total energy turns out to be 

The energy of a bar, subject to one or another of the various 
boundary conditions discussed above, is obtained by substituting 
y = 2A W *„ cos(2W - fl B ) in Eq. (15.12). This results in a super- 
fluity of terms, each containing a product of two trigonometric 
functions of time and an integral of a product of two characteristic 
functions, or two second derivatives of these functions. The kinetic- 
energy terms, containing integrals of the products of two functions, can 
be integrated by means of the first of Eqs. (15.9) and reduced to the 
single sum 

2T 2 r 2 nP lSAlsm 2 (2TvJ - O n ) 

The integrals in the potential-energy terms can be integrated by 
parts twice: 

by using the equation of motion. The terms in the square brackets 
are zero no matter which of the boundary conditions discussed 
above are used. The resulting potential-energy series is therefore 
WvlplSAl cos 2 (2w - 0.). The sin 2 of the kinetic-energy terms 
combines with the cos 2 in the potential-energy terms and gives for the 
total energy the simple series 



W = %™(jf)<Al (15.13) 



164 THE VIBRATION OF BARS [IV.15 

Note the similarity with the corresponding expression for the energy 
of the string, given in Eq. (9.13). 

Nonuniform Bar. — Now that we know the properties of the normal 
modes of vibration, we can solve a number of different sorts of prob- 
lems. For instance, we can find the change in the allowed frequencies 
and characteristic functions when the bar is made somewhat non- 
uniform along its length. If the bar's density or its cross-sectional 
area or radius of gyration changes with x, a review of the derivation 
of the equation of motion shows that the correct equation should be 

iS(s)te»L w te'J Q dt2 

which reduces to Eq. (14.5) when S, p, and k are all constant. To 
study the normal modes of vibration, we set y — Y(x)e~ 2wivt and have 
for the equation for Y, 



1 d 2 



S(x) dx 2 



[«»*,) g]-p=^a]r (1 5.H) 



If the variation in p, S, and k is small, we can treat the problem by 
a modification of the approximation method discussed in Sec. 12. If 
we write 

S = Soil + a(x)], k 2 = 4[l + °{x)], P = Poll + g(x)] 

where So, k , and p are constants and a, <r, and g are small quantities, 
then, by neglecting the products of small quantities, (15.14) can be 
rewritten as 

d * Y to ^ ■ n d(a + <r) d*Y d 2 {a + <y) d 2 Y 
~dx~*~ {2Tfi) Y + l dx ~dx^ + dx 2 ~dx^ 

- (2x/x)%-<r)F = 

where ju 2 = (v/2v) VpoA&l- If a > °"> and g were all equal to zero, 
the solution of this equation would be one of the characteristic func- 
tions ^ n given in Eq. (15.8), and the allowed values of n would be 
the values (j9 n /2Z) given in Eq. (15.6) or (15.11). Since a, <r, and g 
are small, we assume that Y n = ZA^*,, where all the Akn's are small 
except A nn , and that n n = (j8»/20 4 (l + Vn), where r\ n is small. Sub- 
stituting this in the equation, and again neglecting products of small 
quantities, leaves 



IV.15] 



SIMPLE HARMONIC MOTION 



165 



d 2 {a + a) d 2 1 

2 J &» = 



+ 2 d(a + cr) d*_ 
dx dx 3 



dx 2 dx 2 



But (dtyk/dx*) = (Tfa/l)y k . Multiplying through by ^ n or ^ and 
integrating gives the equations for the correction factors: 



-ij:4-'*+k0[»^& 



+ 



dx 3 
d 2 (a + <r) c?V« 



•<*&» — 



2A nnJ 8 



S)Jo**{ 



da; 2 



da; 2 



} 



da; 



2081 - ft) Jo ^ ~ g) ^ 

+ (_ly r 2 <*<« + «o ^v» 

W«/ L dx dx 3 



(15.15) 



+ 



d 2 (a + <r) dV 



da; 5 



da^J/^ 



As an example, consider the case of a tapered bar of rectangular 
cross section having a constant thickness a in the direction of the 

vibration and a varying width b (l - y |Y j n this case, « and p 
are constant, and S(x) = a b(l - y |\ so that g = a = and 
a = —(yx/l). The correction to m» is 

Owing to the boundary conditions, the quantity in the brackets 
reduces to - (i) (d^/dx)^ when the bar is clamped at one end and 
free at the other. The corrected value of the frequency is therefore 

"* V Po 4P V3po L + ir'flt V"da7A=J 



7ra(n + i) 2 
4l 2 



} r 27 l 

^ L 1 + T . (n + i )2 J (n > 1) 



This equation states that tapering the bar so that its free end 
is narrowest raises the natural frequencies over those for a bar whose 
width is everywhere the same as at the clamped end. (This is hardly 



166 THE VIBRATION OF BARS [IV.16 

a surprising result.) If it is tapered so that the free end is three- 
quarters the width of the clamped end (7 = t), the fundamental 
frequency is 1.077 times the fundamental of the nontapering bar 
of the same width as that of the clamped end; the first overtone is 
1.023 times the corresponding first overtone; the second, 1.008 times 
the corresponding second; and so on. 

Forced Motion. — Besides the case of the nonuniform bar, we can 
discuss the forced motion of a uniform bar, now that we know the 
characteristic functions. If the bar is driven by a force F(x)e- 2 ' i " t 
dynes per cm length, the equation for the shape of the bar during 
the steady state is 



&! ** - 4,rV7 = f{x), f = %&, y = Y(x)e-**« (15.16) 
3 expand Y and / in series of characteristic functions, 

Y = Zhntnfr), f = ^Qntnix), Q n = \^jj J q F{x)^ n {x) dx 



we obtain for the steady-state motion 



y = & 



^-^tv^n(x)]e-^ (15.17) 

.n= 1 J 

Resonance occurs whenever the driving frequency v equals one 
of the natural frequencies v n , unless the corresponding g n is zero. 

These results are analogous to those discussed at the beginning of 
Sec. 11. The equation determining ^„ here differs from that used in 
Eq. (11.1) ; but Eq. (15.17) has the same form as Eq. (11.6). In fact, 
if we use the new characteristic functions we can utilize Eq. (11.9) to 
determine the general transient behavior of the bar (provided we can 
perform the integrations). The effects of friction can be handled as 
they were handled for the string on page 104 and in Sec. 12. The 
results are so similar that we do not need to go into detail about them 
here. Some cases will be taken up in the problems. 

16. VIBRATIONS OF A STIFF STRING 
When a string is under a tension of T dynes, and also has stiffness, 
its equation of motion is 

1 dx 2 V dx* P dt 2 K 

This equation can be obtained by combining the derivations in Sees. 



IV ' 16 J VIBRATIONS OF A STIFF STRING 167 

9 and 14. The constant 8 is the area of cross section, k its radius of 
gyration, and p and Q are the density and modulus of elasticity of the 
material. 

Wave Motion on a Wire.— Sinusoidal waves can travel along such 
a wire, for if we set y = Ce™^-"> we obtain an equation relating 
p and v: 



or 



TV + 47r 2 Q£/cV = P Su 2 



n^a I r ?a enCy iS VeiT Sma11 (ix -' if v ' is ver y much small er than 
I /16tt P Q£V), we expand the radical by the binomial theorem and 

keep only the first two terms, and p 2 turns out to be approximately 
equal to Q>Sv*/T). The phase velocity {v/p) is practically equal to 
the constant value VT/p~S, which would be the velocity of every 
wave if the wire had no stiffness (if QSk 2 were zero). The phase 
velocity for the wire is not constant, however. It increases with 
increasing v until for very high frequencies it is \/2lTv (Q* 2 / P )* the 
phase velocity given in Sec. 14 for a stiff bar without tension.' In 
other words, the wire acts like a flexible string for long wavelengths 
and hke a stiff bar for short ones. This is not surprising. 

The Boundary Conditions.— The usual boundary conditions cor- 
respond to clamping the wire at both ends, making y = and also 
(dy/dx) = at x = and x = I. Setting y = Y(x)e-^\ we have for Y 

> (16.2) 

7 2 = (v/2t) Vp/Qk> ) 

Setting Y = Ae 2 *»*, we obtain an equation for the allowed values 
of p.: M 4 - 2/3 V - Y 4 = 0. This equation has two roots for p 2 and 
therefore four roots for p: 

P = ±p h p\ = V/3 4 + t 4 + 2 \ 
or / 

P = ±ip2, p\ = V/3 4 + t 4 - |8 2 ( ^ 16 ' 3 ^ 

p\ = 2/3 2 + „l MlM2 = 7 2 / 

The general solution of Eq. (1G.2) can then be written 

tee 2 *" 1 * + be~ 2r > iix -f- cc 2ri " 2X -f- de~ 2Ti c* x } 

Y = < A cosh (2ttp iX ) + Bsmh(2Tp 1 x) > (16.4) 

' + C cos (2tp 2 x) + D sin (27r/i 2 a;) 1 . 



168 THE VIBRATION OF BARS PV.16 

If the boundary conditions are symmetrical, it will be useful to place 
the point x = midway between the supports. The normal functions 
will then be even functions, yp(-x) = t(x)', « r the y w 111 be °^ ones > 
$(-x) = — iKz). In either case, if we fit the boundary conditions at 
one end, x = (1/2), they will also fit at the other end, x = —(2/2). 
The even functions are built up out of the combination 

Y = A cosh (2irmx) -f C cos (2wn$x) 
and the odd functions from the combination 

Y = B sinh (2ir/*i^) + D sin (2x/x 2 x) 

The boundary conditions F = and (dY/dx) = at x = (2/2) corre- 
spond to the following equations for the even functions: 

Acosh(xMiO = — Ccos(7r/x 2 0, mi^ sinh (tt/xiO = M2C sin (71W) 

The Allowed Frequencies.— By the use of Eq. (16.3), these reduce 
to 

tan fcrW = - >/l + (^r) tanh (ri VmI + 2/8*). (16.5) 

which can be solved for the allowed values of M2. The allowed values 
of the frequency are obtained from the equation 

v = 2X7 2 * /55 = 2^ 2 J(/*3 + 2/3 2 ) %- 2 
\ P \ \ P 

These allowed values of v can be labeled v h v s , v 6 ... , in order of 
increasing size. 

The corresponding equation for /* 2 for the odd functions is 

^l + (?^ tan (xZ M2 ) = tanh fcrf VmI + 2/3 2 ) (16.6) 

The allowed values of v obtained from this equation can be labeled 
vs, vi, v 6 . . . ; and then the whole sequence of allowed values will 
be in order of increasing size, v x being the smallest (the fundamental), 
v 2 the next (the fi rst overton e), and so on. 

When 0(= VT/8t 2 QSk 2 ) goes to zero (i.e., when the tension is 
zero) the equations reduce to those of a bar clamped at both ends, 
with the following allowed values of v. 



IV -16] VIBRATIONS OF A STIFF STRING 169 

v% = 2.7565V1 

= 3.5608 [Q? v 3 = 5.4039*! (16.7) 

Vx P V p ' v 4 = 8.9330vi 

etc. 

obtained from Eq. (15.11) of the previous section. When is infinity 
(i.e., when the stiffness is zero) the equations reduce to those of a 
flexible string, and the allowed values of v are 

v% = 2v\ 
0-5000 PF v z = 3vi (16.8) 

I \ pS Vi = 4j>i 
etc. 

If is large but not infinite (i.e., if the wire is stiff, but the tension 
is the more important restoring force), it is possible to obtain an 
approxima te expres sion for the allowed frequencies. When is large 
tanh (tI \/20 2 + n\) is very nearly unity, and Vl + (2/37/4) is a 
large quantity for the lower overtones (i.e., as long as /*f is small com- 
pared with 0). Equation (16.5) for the even functions then states 
that tan (trim) is a very large negative number, which means that (irl^) 
is a small amount larger than (t/2) or a small amount larger than 
(3x/2). In general, (tIh 2 ) = (m + iV + 5, where 5 is a small quan- 
tity and m is some integer. Expanding both sides of Eq. (16.5) and 

retaining only the first terms, we obtain (1/5) = V20 2 - — l or 

8 = tt(2w + 1) VQSk 2 /T1 2 , where (2m + 1) is any odd integer. 

Equation (16.6) for the odd functions states that tan (W) 
must be very small, which means that for these cases (irlfi 2 ) is a small 
amount larger than t or a small amount larger than 2w, or, in general, 
(irl/i 2 ) = kir + 8, where 8 is small and k is some i nteger. The corre- 
sponding equation for 8 is 8 = ir(2k) VQSk 2 /T1 2 , where (2k) is any 
even integer. 

We have shown that for either even or odd functions we must 
have (irlm) = (nir/2) + 8, where n is any integer and where 5 = 
im VQSk 2 /T1 2 . The even solutions correspond to the odd values of 
n (n = 2m + 1), and the odd functions correspond to the even values 
of n (n = 2k). The a llowed values of /z 2 are given approximately by 

\ ~^~ T \~T~j an( * ^ e anowec * values of the frequency are 



n 



21 
" n ~ 21 \J7S K 1 ~*~ 1 \l :±: f L )' A more accurat e formula can be 



" S( i+ ?^> 



170 THE VIBRATION OF BARS [IV.16 

obtained by retaining the next terms in the series expansions that 
we have made: 




? + (* + =£)$=] 




• • ) (16.9) 

This formula is valid as long as n 2 is smaller than (l 2 T /tt 2 QSk 2 ) . 

When the "stiffness constant" {QSk 2 ) becomes negligible com- 
pared with l 2 T this formula reduces to (16.8), the equation for the 
allowed frequencies of a flexible string. As the stiffness is increased 
the allowed frequencies increase, the frequencies of the overtones 
increasing relatively somewhat more rapidly than the fundamental, 
so that they are no longer harmonics. 

To obtain values for the higher overtones or for the cases where 
(QSk 2 ) is the same size as or larger than (Z 2 T), we must solve Eqs. 
(16.5) and (16.6) numerically. 



Problems 

1 Which of the materials given in Table 3 will give the highest fundamental 
frequency for a bar of given size? Which will give the lowest frequency? 

2. An annealed steel bar 20 cm long is clamped at one end. If its cross section 
is a square 1 cm on a side, what will be the lowest four frequencies of vibration? 
If the cross section is a circle 0.5 cm in radius, what will they be? If the cross 
section is a rectangle of sides a and 2a, what must be the value of a to have the 
bar's lowest frequency be 250 cps? 

3 A bar of nickel iron of length 10 cm, whose cross section is a rectangle ol 
sides 1 cm and 0.5 cm, is clamped at one end. It is struck at the mid-point of one 
of its wider sides, so that v (x) = except at x = (1/2), and Jv dx = 100. What 
is the shape of the bar a time (Ti/4) after the blow? (Z\ is the period of the bar s 
fundamental.) What is its shape at a time (7V2)? What is the motion of the 
end point of the bar? What is the amplitude of that part of the motion of the 
end point corresponding to the fundamental? To the first overtone? To the 

second overtone? . , 

4. Plot the shapes of the first three normal modes of vibration of a bar clamped 
at both ends. Of a completely free bar. ■,,-,• 

5 What is the energy of vibration of the bar, struck at one end, which is 
shown in Fig. 32? What is the ratio between the energy of the fundamental and 
that of the first overtone? Of the second overtone? 

6 A cylindrical bar of radius a, clamped at one end, is damped by the air 
by a force equal to 1.25 X 10"* a*v* (dy/dt) dynes per cm length of bar where , is 
the frequency of vibration and (dy/dt) the bar's velocity. Show that the modulus 
of decay of the nth allowed frequency is 16 X 10* ( »p/o«,«), where ,„ is given 
by Eq. (15.7). 



IV.16] VIBRATION OF A STIFF STRING 171 

7. A cylindrical bar of glass, 0.5 cm in radius and 10 cm long, is clamped at 
one end. If it is struck at its free end, how long will it take for the amplitude of 
the first overtone to diminish to one-thousandth of the amplitude of the funda- 
mental? (Use the results of Prob. 6 to compute this.) 

8. What are the first four allowed frequencies of the bar of Prob. 3? What 
are the first four frequencies if the bar is loaded at its free end with a mass of 1 g? 
The vibration is supposed to be perpendicular to the wider faces. 

9. What are the shapes of the first two normal modes of vibration of the bar 
of Prob. 3 when its free end is loaded with a mass of 1 g? 

10. A bar of phosphor bronze, of length 10 cm and thickness 0.5 cm, is tapered 
from a width' of 2 cm at its clamped end to a point at its free end. What are 
approximate values for its first three allowed frequencies? Plot the shapes of 
the normal modes corresponding to these frequencies. 

11. An oscillating driving force of frequency v is applied to the free end of a 
bar, clamped at the other end. Show that the mechanical impedance of the bar is 



. npSl 

\n = l 



2^ 



12. Plot the mechanical impedance of the bar of Prob. 3 when driven at its 
free end, as a function of v from v = to v = 2,000. The motion is perpendicular 
to the wider faces. 

13. The bar of Prob. 3 is driven electromagnetically at its free end by a coil of 
coupling constant G = 1,000, resistance 10 ohms, and negligible inductance. 
An alternating emf of 1 volt is impressed across the coil. Plot the amplitude of 
motion of the end of the bar as a function of v, from v = to v = 2,000. 

14. A bar is whirled about one end with an angular velocity «, in a plane per- 
pendicular to the axis of rotation. Show that the simple harmonic vibrations of 
the bar from its position of steady rotation are given by y = F(z)e- 2 *<>", where 
Y is a solution of 



-£ + £[<'--> £] + *r-o 



where «* = (Q K y P £«»Z«), z = (x/l), and M 2 = (&rV/«»). Discuss the solutions 
when co is small; when w is large. 

15. Calculate the transient motion of a clamped-free bar acted on by an impul- 
sive force S(t) concentrated at x = $. Obtain the equation corresponding to 
(.lU.^&i )* 



CHAPTER V 
MEMBRANES AND PLATES 

17. THE EQUATION OF MOTION 

We must next consider the vibrations of systems extended in two 
dimensions, whose equilibrium shape is a plane sheet. Surfaces whose 
stiffness is negligible compared with the restoring force due to tension 
are called membranes (examples are drumheads and the diaphragms 
of condenser microphones). When the stiffness is the important 
factor, the surfaces are called plates (examples are the diaphragms of 
ordinary telephone transmitters and receivers). 

The analysis of the motions of a membrane is more complicated 
than that of the motions of the corresponding one-dimensional sys- 
tem, the flexible string; for the membrane has much more freedom 
in the way it vibrates than the string has. The form of the curve 
giving the displacement of a given point on a string as a function of 
time has a close connection with the form of the same string at some 
instant, as we have seen in Sec. 8. The corresponding curve for the 
motion of a point on a membrane can have no similar relationship to 
the shape of the membrane at some instant, for the curve showing 
the displacement of a point as a function of time is a one-dimensional 
line, whereas the shape of the membrane is a two-dimensional surface. 

It is possible for a membrane to behave like an assemblage of 
parallel strings, having waves whose crests are in parallel lines, per- 
pendicular to their direction of propagation (like waves on the ocean). 
The behavior of such waves is exactly like the behavior of waves on a 
flexible string; the waves travel with unchanged shape (when friction 
is neglected), and every such wave travels with equal speed. But 
this is only the simplest form of wave motion that the membrane can 
have. It can have circular waves, radiating out from a point or in 
toward it; it can have elliptical ones, going out from or in to a line 
segment; and so on. This more complicated sort of wave motion does 
not conserve its form as it travels, nor does it have a speed independent 
of its form. Much of it is so complicated in behavior that we are 
unable to analyze it at all. In this book we shall content ourselves 
with a treatment of parallel and circular waves. 

172 



V.17] 



THE EQUATION OF MOTION 



173 



Tdy- 




Tdy 



If the sheet of material is not perfectly flexible (i.e., is a plate), 
then the motion is still more complicated than that for a membrane.' 
We shall deal only with the simplest possible case of such motion, in 
the last section of this chapter. 

Forces on a Membrane.— Our first task is to set up the equation of 
motion for the membrane. The procedure is similar to that for the 
string. We shall find that any 
part of the membrane having a 
bulge facing away from the equilib- 
rium plane will be accelerated to- 
ward the plane, and vice versa. 
In general, the acceleration of any 
portion is proportional to its "bulg- 
iness" and opposite in direction. 
We must find a quantitative mea- 
sure of this bulginess. 

Suppose that the membrane 
has a density of <r g per sq cm and 
that it is pulled evenly around its 
edge with a tension T dynes per 
cm length of edge. If it is perfectly 
flexible, this tension will be dis- 
tributed evenly throughout its 
area, the material on opposite 
sides of any line segment of length 
dx being pulled apart with a force of T dx dynes. The displacement 
of the membrane from its equilibrium position will be called 77. It is a 
function of time and of the position on the membrane of the point in 
question. If we use rectangular coordinates to locate the point, v will 
be a function of x, y, and t. 

Referring to the analogous argument on page 82 for the string 
and to the first drawing of Fig. 34, we see that the net force on an 
element dx dy of the membrane due to the pair of tensions T dy will be 




T(r+dr)dy 



Fig. 34. — Forces on an element of a 
membrane in rectangular and polar 
coordinates. 



T 



V [\dx) x+dx \dx)z. 



-T^dxdy 



and that owing to the pair T dx will be T(d* v /dy 2 ) dx dy. The sum 
of these is the net force on the element and must equal the element's 
mass adxdy times its acceleration. The wave equation for the 
membrane is therefore 



174 MEMBRANES AND PLATES [V.17 

The Laplacian Operator. — The left-hand side of this, equation is the 
expression giving the measure of the bulginess (or, rather, the negative 
bulginess) of the portion of the membrane under consideration. 

We now encounter one of the extra difficulties of the two-dimen- 
sional analysis; for we find that if we had picked polar instead of 
rectangular coordinates, so that 77 was a function of r and <p, then the 
resulting wave equation would have a different aspect. The net 
force due to the tensions perpendicular to the radius (shown in the 
second drawing of Fig. 34) is 

|_V d<p/ v+d<P \r d<P/<pJ r 2 dip 2 
and that due to the tensions parallel to the radius is 

^[('SL-('S)J-?K'S)'** 

The resulting equation of motion in polar coordinates is 

i±( r *i\ + L*i = L*! (172 ) 

r dr \ drj ^ r 2 dip 2 c 2 dt 2 K - 

The left-hand side of this equation has a different form from the 
left-hand side of (17.1). This does not mean that it represents a 
different property of the membrane; for the value of the left side of 
(17.2) for some point on a given membrane will have the same value 
as the left side of (17.1) for the same point on the same membrane. It 
simply means that when we wish to measure the bulginess of a portion 
of a membrane by using polar coordinates, we must go at it differently 
from the way we should have gone at it if we had used rectangular 
coordinates. 

We often emphasize the fact that the left side of both equations 
represents the same property of the surface 17, by writing the wave 
equation as 

V*„4|? (17.8) 

where the symbol V 2 is called the Laplacian operator, or simply the 
Laplacian. It stands for the operation of finding the bulginess of the 
surface at some point. In different coordinates the operator takes 
on different forms: 



V.17] THE EQUATION OF MOTION 175 

d 2 d 2 

r~2 + ^~2 (rectangular coordinates) 



^ 2 — \l d( d\ Id 2 

r dr\ r d~r)~ i ~?dv* (p ° lar coordinates ) 



etc. 

The fact that we have different forms for the Laplacian operator 
corresponds to the fact that the membrane can have different sorts 
of waves. The form for rectangular coordinates is the natural one 
for parallel waves; that for polar coordinates is best for circular waves; 
and so on. Although the Laplacian is a measure of the same property 
of the membrane, no matter what coordinate system we use, never- 
theless there is a great variation in the facility with which we can 
solve the wave equation in different coordinate systems. In fact, 
the known methods for its solution are successful only in the case of a 
few of the simpler coordinate systems. 

Boundary Conditions and Coordinate Systems. — It should be 
realized, however, that the difficulties with coordinate systems are not, 
in one sense, a complete barrier to the discussion of waves of com- 
plicated form. For just as with the string, we can make up compli- 
cated wave motions by adding together simple ones. Circular waves 
can be made up by adding together a large number of parallel ones, 
each going in a different direction; parallel waves can be made up of a 
suitable sum of elliptic waves; and so on. We shall see that it is 
possible to study any sort of wave motion on a membrane of infinite 
extent, by expressing the waves in terms of a suitable sum of parallel 
waves and studying the properties of the sums (or rather, integrals, 
since the sums usually turn out to be integrals). It often turns out 
that the integrals are very difficult to evaluate, so that an immense 
amount of numerical integration is required to find the wave form. 

When the membrane is bounded, the difficulties encountered often 
become insuperable. For the greater generality of the problem of the 
membrane over that of the string is again apparent when we deal 
with the effect of boundary conditions. In the case of the string we 
needed only to specify boundary conditions at two points; but in the 
case of the membrane we must specify conditions all along a boundary 
line — and, in addition, must specify the shape of the line. A change 
in shape of the boundary line will have as much effect on the motion 
of the membrane as a change in the boundary conditions along the line. 

It turns out that the only way we can deal with the effect of 
boundary conditions is to solve the problem in coordinates suitable to 



176 



MEMBRANES AND PLATES 



[V.17 




the shape of the boundary: rectangular coordinates for a rectangular 
boundary, polar coordinates for a circular boundary, and so on. And 
if we cannot solve the wave equation in coordinates suitable to the 
boundary chosen, we shall not be able to obtain the numerical results 
that are necessary for our study of the vibrations. 

Reaction to a Concentrated Applied Force. — Another point of 
difference between the string and the membrane is in the reaction to 

an applied force. A string of 
length I pushed aside by a force 
concentrated at the point x has a 
form composed of the two line 
segments shown in Fig. 35. The 
shape is so arranged that the 
vertical components of the ten- 
sion at the point of application 
Th/x and Th/(l - x) add up to 
equal the force F. The displace- 
ment h = Fx(l - x)/Tl of the 
point of application is a finite 
amount and is proportional to F. 
The membrane, on the other 
hand, cannot support a force 
concentrated at a point, and the 
displacement of the point of appli- 
cation is infinite, no matter how 
small the force is. For instance, if the force is concentrated at the 
center of a circular membrane of radius a, the displacement 17 of a 
point a distance r from the center will be r\ — (2F/T) In (a/r), where 
the symbol In means "natural logarithm of." This expression is a 
solution of Eq. (17.2) for equilibrium conditions [i.e., when the right- 
hand side of (17.2) is zero]. It becomes zero at r = a and infinite 
at r = 0. 

This result means that the simplification of considering the force 
to be applied at a point, which is allowable in the case of the string, 
is too much of a simplification for the membrane. Actual forces are 
not applied at a point but along a length (for the string) or over an 
area (for the membrane), although the length or area can be small. 
The point of the foregoing discussion is that, while small changes in 
the length over which the force is applied make, very small changes in 
the shape of the distorted string, changes in the size of the area of 
application of a force to a membrane can make large changes in the 




Fig. 35. — Equilibrium shapes of a string 
and a circular membrane each acted on by 
a constant force concentrated on a small 
portion of the string and the membrane. 



v - 18 l THE RECTANGULAR MEMBRANE 177 

value of the maximum displacement of the membrane, so that we must 
take into account the size of this area. 

We shall find it best, however, to solve problems of forced motion 
where the force is concentrated in a small area by first solving the 
problem with the force concentrated at a point. This will result in 
a form for the membrane having a sharp peak of infinite height at the 
point of application. Then we shall cut off the top of the peak to a 
height such that the area of the top of the truncated peak is equal 
to the actual area of application of the force. This is shown in 
Fig. 35. 

This discussion shows that it is not efficient to drive a membrane 
by a force that is concentrated in a small area, for the amplitude of 
oscillation of the rest of the membrane will be very much smaller than 
that of the area of application. 

If the sheet of material has a certain amount of stiffness, then a 
force concentrated at a point will not produce an infinite displacement. 
But unless the stiffness is considerable, the displacement of the point 
of application will be considerably larger than the displacement of the 
rest of the sheet. 

18. THE RECTANGULAR MEMBRANE 
The wave equation in rectangular coordinates is 

w + w=7>dF' C = v^> (18.1) 

A solution of this equation is v = F(x - ct), just as for the string. 
It represents a wave moving with velocity c in the direction of the 
positive a>axis, with its crests parallel to the y-axis. Its shape is 
independent of the value of y, and the membrane behaves as if it were 
made up of an infinite number of strings, all parallel to the z-axis. 
Another solution is v = F(y - ct), having similar properties, except 
that the direction of travel is parallel to the y-asaa and the crests are 
parallel to the z-axis. A still more general form is 

V = F(x cos a + y sin a — ct) (18.2) 

which represents a parallel wave traveling with velocity c in a direction 
at an angle a to the z-axis. 

Combinations of Parallel Waves.— In all these cases the mem- 
brane behaves like an assemblage of flexible strings, and the analysis 
of the motion is the same as that given in Sec. 8. For instance, the 



178 



MEMBRANES AND PLATES 



[V.18 



reflection of such waves from a straight boundary along the z-axis is 
given by the solution 

7} = F(x cosa + y sina — ct) — F(x cosa — y sina — ct) 

which is always zero at y = 0. This shows that when the angle of 
incidence is a the angle of reflection is — a. 

As soon as two or more waves going in different directions are 
superposed, the membrane has to bend in more than one direction, 
and the corresponding motion indicates that the membrane is more 
than an assemblage of parallel strings. As we mentioned in the 
previous section, every possible sort of wave can be built up out of a 
suitable sum of simple waves. The mathematical form of this state- 
ment is that every possible motion of the membrane can be expressed 
in terms of the integral 

v = f 2 * F a (x cosa + y sina — ct) da (18.3) 

where F a can have a different form for each different direction a. 
The problem is to find the forms F a to correspond to a given motion 
of the membrane. This will be done for a few cases, later in this 

chapter. 

The simple harmonic solutions of Eq. (18.1) can be built up out 

of a sum of waves of the type 



A a cos 



'Ittv 



(x cosa + y sina — ct) 



over different values of a. This sum can be chosen so that it reduces 
to a set of standing waves suitable for fitting to rectangular boundaries: 

v = ^ i cos — (x cosa + y sina - ct) - $i 
\2ttv 



+ cos 

+ COS 
+ COS 



(x cosa — y sina — ct) — *2 

— (x cosa + y sina + ct) — $3 
c 



2irv 
c 



(x cosa - y sina + ct) - $2 - *s + *i|| 



= A cos fejx cos a - J cos (^ 2/ sin a - Q y J cos (frrvt - 
where Q t = K* a + *»), fi * = ^ _ * z) ' * = ^ 3 ~ * l) ' 



(18.4) 



V.18] THE RECTANGULAR MEMBRANE 179 

Separating the Wave Equation. — Another, and perhaps more 
straightforward, way of obtaining (18.4) is by separating the wave 
equation (18. 1). If we wish to fit rectangular boundaries, we must use 
a standing wave having a set of nodal lines (lines along which the dis- 
placement is zero) parallel to the x axis and another set parallel to 
the y axis. Now, the only way that we can have a nodal line parallel 
to the y axis {i.e., for i\ to be zero for a given value of x for all values 
of y) is to have a factor of rj which is a function of x only and which 
goes to zero at the value of x corresponding to the nodal line. The 
nodal lines parallel to the x axis would require a factor depending on 
y only. Finally, if the motion is simple harmonic, the time depend- 
ence must come in the factor e~ 2vivt . Therefore an expression for the 
form of the membrane satisfying our requirements must be 

r] = X(x)Y{y)e~ 2 * ivt 

where substitution in (18.1) shows that 

y^ = _±[V _ d*Y 

dx 2 c 2 dy 2 



or 

1 d 2 X = _ i_ 

X dx 2 Y 



/ 4*-V d 2 Y\ 

\ c 2 dy 2 / 



The left-hand side of this equation is a function of x only, whereas 
the right-hand side is a function of y only. Now, a function of y 
cannot equal a function of x for all values of x and y if both functions 
really vary with x and y, respectively; so that the only possible way 
for the equation to be true is for both sides to be independent of both 
x and y, i.e., to be a constant. Suppose that we call the constant 
— (4x 2 f 2 /c 2 ). Then the equation reduces to two simpler equations: 

dx 2 c 2 ' dy 2 c 2 K U 

The solution of this pair of equations is 

V = A cos( -~ x - Q x ) cosf — y - Qy\ cos (frtt VFT7 2 - 3>) 

(18.5) 

where we have set t = s/v 2 — f 2 . This solution is another way of 
writing (18.4), with f instead of v cos a and r instead of v sin a. 

The Normal Modes. — If the boundary conditions are that t\ must 
be zero along the edges of a rectangle composed of the x and y axes, 



180 



MEMBRANES AND PLATES 



[V.18 



the line x = o, and the line y = b, it is not difficult to see that fi* - 
8„ = tt/2, that (2f a/c) must equal an integer, and that (2t6/c) must 
equal an integer (not necessarily the same integer). The character- 
istic functions, giving the possible shapes of the membrane as it 
vibrates with simple harmonic motion, are 



rj = A\f/ 



(x,y) cos (2Tv m J — 3>) 
:\ • (nmy\ 

T XU \T) 



. (irmx 
(x,y) = sm^— 



(18.6) 



where 



Vmn 



=w!^/fe^w 



(m = 1, 2, 3 
(n = 1, 2, 3 




m = l n = l 




m=l n=2 m = 2 -^0=2 

Fig. 36.— Shapes of the first four normal modes of a rectangular membrane. Arrows 

point to the nodal lines. 

The shapes of the first four normal modes are shown in Fig. 36. We 
notice that the number of nodal lines parallel to the y axis is (m - 1) 
and the number parallel to the x axis (n - 1). 

The Allowed Frequencies.— The fundamental frequency is vu, 
depending on T and a in a manner quite analogous to the case of the 
string. Among the allowed frequencies are all the harmonics of the 
fundamental, v 22 = 2vn, v 33 = 3vu, etc. But there are many more 
allowed frequencies which are not harmonics. When a is nearly 
equal to b, 2 extra overtones, v u and v 21 , come in between the first and 
second harmonic; 6 extra, v 13 , v 3h v 23 , v S2 , v u , v A i, all come between the 
second and third; 10 extra come between the third and fourth; and 
so on. No matter what the ratio between a and b is, it is possible to 
show that the average number of overtones between the nth and the 
(n + l)st harmonics is hm(a 2 + b 2 )/ab, or that the average number 



V.18] 



THE RECTANGULAR MEMBRANE 



181 



of allowed frequencies in the frequency range between v and v + Av 
is (2rvab/c 2 )Av. 

In the case of the string the allowed frequencies are equally spaced 
along the frequency scale, but in the case of the membrane the allowed 
frequencies get closer and closer together the higher the pitch. The 
higher the pitch, the more overtones there are in a range of frequency 
of a given size. This property is true of all membranes, no matter 
what shape their boundary has. It can be shown, for any membrane, 
that the average number of allowed frequencies between v and v + A? 
is (2tv/c 2 )Av times the area of the membrane. We shall show in 
Sec. 32 how this can be proved. 




X ^3Z^ y^T-V 

Fig. 37. — Various modes of simple harmonic vibration of a square membrane, for the 
degenerate cases v\% = vi\ and v™ — vz\. Arrows point to the nodal lines. 

The Degenerate Case. — An interesting phenomenon occurs when 
the rectangular membrane is a square one {i.e., a = b), for then the 
allowed frequencies become equal in pairs, v mn being equal to v nm . 
There are fewer different allowed frequencies, but there are just as 
many characteristic functions as there are when a is not equal to 6. 
This is called a condition of degeneracy. There are two different 
functions \l/ mn and \l/ nm , each corresponding to the same frequency 
(except for the cases n = m, which are not degenerate). In such 
cases the membrane can vibrate with simple harmonic motion of fre- 
quency v mn with any one of an infinite number of different shapes, 
corresponding to the different values of y in the combination 

rpmn COST + ypnm SU1 J 



182 



MEMBRANES AND PLATES 



[V.18 



Figure 37 shows the shapes of the normal modes of vibration of 
the square membrane corresponding to v 12 and v 13 , for different values 
of 7. The vibrations can be standing waves, corresponding to 

V = {i>mn cos 7 + \f/ nm sin 7) cos (2rvmJ) 

The nodal lines have a different shape for each different value of 7. 
It is also possible to have traveling waves, corresponding to the 
expression 

V — bPmn cos {2irv mn t) + ^ nm sin (2irv m J)] 




Fig. 38. — Successive shapes of a rectangular membrane struck at its center. Times 
are given in terms of fractions of the fundamental period of vibration. 

In this case the nodal lines go through the whole range of possible 
shapes during each cycle. It is only in degenerate cases that it is 
possible to have traveling waves of simple harmonic motion in a mem- 
brane of finite size. 

The Characteristic Functions. — The characteristic functions for 
the rectangular membrane have the following integral properties: 



XT*- 



n&m'n' dx dy = 



fO 

{(ab/4) 



(rn' = to 



or n 7 
and n' 



n) 



n) 



(18.7) 



V.19] THE CIRCULAR MEMBRANE 183 

The behavior of a membrane having an initial shape y]o(x,y) and an 
initial velocity v (x,y) is therefore 



co oo 

V = 2 X ^ n {x,y)[B m n aos{2icv mn t) + C mn sin (2irv mn t)] (18.8) 

m= 1 n— 1 
#m» = -T I I l/O^nm dc <%, Cmn = ft& J I V<$ mn dx dy 

This expression is the correct one for both the degenerate and the non- 
degenerate cases. 

Figure 38 shows the motion of a membrane that has been struck 
so that a small area around its center is started downward at t = 0. 
In this case r?o = 0, and v is zero except near the point x = a/2, 
y = 6/2, where it has a large enough value so that jjv dx dy = U. 
The integrations for the coefficients C mn become simple, and the result- 
ing expression for 17 



2U 

v = 



^ 7~~ tmnl g' 2) ^ mn ( X,V ^ Sln ( 2TV «» < ) 



xa6 

The series does not converge if m and n both run to infinity, a corollary 
of the fact that a concentrated force produces an infinite displacement 
of the membrane. However, if the series does not continue for m 
larger than some number M or for n larger than N, then there will be 
only a finite number of terms to be added together, and the result 
will never be infinite. Such a finite series corresponds approximately 
to a starting area of dimensions (a/M) by (b/N). 

The case shown in Fig. 38 is f or M = N = 10. The initial shape 
and the shapes for successive eighths of the fundamental cycle are 
shown. We see the initial pulse spread out and then reflect back to 
pile up in the center in a "splash." Since not all the overtones are 
harmonics, the motion is not periodic. We notice that the shape 
of the pulse changes as it spreads out to the edge. More will be said 
about this change of shape in the next section. 

19. THE CIRCULAR MEMBRANE 

Rectangular coordinates are useful to describe parallel waves and 
normal modes for a rectangular boundary, but for the discussion of 
circular waves and for the study of the normal modes for a circular 
boundary we shall find it easiest to use polar coordinates. In this 
study we shall encounter more forcibly than before the essential 



184 



MEMBRANES AND PLATES 



[V.19 



differences between wave motion on a string and that on a membrane. 
To bring out these differences we start with a discussion of the general 
vibrations of a membrane of infinite extent. 

Wave Motion on an Infinite Membrane. — Let the initial shape and 
velocity shape of the infinite membrane be vo(x,y) and v (x,y). By 
letting a and b go to infinity in Eq. (18.8), we find that we can express 
the subsequent shape of the membrane by the Fourier integral [see 
Eq. (2.20)]. 

v(x,y,t) = -j I I d$ dr e ( 2 ™/ c ><r*+^> 

C J — 00 J — 00 

[/» 00 /» 00 

cos (2xVr 2 + t 2 I I da' dy' rj (x' ,y')e-^ i/c ^^'+^ 

2xVr 2 + r 2 J--J- 



+ 



da;' dy 7 v (z',2/')e- (2iri/c)(f *' +TI/ ' ) 



By a series of transformations too involved to discuss here, this 

integral can be transformed into the 
following simpler and much more in- 
teresting form : 




v(x,y,t) = ^ 

RdRdd 
y/cH* - R 2 



* c l dt Jo Jo 



J*ct flit 
Vo(x',y') 
o Jo 
RdRdd 



VcH* 



ld& I 



(19.1) 



r 



-ct 



~r 



-ct- 



Fig. 39. — Dependence of the wave 
motion of a string and membrane on 
initial conditions. The displacement 
of the point (x,v) on the membrane at 
the time t depends on the initial con- 
ditions for that part of the membrane 
enclosed by a circle of radius ct with 
center (x,y). The displacement of 
point x on the string depends on the 
initial conditions for the portion 
within a distance ct on either side of x. 

, Impermanence of the Waves, 

corresponding expression for the 



where R is the distance between the 
point (x,y) and the point (x',y'), as 
shown in Fig. 39. 

Equation (19.1) shows that the 
displacement of the membrane at the 
point (x,y) at the time t depends on 
the shape and velocity at t = of 
all those parts of the membrane with- 
in a circle about (x,y) of radius ct. 
The area that affects the displace- 
ment r](x,y,t) spreads out as t in- 
creases. 

, — Let us compare this result with the 
infinite string given in Eq. (8.3): 



V.19I THE CIRCULAR MEMBRANE 185 

y(x,t) = ^ \j t I Vo(x') dx' + I v (x') dx' 

where we have expressed the equation in a form like (19.1). The 
esse ntial differ ence between this expression and (19.1) is the factor 
l/\/c 2 t 2 — R 2 in the integral for the membrane, which is missing in 




Fig. 40. — Comparison between the behavior of a plucked string and a plucked 
membrane. The first sketches show the initial shapes; the lower ones the shapes at 
successive instants later. One quarter of the membrane has been cut away to show 
the shape of the cross section. 

the integral for the string. That the difference is important can be 
seen by the following argument: Suppose that the initial disturbance 
on the string is confined to a small range, from a; = X-Atoa; = 
X + A, y and v being zero elsewhere. Then, by the preceding 
equation, the string at the point x (we assume that x is smaller than 
X — A, to make the discussion less verbose) will be undisturbed as 
long as ct is less than (X — x — A). The wave will reach the point x at 
t = (l/c)(Z - x - A), and from this time to t = (l/c)(X - x + A) 



186 



MEMBRANES AND PLATES 



[V.19 



the displacement at x will be changing. However, from the time 
t = (l/c)(X — x + A) on, the string at the point x will again be undis- 
turbed. This corresponds to the fact that the wave, as it travels, does 
not change its shape; the pulse, as it passes the point x, leaves no 
"wake" trailing behind it. 

For comparison, let us suppose that the initial disturbance of the 
membrane is confined to a small circular area of radius A, a distance 
D away from the point (x, y). As with the string, no disturbance will 




Fig. 41. — Comparison between the behavior of a struck string and struck membrane. 
The initial displacements are zero, and the center portions of both string and membrane 
are given an initial upward velocity. 

occur at (x, y) as long as ct is less than (D — A). The wave will reach 
(x, y) at t = (l/c)(D — A), and from this time until t = (l/c)(Z> + A) 
the displacement at (x,y) will change. However, unlike the string, 
the displacement at {x, y) will continue to change after t = (l/c)(Z> + A), 
owing to the factor l/\^c 2 t 2 — R 2 in the integral expression. The 
pulse, as it spreads out from its source, leaves a wake trailing behind 
it. The wave therefore differs from the waves on a string and also 
differs from waves on a bar, where ripples travel ahead of the "crest." 
Figures 40 and 41 illustrate this property of the membrane and 



V.19] THE CIRCULAR MEMBRANE 187 

contrast it to that of the string. Figure 40 gives the shapes of the 
membrane at successive instants, after it has been pulled out over a 
small area and let go at t = 0. Figure 41 shows the resulting motion 
when the central area in the first sketch is given an upward velocity 
at t = 0. The corresponding motions of a string with an equal value 
of c are also shown. In each case, the outermost part of the wave 
on the membrane keeps its shape as it moves outward, being similar 
to the shape of the corresponding part of the wave on the string. The 
rest of the membrane, however, changes shape as it moves, more and 
more of the crest being left behind. The initial conditions for these 
two figures have been chosen to have very exaggerated forms, just 
in order to show the above properties as clearly as possible. 

Simple Harmonic Waves. — Now let us turn to the circular waves 
that vibrate with simple harmonic motion. As in the rectangular 
case, we separate the wave equation. In polar coordinates the equa- 
tion is 



r dr\ dr/ 



r 2 d<p 2 c 2 dt 2 



and by setting rj = R(r)${<p)e~ 2vivt we obtain the equations for R 
and <£: 

d 2 $ 

—— = — /z 2 <£>, <£ = cosO<p) or sin(jit^) (19.2) 

a<p 

W + r dF + V^" "" T 2 ) R ~ ° (19 - 3) 

For these circular waves one "boundary condition" is required 
even before the shape of the boundary line is decided upon. The 
requirement is simply that the displacement rj be a single valued 
function of position; for the coordinate <p is a periodic one, repeating 
itself after an angle %r, and we must have rj(r, <p) equal to r\(r, <p + 2r). 
This restricts the allowed values of /x to integers: 

$ em = cos (m<p), $om = sin (m<p) (m = 0, 1, 2, 3 • • • ) 

This is not true for a membrane whose boundary is shaped like a 
sector of a circle, where <p cannot go from zero clear around to 27r, but 
such cases are of no great practical importance. 

The foregoing requirement is the third different type of boundary 
condition that we have encountered, the first type being the fixing 
of the displacement or its slope (or both) at some point or along some 
line (as with the string, the bar, and the rectangular membrane), the 



188 MEMBRANES AND PLATES [V.19 

second being simply that the displacement have no infinite values in 
the range of interest (as with the whirled string). The third, the 
condition of periodicity, will be used whenever any of the coordinates 
are angles that repeat themselves. 

Bessel Functions. — Equation (19.3) for the radial factor is Bessel's 
equation. We have solved it for n = 1 in Sec. 2 and for n = in 
Prob. 3 of Chap. I. The general solution J m (z), where z = (2wvr/c) 
and where m is an integer, has the following properties : 

Jm(z) = m\ \$) * (m+1)! W 

+ I . (*) m+ * _ . . . \ (19.4) 

^2!(m + 2)!\2/ / v 

t ,\ 1 2" ( 2m + 1 \ 

i C 2w 

Jm(z) = 7T~^ I e lzcoBW cos (mw) dw 
2iri m Jo 

Jm-l(z) + J m +l(z) = — J m (z) 

z 

-T- Jm(z) = i [J m -\(Z) - J m +l(z)] 

^[z m J m (z)] = z m J m ^(z), ^z- m J m (z) = -z- m J m+1 (z) 



I Ji(z) dz = —J (z), I zJ (z) dz = zJx(z) 



I 



s 



I 



Jl{z)zdz = Z ^[Jl(z)+J\(z)] 



Ji(z)z dz = 2 [Jl(z) - J m -i(z)J m+ i(z)] 



2 

J m (az)J m (pz)z dz = — — % [pJ m (az)J m _ 1 (fiz) 



2 _ R2 

- aJ m (pz)J m -i(az)] 



(19.5) 



All these properties are proved in books on Bessel functions. Values 
of Jo, J i, and J 2 are given in Table V at the back of this book. 

The function J m {2irvr/c) is not the only solution of Eq. (19.3), for 
it is a second-order equation, and there must be a second solution. 
This other solution, however, becomes infinite at r = and so is of 



V.19] THE CIRCULAR MEMBRANE 189 

no interest to us at present. It will be discussed in the next section, in 
connection with forced vibrations. 

The Allowed Frequencies. — Coming back to our problem, we can 
now say that a simple harmonic solution of the wave equation which 
is finite over the range from r = to r = <», which is single valued 
over the range from <p = to <p = 2t, is 



- cc)J m (^J 



r] = cosm(<p — a)J m [ J cos(2-n-*Z — Q) 

If the membrane is fastened along a boundary circle of radius a, the 
allowed frequencies must be those that make J m (2Tva/c) = 0. For 
each value of to there will be a whole sequence of solutions. We shall 
label the allowed values of the frequency v mn , so that *oi, *02, *oi, etc., 
are the solutions of J (2Tva/c) = 0; *n, * X2 , *i3, etc., are the solutions 
of J i(2t va/c) = 0; and so on. The values of * OTn are given by the 
equations 

Vmn =£-0mn, 001 = 0.7655, 002 = 1.7571, 003 = 2.7546 • • • 



2a 



0n = 1.2197, 12 = 2.2330, 0i 3 = 3.2383 
02i = 1.6347, 022 = 2.6793, 23 = 3.6987 



0m« ^ n + -7T- — t, if n is large 



(19.6) 



The frequency p i is the fundamental. Another way of writing these 
results is 

voi = 0.38274 ij^, vn = 1.5933poi, *2i = 2.1355*01 I ( 19>7 ) 

* 02 = 2.2954*oi, *3i = 2.6531*oi, v 12 = 2.9173* i • • • ) 

It is to be noticed that none of these overtones is harmonic. 

The Characteristic Functions. — Corresponding to the frequency 
*o« is the characteristic function 

Wr, fp) = j(~f^j (19.8) 

and corresponding to the frequency v mn (to > 0) are the two charac- 
teristic functions 

*«- = cos (m<p)J Jfe^J, tfw = sin (m<p) J m (^f\ (19.9) 



190 



MEMBRANES AND PLATES 



[V.19 



Except f or m = the normal modes are degenerate, there being two 
characteristic functions for each frequency. The shapes of a few of 
the normal modes are shown in Fig. 42. We notice that the (ra, n)th 
characteristic function has m diametrical nodal lines and (n - 1) 
circular nodes. 





where 



Fig. 42. — Shapes of some of the normal modes of vibration of the circular membrane. 
Arrows point to the nodal lines. 

The integral properties of these functions are obtained from 
Eqs. (19.5): 

|T ° f ^ temntem'n-r dr d V = Jq Jq' ^Omnlfc>»V T dr d<p 

( (m 7^ m' or n ?± n') 

= \ra 2 A mn (m = m! and n = n') (19.10) 

A „ = [J"i(ir/3 OTC )] 2 , A TO „ = i[J TO _i(x/3 m „)] 2 (m > 0) 
The values of the constants A mn can be computed from the following 
values of J m : 

JfrM = +0.5191, Ji(tM = -0.3403, 

Ji(7t/3o 3 ) = +0.2715, JiOtfoO = -0.2325 
JoOnSii) = -0.4028, JoOnM = +0.3001, 

JoWh) = -0.2497, JoOtfu) = +0.2184 
/iCir/Sai) = -0.3397, J r i(x/3 22 ) = +0.2714, 

J^&s) = -0.2324, J" 1 (7r i 824) = +0.2066 

With the values of the constants A known, it is possible to com- 
pute the behavior of a circular membrane when started with the 



V.19] THE CIRCULAR MEMBRANE 191 

initial shape vo(r,(p) and the initial velocity v (r,<p). By methods 
that we have used many times before, we can show that this behavior 
is governed by the series 



00 / 00 

n= 1 v m = 



where 



[Bemn COS (2lTV mn t) + C emn Sm (2lTV mn t)\ 

+ J) tom n [B 0mn cos(2nrv mn t) + C 0m „ sin (2rv mn t)] > (19.11) 

TO = 1 / 

1 f ° f ^ 

■B emn = —-T- — I I r)Q\p emn r dr d<p 

Td'Amn Jo Jo 

i /»a /»2tt 

C«mn = o~i — ^2l — I I V($ emn r dr d<p 
4ir z v mn a 2 A mn Jo Jo 

A similar set of equations holds for B Qmn and C Qmn . 

Relation between Parallel and Circular Waves. — In Sec. 18 we 
stated that a circular wave could be built up out of a series (or rather 
an integral) of different parallel waves, and we set up a general expres- 
sion (18.3) for the form of the integral. We must now verify our 
statements, by finding the form of F a which is to be used in (18.3) to 
give circular waves. If the waves are to be simple harmonic, and if 
we decide to make the dependence of F on a also periodic, Eq. (18.3) 
becomes 

it = I cos (ma) e (25riy/c) ( * °°* a+v Bin a-ct) da 

Since x — rcos<p and y = rsin^>, a; cos a + ysina becomes r(cosa 
co8<p + sin a sin <p) = r cos (a — <p). Changing to the variable = 
a — <p, the integral becomes 

/»2* 

V = e- 2Tiyt I [cos (m<p) cos(m/3) — sin(m^) sin (m(3)]e 2irivr <*» v* dp 

The term with sin (ra/3) integrates to zero, but the term with cos (m$) 
can be transformed, by the aid of the last of Eqs. (19.4), into 

v = e -2ri« C0S ( m(p )(2Tl m ) J J— J 

Therefore the expression for a circular wave in terms of parallel waves 
is 



192 MEMBRANES AND PLATES [V.19 

cos (m<p) Jrn\^~) e~ 2 * M 



\2Tri m ) Jo 



2ir 

cos (ma)e (2ri ' /c) ( * C03 a+y sia a - ct) da. (19. 12) 



There is an equivalent equation with sin (m<p) and sin (ma) in place 
of the respective cosines. 

Similarly, it is possible to express the simple harmonic parallel 
wave e (2«»/c)o- c «) _ e (2wiv/c) (r cos a-ct) m terms of a series of circular 

waves SA m cos (m<p)J m (2Trvr/c)e~ 2vi '' t . We use the series with the 
cosine terms only, because the parallel-wave function is symmetrical 
in <p; a change in the sign of <p will not change the value of the expo- 
nential. Now, from the point of view of the dependence on <p, the 
series of circular functions is a Fourier series, and by the Usual methods 
of Sec. 9 we obtain 

00 

e (8»ir/e>(roo.r-eD — Q-^ivi V C m {r) COS (nip) 
m = Q 

where 



C °-2* 



i c 2 ' i r 2 " 

— I e iMwr/e)m *da, C m = - I e (2 " v/c)ooa ^cos(m^) dtp 
wr Jo t Jo 



(m > 0) 



From the last of Eqs. (19.4) we see that the first integral is just 
Jo(2irvr/c) and the second 2i m J m {2irvr/c). Therefore, the series of 
circular waves that build up into a parallel wave is 



e (2 r i,/c)(x-ct) — | J | ~ n " r , j 



+ ^2i m cos (m<p) J m r^ J e- 2 ™' (19.13) 

m = 1 \ / J 

The real part of this series is 

2p'(*-e*)] 

= W^p) -2coa(2<p)jjfe^-\ + • • • cos(2rvt) 



+ [ 2 cos (<p)Jr(^) ~ 2 cos (3*)J 3 (^) 



+ 



sin (2tt^) 



V.19] THE CIRCULAR MEMBRANE 193 

The Kettledrum. — It sometimes happens that a circular membrane 
is stretched over one end of a vessel that is airtight. This is the case 
for the kettledrum and for some types of condenser microphones. 
Here the tension is not the only restoring force, for the motion of the 
membrane alternately compresses and expands the air in the vessel, 
and this reacts back on the membrane, changing its natural frequencies 
and its general behavior. If the diaphragm is very light and the 
tension is extremely large, the speed of wave motion in the membrane 
will be as large as the speed of sound in air, or larger; and the problem 
would require for its solution all the techniques that will be developed 
in Chap. VII. If the velocity of transverse waves in the membrane 
is considerably less than the speed of sound, however, the problem 
is much simpler, for then the compression and expansion of the air in 
the vessel is more or less the same over the whole extent of the mem- 
brane and will depend on the average displacement of the membrane. 

When the membrane is displaced from equilibrium to the shape 
expressed by the function -q, the volume of the vessel is diminished 

by an amount I I rjr dr d<p. If the equilibrium volume inside 

the vessel is TV and the equilibrium density of the air is p , then when 
the alternations of pressure are rapid enough to be adiabatic changes, 
the excess pressure inside the vessel will be 

p = — I fj^r? J \ I rjr dr d<p dynes/sq cm 

where c a is the velocity of sound waves in air at the equilibrium pres- 
sure and temperature in the vessel. At normal pressures and tem- 
peratures (temperature 20°C, pressure 760 mm of mercury) in air 
the value of p cl is 1.44 X 10 6 ergs per cc. This expression for the 
excess pressure will be proved in the next chapter. The pressure is 
given with a negative sign because it is always in the direction opposite 
to that of the average displacement of the membrane. 
The equation of motion for the membrane is therefore 

and the equation for simple harmonic vibrations is 

V = Y(r,(p)e- 2vivt 



_2 7 ,2 / -»\ l' a l* 2r 



\Vot) Jo Jo 



V*Y + ^Y=\$%)\ I Yrdrd<p (19.15) 



194 



MEMBRANES AND PLATES 



[V.19 



If Y = c ? 8 (m<p) J m (2rvr/c) (m > 0), the integral on the right-hand 



sin 



side of this equation will be zero (owing to the integration over <p), 
and the solution that satisfies the boundary conditions Y = at 
r = a will be the characteristic functions given in Eqs. (19.9), with 
the corresponding allowed frequencies. The presence of the airtight 
Vessel, therefore, has no effect on the normal modes of vibration which 
have one or more diametrical nodal lines {i.e., which have a factor 
cosine or sine of (m<p), where m is not zero). 

For the case where m = the integral on the right is not zero. 
Since the solution of the equation without the integral is J (2rvr/c), 
we try the function Y = J (2Tvr/c) - J (2Tvva/c), which satisfies 
the boundary condition Y = at r = a. The integral then reduces to 

where x = (irpocla 4 /VoT). Inserting this into Eq. (19.15) and utiliz- 
ing Eqs. (19.5) results in the equation 



Jo(w) = — 2 
■ vr 



J (w) Ji(w) 

w = (2irva/c) 



) 



-4w 

w 2 



(19.16) 



which determines the allowed values of the frequency for those normal 
modes that are independent of <p (i.e., have no diametrical nodal lines). 





Table 4. — Frequencies op the Kettledrum 




X 


Yoi 


702 


703 


704 





0.7655 


1.7571 


2.7546 


3.7534 


0.5 


0.7880 


1.7590 


2.7550 


3.7535 


1 


0.8097 


1.7610 


2.7555 


3.7537 


2 


0.8510 


1.7651 


2.7566 


3.7541 


3 


0.8899 


1.7694 


2.7576 


3.7545 


4 


0.9265 


1.7739 


2.7587 


3.7549 


5 


0.9604 


1.7787 


2.7598 


3.7553 


6 


0.9914 


1.7837 


2.7609 


3.7557 


8 


1.0445 


1.7945 


2.7632 


3.7565 


10 


1.1101 


1.8065 


2.7657 


3.7573 



Frequencies of the symmetrical normal modes of vibration of a circular mem- 
brane closing an airtight vessel, as functions of x the effective restoring force of the 
enclosed air. The nonsymmetrical modes (to > 0) are all independent of x- The 
frequencies in cps are given by the formula v = (y on c/2a). 



V.20] FORCED MOTION. THE CONDENSER MICROPHONE 195 

The constant x is a measure of the relative importance .-of the air 
confined in the vessel with respect to the tension, as a restoring force 
on the membrane. It is small if the tension is large or if the volume 
of the vessel is large. The limiting case x = is the one studied 
earlier, and the allowed frequencies are given in Eqs. (19.6) and (19.7). 
The allowed values v 0n of the frequency [or, rather, of y 0n = (2av 0n /c)] 
for other values of x are given in Table 4. This table shows that the 
presence of the vessel tends to raise the values of the allowed fre- 
quencies v 0n . 

The Allowed Frequencies. — The allowed frequencies and corre- 
sponding characteristic functions for the membrane plus vessel are 
therefore 

^eOn = Joi T 2- J — Jo(iryOn) \ 

f emn = CO s(mv)J m (?^y ^ n = Bm{m^) jj^f^\ (19.17) 

*--(&)$ --fe)^ (W>0) ) 

where the values of y 0n are given in Table 4 and the values of /3 m „ 
are given in Eq. (19.6). An approximate formula for y „, valid for 
small values of x, is 7o« ^ jSo« + (2xAr 4 /3jjJ, and the corresponding 
approximate formula for the allowed frequencies is 



v 0n c* (vji.) [l + o_ 3T 71 a 2 ;..o ^4 I (19.18) 



8T*Vo<rK4ny 



where v^ n is the value of v 0n when x is zero, given in Eq. (19.7). 

A series analogous to that of Eq. (19.11) can be set up to represent 
the motion of the membrane started in any manner, by the use of 
the integral properties of the characteristic functions given in Eq. 
(19.10) and in the following equation: 



j: 



tf'eontfwr dr = < 2 |_ (tt7o«) 2 




{n f = n) 

(n' t* n) 

(19.19) 



20. FORCED MOTION. THE CONDENSER MICROPHONE 



So far, we have been dealing with circular solutions of the wave 
equation which are finite everywhere. When we have to deal with 
circular waves sent out by a simple harmonic driving force concen- 



196 MEMBRANES AND PLATES [V.20 

trated at a point, we must expect to use solutions that become infinite 
at the point of application of the force (which we set at r = 0). These 
solutions are the other solutions of BessePs equation (19.3), for this 
equation is a second-order differential equation and must have two 
independent solutions. 

Neumann Functions. — The second solutions of Bessel's equation 
are called Neumann functions and are given the symbol N m (z). They 
have the following properties : 

I d ( dN m \ , ( \ m 2 \ 

N (z) ^ (-) In (0.890536z) = (?) Qnz - 0.11593) \ (20.1) 

Ar , . (m - 1)1 /2\ w , ^ m 
N m (z) -> - ^ - ( - 1 (m > 0) 

z-»0 7T \Z / 



o 



N m -i{z)J m (z) - N m (z)J m _!(z) = I — 

The properties given in Eqs. (19.5) for the functions J m (z) are also 
true for the corresponding functions N m (z). 

The function representing a circular outgoing wave caused by 
a force Fe~ 2vivi concentrated at r = is 



'-{$[<*?) + <*r). 



(20.2) 



From Eqs. (19.4) and (20.1) we can show that when r is very small 
t? becomes very large, having the value — (2F/T)\nr er 2rirt , which 
it must have in order to balance the force, as we showed on page 
176. When r is very large, r\ will approach in value the function 

{{ J / — e (2«i-/c)(r-c«)-tv/4 j representing a circular simple harmonic 

wave spreading outward with velocity c. The real part of this 
function is 

-~- J ln(r) cos (2-irvt) 



r->0 

^(l)V^ cos 



*Z lr -cO+l 



V.20] FORCED MOTION. THE CONDENSER MICROPHONE 197 

Although the motion of the whole wave with time is sinusoidal, 
only the part of the wave at large distances from the driving force 
approaches a sinusoidal dependence on r, and even at these distances 
the amplitude of the waves diminishes with increasing r, owing to 
the factor y/c/vr. Note, also, the phase lag (tt/4) of the outlying 
wave behind the driving force, which is to be compared with the lag 
of (ir/2) for the string discussed on page 92. These properties again 
emphasize the difference between one- and two-dimensional waves. 

Unloaded Membrane, Any Force. — The coupling between a mem- 
brane and the surrounding medium is generally much more effective 
than is the corresponding coupling for a string. Here the vibrating 
system is a surface, which must move a sizable portion of the medium 
every time it moves, whereas strings are usually thin enough to avoid 
disturbing the medium much by their motion. Consequently, a 
calculation of the free and forced motions of a membrane in vacuum 
is not so satisfactory an approximation to its behavior in air or water 
as are the analogous formulas for a string. 

The calculation of the effect of the medium on the motion of the 
membrane is correspondingly more difficult than is the case for the 
string, where the effects are always small and can be treated as per- 
turbations. Only when the membrane is heavy and the medium 
light are the effects of the medium small enough to treat successfully 
by the methods of Sees. 10 and 12. This case will be considered first, 
however, since it is more straightforward. 

As a matter of fact, we shall start with the simplest case, where 
the effects of the medium are negligible, even though this is not often 
applicable. By methods similar to those used in deriving Eq. (10.16), 
we obtain an expression for the steady-state motion of a membrane 
under the influence of a distributed force F(u ) v)e~ iat per unit area at 
(u,v) (or r,<p) : 



F(u,v)\f/ mn (u,v) du dv 



«L — O) 2 



ba ^J A mn IJ J 

m,n 

rc . , , , /0 (m! 9^ m or n' 9^ n) 

II tmntmV du dv = < _ ' 

\*3*-*-<mTl \ffl/ lib ctllLL lb — ft 



(20.3) 



where the integration is carried out over the surface of the membrane. 
The functions \f/ mn are given by Eq. (18.6) for the rectangular mem- 
brane and by Eq. (19.9) for the circular membrane; A mn is given by 
Eq. (18.7) (i.e., it equals -J-) or by Eq. (19.10); and w mn = %rv mn is 
given by Eq. (18.6) or (19.6). S is the area of the membrane and 
S<r its total mass. 



198 MEMBRANES AND PLATES [V.20 

When the force is concentrated near the point (u,v) and its total 
value is Pe~ iat , this series becomes 

SO- ^J K m n 0>l n - CO 2 

mn mn 

When the frequency is zero, the sum gives the shape of the membrane 
when pushed by a steady force P concentrated at the point (u,v). 

m = ^Y(u,v;x,y); Y = ^ ^ n M^ n (x,y) 

The function Y goes to zero at the boundary of the membrane and 
approaches the function - (Sa/^T) \n[(x — u) 2 + {y — v) 2 ] when 
(x,y) is close to (u,v). Using Y, we can reexpress the series for the 
steady-state driven motion due to a concentrated force 



" = s 



[ Y(u,v, x,y) + ^ jr *%-J? r~ ( 20 ' 4 > 

L mn -■ 



The dependence of the shape on frequency is given by the second 
sum, which converges much more rapidly than the sum written first. 

Figure 43 shows the shapes of a square membrane driven by a 
force concentrated near its center, for different values of the driving 
frequency. The resonance frequencies are the fundamental ^n ajid 
the odd-numbered overtones pi 3 = vu v5, vw = 3^n, vn = vu vl3, 
etc. The even-numbered overtones do not appear, since the corre- 
sponding characteristic functions have a node at the mid-point of the 
membrane and the terms in the sum vanish. 

The nodal lines are shown in the figure by dotted lines. These 
change their shape, spreading outward as v is increased, a new node 
being introduced, near the center, after each resonance frequency has 
been exceeded. 

Localized Loading, Any Force. — When we begin to consider the 
effects of the medium surrounding the membrane, we must first ask 
whether the wave motion in the medium is faster or slower than trans- 
verse waves in the membrane. If the wave velocity in the medium 
is much slower than that in the membrane, then the reaction of the 
medium on any given portion of the membrane depends entirely on 
the motion of that part of the membrane; for the different parts of 
the medium are relatively slow in letting each other know what 
motions they are undergoing. 



V.20] FORCED MOTION. THE CONDENSER MICROPHONE 199 



This case is seldom met in practice, but the calculations of driven 
motion are not difficult. The load of the medium is expressible in 
terms of a resistive term R per unit area and an additional reactive 
load that can be added to the mass per unit area of the membrane to 
give an effective mass <r e . Both R and <x e may vary with the driving 
frequency; however, when the wave velocity of the medium is very 
much less than that for the membrane, R is approximately constant, 
equal to the product of the density of the medium with the wave 
velocity in the medium, (pc a ), and the additional mass loading becomes 
negligible. 




Fig. 43. — Shapes of a square membrane driven by a force of frequency v concen- 
trated on a small area near the mid-point of the membrane. The resonance fre- 
quencies are the fundamental vu, P13 = 2.2361yn, v\z = 3vu, etc. At these frequencies 
the amplitude will be infinite, since friction has been neglected. The nodal lines are 
indicated by the dotted lines. 

The equation of motion for a distributed driving force is 



dt 



dri 



t =TVh,-R^+ F{x,y)e- 



We substitute the usual series of characteristic functions, appropriate 
to the boundary, for 17 and eventually obtain the familiar equation 
for the steady-state motion: 






lOiZn 



(20.5) 



where 



fmn = ( or — ) I I F(x,y)\p mn dx dy over the membrane 



200 MEMBRANES AND PLATES [V.20 

and where 

(2nrv mn ) 2 <r e 



= R — t\ OXTe 



(for localized loading) 



is the effective impedance of the (m,n)th mode for transverse motion 
of the membrane in the medium (at the frequency <a/2ir). As before, 
the characteristic functions ^ m „ are given by Eq. (18.6) or (19.8). 

The methods discussed in Sec. 10 can be used to compute the 
transient motion of the membrane. No further" details need be dis- 
cussed here. 

Uniform Loading, Uniform Force. — The other extreme, where the 
wave velocity in the medium is much greater than that of the mem- 
brane, is more often encountered but is more difficult to solve. Here 
the effect of the motion of one part of the surface is transmitted rapidly 
through the medium to affect the other parts, so that in the limit the 
reaction of the medium is uniform over the membrane, proportional 
to its average displacement. This limiting case has already been dis- 
cussed in connection with the kettledrum behavior. Here we wish 
to discuss the driven motion. 

The case of greatest practical interest is that of the circular dia- 
phragm, which is related to the problem of the condenser microphone. 
The constructional details of the microphone will be discussed some- 
what later. All that is necessary to know here is that the driving 
force due to an incident sound wave is Fe~ i01t per unit area, approxi- 
mately uniform over the membrane, and that the reaction force per 
unit area of the medium, on both sides of the membrane, is propor- 
tional to the average displacement of the membrane, the proportionality 
factor being —iuz, where z is the effective specific acoustic impedance 
of the medium (counting both sides of the membrane). 

The real part of z, R, is composed in part of the radiation resistance 
of the medium next to the outer part of the membrane. Formulas 
for the dependence of this part of R on frequency and membrane size 
are worked out in Chap. VII. The side facing the inside of the 
microphone case may also have a resistive part in its reaction, par- 
ticularly if the case is pierced with small holes to equalize the pres- 
sure inside and out. Motion of the air through these holes produces 
viscous friction. The reactive part of the impedance z due to the 
outer air is masslike (i.e., X is positive), whereas the reaction of the 
air inside the casing is usually stiffness controlled, as was the case 
with the kettledrum. 



V.20] FORCED MOTION. THE CONDENSER MICROPHONE 201 



The equation of motion of the membrane is therefore 

d 2 7} 



dt 2 



= TV 2 v + iazv + Fer* 1 



(20.6) 



where 



' = («*) I dv i nTdr 

By analogy with Eq. (10.17) we set down the following expression 

'-4 / {t)-'{t)]'~ ^ 

for the steady-state motion of the membrane. This expression goes 
to zero at the edge of the diaphragm (r = a). The average displace- 
ment rj can be found by using the integral formulas and the recursion 
formulas given in Eq. (19.5) : 



= AJ 



(?) 



Setting these expressions in Eq. (20.6) serves to determine the value 
of the constant A 



A = 



(Fa 2 /T/JL 2 ) 



Join) + Q/itiJM 



(x = (ua/c) = ir/3oi(v/i'oi) ) 

r = (za/ac) = (z/W^oi) > (20.8) 
= 6 -ix= (a/<rc)(R - iX)) 



The amplitude A is therefore proportional to the driving force F 
and inversely proportional to the tension T. The dependence on 
frequency is through the parameter /*. When the driving frequency 
is equal to one of the resonance frequencies v 0n , of Eq. (19.6), the 
quantity Jo(m) is zero, and if the air impedance parameter £* is small 
compared to n, A will have its largest values at these frequencies. 

In calculating the response of a condenser microphone, we must 
compute the average amplitude of motion of the diaphragm. From 
the results we have obtained, this is 

(Fa 2 /Tti 2 )J»(n)e- i0,t _ Fa 2 



Join) + {$/iv)JM T 



i/(ju)e- i(a "- Q) 



(20.9) 



where 



and 



H(n) = 



_ JM J 

M 2 I 






+ " 



©■*>r 



tan £2 = 



6JM 



nJo(») — %JM 



202 MEMBRANES AND PLATES [V.20 

Figure 44 shows curves of the average amplitude function H and the 
average phase lag £2 for x = 0, for different values of 0, plotted as" 
functions of the frequency parameter n = (<aa/c) — irfio n (v/von)- 

These curves are interesting and important, for they are typical 
of average response curves for membranes of any shape. At low- 
frequencies (m < 1) the average response is fairly independent of fre- 
quency. Using Eqs. (19.4) and neglecting all but the first power of 
n, we have 

* a * F er*" [a < (c/a)] 



~ i ° 3Zm / v , x > (20.10) 

This result is similar to that for a simple driven oscillator. The 
effective driving force is the area of the diaphragm times the force 
per unit area. The mechanical impedance is the area of the dia- 
phragm times the total specific acoustic impedance z of the medium 
on both sides of the diaphragm, plus the equivalent mechanical 
impedance of the diaphragm itself for low frequencies. This latter 
corresponds to an effective mass of four-thirds of the total mass of the 
membrane and an effective stiffness constant of 8t times T, the mem- 
brane tension in dynes per centimeter. The factors (i) and 8w come 
in because all the membrane does not vibrate with the same amplitude 
of motion, as does a mass on the end of a spring. 

Formula (20.10) breaks down when the frequency comes near the 
first resonance, and the exact formula (20.9) must be used. This 
first resonance comes at n = 2.405, or v = v i [see Eq. (19.7)], and 
the average amplitude is large unless R is large. As the frequency 
is increased still further, the average amplitude decreases rapidly, 
becoming zero at \x — 5.136, or c= vzi, no matter what value x and 
6 have. 

Just above the first resonance v i the membrane vibrates nearly 
out of phase with the driving force (if 6 is small) ; and as the frequency 
is increased still further a circular nodal line appears at the outer 
edge and shrinks in toward the center, the part of the membrane 
inside the node remaining out of phase and the part outside the node 
being nearly in phase with the force. As the driving frequency 
increases and the nodal circle shrinks, the motion of the outer part 
cancels more and more of the motion of the inner part in the average 
displacement, until at n = 5.136 the two parts completely cancel 
each other out, and the average displacement is zero. At this point, 



V.20] FORCED MOTION. THE CONDENSER MICROPHONE 



203 



«/ 2 (m) is zero and therefore the effect of the reaction of the medium 
also goes to zero (as long as this reaction depends only on the average 
displacement fj). For this reason the frequency of zero average dis- 
placement is not affected by the reaction of the air, represented by 
the quantities x and 0. 



180" 



/I 90° 



H 0.2 




Fig. 44. — Response curves and curves of angle of lag of displacement for a damped 
condenser microphone as functions of driving frequency, for different values of the 
air resistance parameter 0. The quantity fj. is 2.405 times the ratio of the driving 
frequency to the fundamental frequency of the diaphragm. 

A little beyond this antiresonance frequency, -when n = 5.520, is the 
second resonance, and so on. Between each of the resonance frequen- 
cies v 0n and the next successive one is an antiresonance frequency v 2n . 

The Condenser Microphone. — The condenser microphone corre- 
sponds approximately to the case just discussed. The diaphragm 



204 



MEMBRANES AND PLATES 



[V.20 



*-l- 



P 
J 



N -i, 



B 

Fig 45. — Simplified 
cross section of a con- 
denser microphone. 



of the microphone is metallic, and therefore it has stiffness; but the 
diaphragm is often so thin and is under such great tension that the 
effects of stiffness can be neglected (though we can take the effect 
into account if necessary, as we shall see in the next section). 

Figure 45 shows a simplified cross-sectional view of a condenser 
microphone. D is the diaphragm, which is usually thin enough and 
under large enough tension to be considered as a membrane. This is 
stretched over the end of a vessel B, which usually has vent holes 
represented by H, so that it is not airtight. 
Behind the diaphragm a short distance is a 
plate P, sometimes pierced with holes so that 
the air can penetrate easily, and insulated elec- 
trically from the diaphragm. This forms the 
other plate of the condenser. 

The driving force is the excess pressure, on 
the outside of D, caused by a sound wave in 
passing. Unless the wavelength of the sound 
wave is smaller than the size of the microphone 
(which happens only for frequencies higher than 
5,000 cps, for most microphones), the excess 
pressure can be considered to be uniform over the surface of the 
diaphragm. The case for small wavelengths will be taken up in the 
chapter on the scattering of sound. 

We can also assume that the pressure varies sinusoidally with 
time. If the sound is a combination of waves of many different 
frequencies, the resulting motion of the diaphragm will be a com- 
bination of the motions due to each wave separately. 

If the intensity of a simple harmonic wave of frequency v is T ergs 
per sec per sq cm, then the excess pressure on the outside of the 
diaphragm will be F == F e- 2iriH , F = 9.2 y/T dynes per sq cm 
(for air at standard conditions). This expression is correct only if 
the presence of the microphone does not alter the motion of the sound 
wave appreciably. In Chap. VII this question will be discussed in 
detail. We have mentioned earlier that the intensity due to average 
conversation from a person 3 ft away is about 1 erg per sec per sq cm. 
Electrical Connections. — The output voltage of the condenser 
microphone is not proportional to the amplitude A but is proportional 
to the average displacement of the diaphragm. If the equilibrium 
distance between D and P is A, then the equilibrium capacity of the 
microphone is C = (wa 2 /4irA) in electrostatic units. If the displace- 



V.20] FORCED MOTION. THE CONDENSER MICROPHONE 205 

ment from equilibrium of the diaphragm is ri(r,t), then the inverse 
capacity of the microphone is 

1 = i (A - „)„ = i-(l - X J v*) 
Using the expression for rj given above, we have 



Figure 46 shows the usual circuit for connecting the microphone 
to the amplifier. Resistance R is made 
large enough so that the condenser can- 
not charge and discharge rapidly enough 
to follow the alternations of capacitance 
caused by the sound waves. The aver- 
age charge on the plates is E Q C , and the 
average potential difference between the 
plates is E . At any instant, however, 
the potential difference will be the charge 
E C (which will not change if R is large 
enough) divided by the value of the microphone capacitance at that 
instant: 



To 
amplifier 



lililmi 



Fig. 46. — Electrical circuit for the 
condenser microphone. 



tfi = (^-°) = E 



i-i<- 



(¥) 



The difference between E and this quantity is the emf impressed 
on the amplifier: 

E = B, - *, - (M\ j,(?22\ e — 
The magnitude of the output voltage is therefore 



E = 1.348 



Eq 



o-A(i/ i) 2 



Vt#0) 



(20.12) 



where E is the voltage impressed across the microphone, A the dis- 
tance between the diaphragm and backing plate, a the density of the 
membrane in grams per square centimeter, v i its fundamental fre- 
quency of oscillation, and T the intensity of the sound in ergs per 
second per square centimeter. The quantity H is the one given in 
Eqs. (20.9), and the angle of lag of the output voltage behind the 



206 MEMBRANES AND PLATES [V.20 

sound pressure is the angle fl defined in the same equations. There- 
fore the curves of Fig. 44 are also curves of sensitivity and phase lag 
of the microphone signal, provided that its construction corresponds 
reasonably well to the assumptions as to air reaction we have made 
in deriving Eqs. (20.9). 

Transient Response of Microphone. — According to the discussion 
following Eqs. (6.16) and (6.17), we can find the response of the 
uniformly loaded diaphragm to a unit impulsive force spread uni- 
formly over the diaphragm, by integrating (l/27rF) times the 17 given in 
Eqs. (20.9) over to from — ■ °o to + °° : 

- (A - _^L f °° J"2(m) e-^ /o) 

This integral is extremely difficult to compute exactly, but it can be 
evaluated approximately as long as |f | is considerably smaller than 
n(\z\ < < w, i.e., the load per unit area due to the medium on both 
sides of the membrane is small compared with the mass reactance 
of a unit area of the membrane itself). In this case the zeros of the 
denominator of the integrand are very close to /z = ±(T/3 0n ). To 
the first approximation, near \x = irl3on, «/o(m) — — (m — Trfion)Ji(jfio n ), 
which can be obtained from Eqs. (19.5) and from the Taylor's expan- 
sion, J (n) = Jo(irp 0n ) + (m — *Po»)[dJo(n)/dfi] r p. Also, using Eqs. 
(19.5), we can show that in this region J 2 (m) = (2/m)«A(m) — «/o(m) — 

[/i - ir|3on + (2/irj8on)]J"l(ir/3on) ^ (2/lT|8o») Jl^Pon) for JU ~ Tl^On- 

Therefore .the contour integral breaks up into a sum of contour 
integrals, each around the points m = ±7r/3o«. The one about \i = 
+7T/3o« is approximately ( — 2iri) times the residue of 

ca ( 1 Y M - T0on + (2/tt/3 *) e - (icilt/a) 



( 1 V M - x, 



%rT\Tp Q J n - Tpon + [2tf/0r/3 «) s 

at its pole at n = (tj8o») - [2#/(t0o») 2 ]. The contours around the 
points fx = — 7t/3 „ can be similarly approximated, and the final result, 
for the average displacement of the diaphragm in response to a uni- 
form impulsive force of unit impulse per unit area, is 

rj & (t) ~ ^ ^ Q-) 3 e-^w^> sin6^) (20.13) 

where R n is the resistive part of the medium's impedance per unit 
area (z = R — iX) at the frequency v ». The response to a general 
impulsive force F(«) per unit area is, according to Eq. (6.17), 



V.20] FORCED MOTION. THE CONDENSER MICROPHONE 207 



f, = §[ ^ F(T)rj s (t - t) dr 



The corresponding series for the actual displacement t\ of the dia- 
phragm is obtained by multiplying the nth term in the series for m by 
[7r/W2/i(ir/3on)]/o0r/3o«r/a). Figure 47 shows a time sequence of the 
shapes of the diaphragm after being shock excited. 

Unit Impulse at "t=0,spreaol uniformly over Membrane 

I I I 

t=Q • * * * 




Fig. 47. — Cross-sectional view of motion of a circular membrane with damping 
proportional to average velocity (uniform damping), when subjected to a uniform 
impulsive force. Compare with Fig. 38. 

The motion corresponding to this formula is quite different from 
the related motion of a string; for in this case the higher the overtone, 
the less the damping. The damping factor here for the nth mode is 
2R n /<T(irPo n ) 2 , which rapidly decreases as /3 « gets larger. The result is 
due to the uniform nature of the coupling between the diaphragm and 
the medium and is true only if the wave velocity in the medium is large 
enough so that the reaction depends solely on the average displace- 
ment of the diaphragm, being uniform over the diaphragm. For 
then, since the higher modes have a small average displacement com- 



208 MEMBRANES AND PLATES [V.21 

pared with their maximum displacement, not much of their energy 
of vibration gets carried away by the medium. By the same token, 
the coupling between the higher modes and the uniform impulsive 
force is small, so that the amplitude of the higher overtones is quite 
small (it varies inversely as @\ n , as a matter of fact). Thus the series 
converges rapidly. 

The equation predicts that a kettledrum diaphragm, for instance, 
when set into motion by a sound pulse, will have motion as a whole 
(lowest mode) which is rapidly damped, but that small-amplitude 
high-frequency short wavelength ripples in the diaphragm will persist 
for a long time after the pulse. This is actually the way such a dia- 
phragm does behave, so our assumptions must not be far from the 
correct ones for the drum. In the case of the condenser microphone 
it is not quite such a good approximation, for the reaction of the air 
inside the casing is not completely uniform, but is partly localized; 
and the effect of localized damping is to damp out the higher modes 
more rapidly than is the case when no localized reaction occurs. 

The intermediate case, where the wave velocity in the membrane 
is neither much larger nor much smaller than the speed of sound in 
the medium, is too difficult to handle this early in the volume. We 
shall return to it in Chap. VII. 

21. THE VIBRATION OF PLATES 

The study of the vibrations of plates bears the same relation to 
the study of the membrane as the study of the vibrations of bars does 
to the study of the flexible string. The effect of stiffness in both 
cases increases the frequencies of the higher overtones more than it 
does those of the lower overtones and so makes the fundamental fre- 
quency very much lower than all the overtones. However, the 
motions of a plate are very much more complicated than those of a 
bar, so much more complicated that we shall have to be satisfied with 
the study of one example, that of the circular plate, clamped at its 
edge and under no tension. The diaphragm of an ordinary telephone 
receiver is a plate of this type, so the study will have some practical 
applications. 

The Equation of Motion. — The increased complications encoun- 
tered in the study of plates come partly from the increased complexity 
of wave motions in two dimensions over those of one, but also come 
about owing to the complex sort of stresses that are set up when a plate 
is bent. The bending of a plate compresses the material on the inside 
of the bend and stretches it on the outside. But when a material is 



V.21] THE VIBRATION OF PLATES 209 

compressed it tries to spread out in a direction perpendicular to the 
compressional force, so that when a plate is bent downward in one 
direction there will be a tendency for it to curl up in a direction at 
right angles to the bend. The ratio of the sidewise spreading to the 
compression is called Poisson's ratio and will be labeled by the letter s. 
It has a value about equal to 0.3 for most materials. This complica- 
tion was not considered when we studied the vibration of bars, for we 
tacitly assumed that the bar was thin enough compared with its 
length so that the effects of a sidewise curl would be negligible. 

The derivation of the wave equation for the plate involves more 
discussion than is worth while here (it is given in books on theory 
of elasticity). The equation is 

y4„ + 3p(l ~ s 2 ) gg = (211) 

v ^ Qh 2 dt 2 K } 

where p is the density of the material, s its Poisson's ratio, Q its modulus 
of elasticity, and h the half-thickness of the plate. Values of p and 
Q are given for different materials in Table 3 of Chap. IV. 

We shall not spend any time discussing the general behavior of 
waves on a plate of infinite extent but shall simply remark that, like 
the bar, the plate is a dispersive medium; waves of different wave- 
length travel with different velocities. 

Simple Harmonic Vibrations. — To study the simple harmonic 
motion of a plate, we insert the exponential dependence on time and 
separate the factors depending on the individual coordinates. The 
differential operator V 4 is difficult to separate in most coordinate 
systems, but for polar coordinates the results turn out to be sufficiently 
simple to justify our analyzing them in detail. Here, if we set r\ — 
Y{r,(p)e~ 2Tivt , where Y's dependence on <p is by the factor cos or 
sin (m<p), then the differential equation for Y can be written as 

( V2_ 7 2 )(V 2 + t2 ) F = o, T 4 = 12TV ffi, " S2) '(21.2) 

Therefore, Y can be a solution either of (V 2 F + y 2 F) = or of 
(V 2 F - t 2 F) = 0. 

Since V 2 and Y are to be expressed in polar coordinates, the solution 

COS 

of the first equation which is finite at r = is Y = . (nap) J m (yr), 
^ sin 

where m is an integer. This is the usual solution for the membrane, 

with 7 instead of (2tv/c). The solution of the second equation is 

obtained from the first by changing y into iy and necessitates a little 



210 MEMBRANES AND PLATES [V.21 

discussion of Bessel functions of imaginary values of the independent 
variable. Let us call these hyperbolic Bessel functions and define 
them by the equation I m (z) = r^J^iz). The properties of the 
function I m (z) can be obtained from Eqs. (19.4) and (19.5) for J m (z). 
The more useful formulas are 

I m -i(z) - I m+1 (z) = — I m (z), -^ I m (z) = i[Im-i(z) + I m +i(z)] 

fIo(z)z dz = zlriz), J7i(z) dz = I (z) (21.3) 

Values of the functions 7o, Ji, and Z 2 are given in Table IV at the 
back of the book. 

The Normal Modes. — Possible solutions for the simple harmonic 
oscillations of a plate are therefore given by the expressions 

F ( r >*> = 1 S W [AJ m (jr) + BI m (yr)] 

Sill 

The boundary conditions corresponding to a circular plate of radius a, 
clamped at its edges, are that Y(a,<p) = and (dY/dr) r=a = 0. The 
first condition is satisfied by making 



B = -A 



r jm(7«) i 

|_/ OT (7a)J 



and the second condition is satisfied by requiring that y have those 

values that make 

d d 

Im(ya) -T- J m (yr) — J m (ya) -=- I m (yr) =0 at r = a (21.4) 

This equation fixes the allowed values of the frequency, for y depends 
on v. We shall label the solutions of Eq. (21.4) by the symbols 
y mn , where y mn = {ir/a)8 mn , and where 

l8oi =■ 1.015, O2 = 2.007, O3 = 3.000) 
0n = 1.468, 0i 2 = 2.483, 8 13 = 3.490 > (21.5) 

21 = 1.879, 022 = 2.992, 23 = 4.000 ) 

m 

Pmn, *■ n ~J~ ~~jr 

The allowed values of the frequency are therefore 



^ (8 Y 



3p(1 - s 2 ) 



irk / 

Vmn 2a 2 '\ : 

^ • 9342 &)^£? : S [ (2L6) 

rn = 2.091^01, "21 = 3.426^01, "02 = 3.909P01 
^12 = 5.983^01, etc. 



V.21] 



THE VIBRATION OF PLATES 



211 



The allowed frequencies are spread apart much farther than those 
for the membrane, given in Eq. (19.7). The overtones are not 
harmonic. 

The characteristic functions are 



L m \ a ) ImiirPmn) ™ \ « / J 



fmn = COS(m 



and a similar expression for ^ «n (for m > 0) where sin(m«?) is used 
instead of cos (m<p). Some of these functions are shown in Fig. 48. 






Fig. 48. — Shapes of a few of the normal modes of vibration of a circular plate clamped 

at its edge. 

The free vibrations of the plate corresponding to arbitrary initial 
conditions can be expressed in terms of a series of these characteristic 
functions. 

Forced Motion. — As an example of the method of dealing with 
forced motion, let us repeat the calculations for a fully damped con- 
denser microphone with a plate diaphragm instead of a membrane. 
The equation of motion to be used instead of (20.6) is 



dt 2 



4x 



F B 



7 ^^ + ^° e " 2X " 



R 



//GO 



r dr d<p (21.7) 



■jra 2 hp 

where t 4 = [12t 2 v 2 p(1 — s 2 )/Qh 2 ]. We now choose the function r? 
to be 

A p — 2irivt 

V = 77 — r [Ii(ya)J (yr) + Ji(ya)I (yr) - Ii(ya)J (ya) 

- Ji(ya)I Q (ya)] 

which has zero value and slope at r = a and which has the following 
average value: 



212 MEMBRANES AND PLATES [V.21 



TO" J J 



vrdrdtp = 7 , . [Ii(ya)J 2 (ya) - J 1 {ya)l2{ya)]e- Mvt 



In order to compare the results of these computations with the 
ones given in the previous section for the membrane diaphragm, we 
shall use the same variables n = 2A05(v/v 01 ) &nd£ = 0.3828(Z/2hpi>oi) 
that were used in Eq. (20.8); the only difference being that for v i, 
instead of the value given in (19.7), we use the value given in Eq. 
(21.6) for the fundamental frequency of the plate. In terms of these 
variables, (aV) = 4.23 In or ay = 2.057 Vi*. 

Substituting the expression for n into Eq. (21.7), and changing 
to the new variables, we obtain an expression similar to Eq. (20.8) 

(F/a, 2 h P ) 1 

A = 7 1 (2.06 Vm) l "(m) + (TAm)^0*) (2L8) 

L (m) = /i(2.06 VmVo(2.06 Vm) + /o(2.06 V^)J 1 (2.0Q Vm) 
L 2 ( M ) = /i(2.06 Vm)«/ 2 (2.06 Vm> - / 2 (2.06 Vm)«/i(2.06 Vm) 

The calculations for a condenser microphone with plate diaphragm 
go through in a manner similar to that for the membrane, and the 
expression for the output voltage is 

E = 1.348 , , E \ 2A VTHMe-w-™ (21.9) 

flp{voi) A 



r <0 .) = ^ {[lm -Ilm] 2 + ($l>m] 



tanfli = 



\jiLo(ji) - xL 2 (n)] 

Curves for Ht as function of n for different values of 6 are given in 
Fig. 49. These may be compared directly with the curves in Fig. 44, 
for the membrane diaphragm. The units have been so chosen that 
when the two diaphragms have the same mass per unit area and the 
same fundamental frequency then both horizontal and vertical scales 
are the same. We see that below the first resonance frequency the 
response curves are quite similar and that the chief difference above 
this frequency is that the point of zero response is higher on the fre- 
quency scale for the plate than for the membrane — so much higher 
that the second resonance peak for the plate is outside the range 
plotted. 

It is evident that the plate diaphragm has the advantage of having 
a longer range of frequencies below the point of zero response. This 



V.21] 



THE VIBRATION OF PLATES 



213 



advantage, however, is almost neutralized by the disadvantage that 
the average response of the plate is somewhat smaller than that of 
the membrane, owing to the fact that the displacement must be small 
near its edge. The chief disadvantage of the plate, however, is the 
practical one that it is difficult to obtain and properly mount a plate 
thin enough to give a small mass per unit area and yet stiff enough 
to have a high-frequency fundamental (although plate diaphragms 
have been successfully used on miniature condenser microphones). 



0.3 r 




Fig. 49. — Response curves for the damped condenser microphone with plate dia- 
phragm, plotted as a function of the frequency parameter fi for several different values 
of the damping constant 0. 

We shall not go into the details of the solution of the motions of a 
plate under tension, the intermediate case between the membrane 
and the plate without tension. It is obvious that the response curve 
for a damped condenser microphone with such a diaphragm will be 
intermediate between the curves of Figs. 44 and 49. 

Problems 

1. A membrane is made of material of density 0.1 g per sq cm and is under a 
tension of 100,000 dynes per cm. It is wished to have the membrane respond best 
to sound of frequency 250 cps. If the membrane is square, what will be the 
length of one side? What will be the frequencies of the two lowest overtones? 

2. A square membrane, 20 cm on a side, of mass 1 g per sq cm, is under a 
tension of 10 8 dynes per cm. Its motion is opposed by a frictional force of 42(dq/dt) 
dynes per sq cm. Find the modulus of decay of the oscillations. What are the 
first four "frequencies" of the damped motion? 



214 MEMBRANES AND PLATES [V.21 



3. A square membrane, b cm on a side, of density <r and under a tension T, is 
loaded at its center with a mass of M g. Show that the allowed frequencies are 
approximately 



(, T[m 2 , nn [, AM (b b\l) 



as long as (AM/ab 2 ) is small. What will be the expression for the frequency if the 
membrane is rectangular but not square? 

4. Show that the energy of vibration of a rectangular membrane of sides 
a and 6 is 

^-«-jriT[(2) , +(g) , ^®>* 

and that when the motion is given in terms of the series 

« 
t] = 2j A mn sin I J sin I -=— J cos {2irv mn t + * mn ) 

m,n \ a / \ / 

the series for the energy is 



W = — (<rab) ^ 
2 ni,n 



mn mn 



5. A square membrane 20 cm on a side, with a = 1 and T = 10 6 , is started 
from rest at t = with an initial shape r, = lCr b a;(20 - x)y(20 - y). What are 
the energies of vibration corresponding to the fundamental and the lowest three 
overtones, and what is the total energy of vibration of the membrane? 

6. A rectangular membrane is pushed aside at the point (x ,yo) by a. steady 
force F and is then suddenly released at t = 0. What is the expression for the 
subsequent motion of the membrane? Neglect the reaction of the air. 

7. A circular membrane of radius 10 cm, with a = 1, T = 36, is struck so 
that a very small area A»S around its mid-point has an initial velocity (1/A/S) cm 
per sec, all the rest of the membrane being at rest at t = 0. Plot the shape of a 
cross section of the membrane (77 as a function of r) for t = 0, 1, 2, 3, 4, 5, 6 sec. 

8. The tensile strength of aluminum is 2.5 X 10 9 dynes per sq cm, and its 
density is 2.7 g per cc. What is the highest value of fundamental frequency 
that can be attained with an aluminum membrane stretched over a circular 
frame 3 cm in radius? If the aluminum is 0.005 cm thick, what will be the maxi- 
mum tension attainable? 

9. A square membrane 4 cm on a side with a = 0.01 and T = 10 6 , which is 
undamped, is driven by a uniform force of l,000e- 2iriI " dynes per sq cm. Plot the 
amplitude of vibration of the mid-point as a function of frequency from v = 
to v = 2,000. 

10. Suppose that the membrane of Prob. 9 is acted on by a damping force of 
A2(dT)/dt) dynes per sq cm. Plot the amplitude of motion of the mid-point as a 
function of v from v = to v = 2,000. 

11. Find the effect on the allowed frequencies of a circular membrane due to a 
small extra mass of M g attached to its mid-point. 



V.21] THE VIBRATION OF PLATES 215 

12. An undamped circular membrane of radius 2 cm, a- = 0.1, and T — 631,700 
is driven by a uniform force of 40,000e~ 2iriw dynes per sq cm. Plot the amplitude of 
motion of the mid-point of the membrane as a function of v from v = to v = 1,200. 
What are the lowest three frequencies of free vibration of the membrane? What is 
the shape of the membrane (rj against r) when v = 600? 

13. The circular membrane of a kettledrum has a radius of 50 cm, <r = 0.1, 
and T = 10 8 . What is its fundamental frequency without the backing vessel? 
The backing vessel raises the fundamental frequency to 1.45 times this value. 
What is the volume of the vessel? What are the lowest five frequencies of the 
membrane- vessel system? 

14. A condenser microphone diaphragm has a radius a = 5 cm, a = 0.005, 
and T = 3 X 10 6 dynes per cm; A = 0.005 and E = 100 volts. What is the 
lowest frequency of the undamped diaphragm? Suppose that the impedance of 
the air per square centimeter of diaphragm is resistive, z = R = 100, the reaction 
being uniform and proportional to y. Plot the response curve H for the micro- 
phone for the frequency range v = to v = 10,000. 

16. A condenser microphone diaphragm is a membrane of radius a with local- 
ized air reaction, the resistive term being R(drj/dt) per sq cm, and the reactive 
term being Krj (with no additional effective mass). Show that the effective 
impedance to insert in Eqs. (20.5) is 



= R 



-(£)[(*?*)+*] 



This diaphragm is acted on by a distributed force 9.2 \/ye iat dynes per sq cm. 
Show that the steady-state displacement of the diaphragm is 



5.86 Vxr"' 2)- 



Jo(Tr@o n r/a) 



_- fionJi (irl3on) ( — iuzon) 

16. Compute the output voltage of the microphone of Prob. 15 as a function of 
E , A, a, <r, T, R, K and driving frequency. 

17. Calculate the transient response to a general impulsive force for the mem- 
brane of Eq. (20.5). 

18. A membrane with uniform air reaction is acted on by a uniform impulsive 
force 8(0 per sq cm, so that its motion is given by Eq. (20.13). Values of the 
constants are (c/a) = 1, (4ca/T) = 1, (R n /<r) = 1. Plot value of ij(t) as function 
of time t from t = to t = 10. 

19. The diaphragm of Eq. (20.13) is acted on by a uniform force 

(0 (* < 0) 

F(t) = < A sin (ut) [3 < t < (»/«)] 
(0 [t > (wtt/co)] 

Compute the series for the average displacement of the membrane. 

20. A plate diaphragm of steel Q = 19 X 10 11 , p = 7.6, s = 0.3 has dimen- 
sions a = 3 and h = 0.05. What is its fundamental frequency? It is driven by a 
pressure of lOOe -2 *"*''' dynes per sq cm uniformly distributed over the diaphragm. 
The mounting is so designed that the reaction due to the medium is negligible over 
the useful range of v. Plot the displacement of the mid-point of the diaphragm 
as a function of v from v = to v = 5,000. 



216 MEMBRANES AND PLATES [V.21 

21. Suppose the diaphragm of Prob. 20 is to be driven by a force of 100e _2iri,,t 
dynes concentrated at its mid-point. Plot the amplitude of motion of the mid- 
point as a function of frequency from v = to v = 5000. 

22. Plot the shape of the diaphragm of Prob. 21 (tj as a function of r) for v = 
1,000, 3,000, 5,000. 

23. A circular plate diaphragm is loaded at its mid-point by a mass M . Com- 
pute the approximate expressions for the allowed frequencies of the loaded plate, 
valid when M is small. 

24. Compute the transient response of a plate diaphragm to a uniform impul- 
sive force, and obtain a formula corresponding to Eq. (20.13). 



CHAPTER VI 

PLANE WAVES OF SOUND 

22. THE EQUATION OF MOTION 

We now come to the study of wave motion in air, the most impor- 
tant type of wave motion studied in the science of acoustics. Sound 
waves differ from the waves that we have discussed heretofore in 
several important respects. They are waves in three dimensions 
and as such can be more complicated in behavior than waves in two 
dimensions or in one. Sound waves also differ from waves on a string 
or on a membrane by being longitudinal waves. So far, we have been 
studying transverse waves, where the material transmitting the wave 
moves in a direction perpendicular to the direction of propagation of 
the wave. Each part of the string, for instance, moves in a direction 
at right angles to the equilibrium shape of the string, whereas the 
wave travels along the string. The molecules of air, however, move 
in the direction of propagation of the wave, so there are no alternate 
crests and troughs, as with waves on the surface of water, but alternate 
compressions and rarefactions. The restoring force, responsible for 
keeping the wave going, is simply the opposition that the gas exhibits 
against being compressed. 

Since there are so many points of difference between the waves 
discussed earlier and the more complicated forms of sound waves, 
it is well not to introduce all the complications at once. Accordingly, 
we shall first study the motion of plane waves of sound, waves having 
the same direction of propagation everywhere in space, whose "crests" 
are in planes perpendicular to the direction of propagation. They 
correspond to the parallel waves on a membrane. Waves traveling 
along the inside of tubes of uniform cross section will usually be plane 
waves. Waves that have traveled unimpeded a long distance from 
their source will be, very nearly, plane waves. 

Waves along a Tube. — Suppose that we consider the air in a tube 
of uniform cross section of area S. When everything is at equilibrium 
let us color red all the molecules in the plane, perpendicular to the 
axis of the tube, at a distance xi along the tube from some origin; 

217 



218 PLANE WAVES OF SOUND [VI.22 

color blue the molecules in the plane at z 2 ; and so on. When a sound 
wave passes through the tube, these planes will be displaced from 
their equilibrium positions back and forth along the tube. At some 
instant the red plane will be at x x + £Oi), the blue plane at x 2 + £(2:2), 
and so on. Each molecule of the gas originally a distance x from the 
reference plane will be displaced in the x-direction a distance £. This 
displacement depends on the time t and also on the particular mole- 
cule that we are watching {i.e., it depends on x). 

Of course, this is a crude picture of what actually happens; the 
real molecules are bouncing back and forth because of temperature 
agitation, even when there is no sound. The quantity £(x,t) actually 
measures the average displacement, due to the sound wave, of those 
molecules whose average position was originally x. The average 
velocity of this plane in the z-direction, (d£/dt), is called the particle 
velocity u. 

We must now find an equation giving the dependence of £ on x 
and t. This equation will be obtained by combining three equations, 
one of them Newton's equation of motion and the other two repre- 
senting two simple properties of a gas. One of these properties is 
just a restatement of the law of the conservation of matter: The 
amount of gas between the plane of red molecules and the plane of 
blue molecules will remain the same as the planes move. The other 
property is that relating the change in density of a perfect gas with 
its change in pressure in the case when the gas is so rapidly compressed 
that it cannot unload its gain in heat to the surrounding gas. 

To put these properties into mathematical form we must make a 
few definitions. We shall denote the equilibrium density of the gas 
by p and its equilibrium pressure by P - The actual density at the 
point x and time t will be denoted by p(x,t), and the relative change in 
density will be denoted by 8(x,t), where 

P (x,t) = P (l + 5), 8 = i p(x,t) - 1 (22.1) 

The difference between the actual pressure and the equilibrium pres- 
sure will be denoted by p{x,t). It is this excess pressure p that pro- 
duces the motion of a microphone diaphragm. 

The Equation of Continuity. — Returning to the colored planes of 
molecules, we see that unless the gas moves so much that turbulence 
is set up the colored planes will remain planes as the gas moves. 
Since the planes are made up of molecules of the gas, it is clear that, 
unless there is turbulence, the gas ahead of one plane will always be 



VI.22] 



THE EQUATION OF MOTION 



219 



,>j(o) >|(x,)/g(x 2 ) /j(x) 



-dx-* 



ahead of that plane, and the gas between two planes will always be 
between those planes. Suppose that we consider the gas between two 

planes as shown in the second sketch 
of Fig. 50, which at equilibrium are 
at the distances x and x + dx. At 
equilibrium, the mass of the gas be- 
tween the planes will be its density 
p times the volume cut off by the 
planes and the surface of the tube, 
S dx. By the argument above, the 
mass of gas between the planes, when 
they are displaced by the sound wave, 
will still be pS dx. But when the 
planes are displaced, the volume be- 
tween the planes may be altered, for 
the displacement of one plane is |(x) 
and that of the other is £(x + dx) = 
£(x) + dx(d%/dx). The volume en- 
closed is therefore S[dx + £(x + dx) 
- £(*)] = Sdx + Sdx(d£/dx). The density of the gas between the 
planes must be altered so that the total mass pS dx can remain unal- 
tered: p(x,t)[S dx + S dx(d£/dx)] = P S dx. With reference to Eq. 



4(x)-- 



(**> 



N £(x+dx) 



x x+dx 
Fig. 50. — Longitudinal displace- 
ment in a sound wave. Gas particles 
originally in the plane at x are dis- 
placed by an amount l-(x). 



(22.1), this gives pS dx = P (l + 8)Sdx 



+ 2} 



When the change 



in density and displacement are small (and they are small in all but 
the loudest sounds), we can neglect the product of the two small 
quantities 8 and (d£/dx), so that 



Therefore, 



( 1 + g 



(l + S)(l+^)^l + 5 + 



1=1+5+ 



dx 



or 5 = — 



dx 
dx 



(22.2) 



This is a form of the so-called equation of continuity. It states that 
when the gas to the right of a given point is displaced more (to the 
right) than the gas to the left of the point is displaced (i.e., if the two 
colored planes are pushed apart by the displacement) then the density 
of the gas is diminished at the point. We note for future reference 
that if the cross-sectional area of the tube S depends, on x, the correct 
equation will be 



220 PLANE WAVES OF SOUND [VI.22 

* = -s!< s » ■ (22 - 3) 

Compressibility of the Gas.— The second property of the gas which 
is used in deriving the equation of wave motion depends on some 
thermodynamic properties of gases. All perfect gases obey the well- 
known relationship between the total pressure P, the volume occupied 
V, and the temperature T which is PV = RT. Differentiating both 
sides gives R dT = P dV + V dP, and on dividing by the undiffer- 
entiated equation we obtain 

dT dP ,dV , 99 ^ 

T = ~P + T C 

If the volume occupied by the gas is kept constant, and the pressure is 
increased by heating, dV is zero and dP = (P/T) dT. The increase of 
the heat energy of the gas during this process will be dQ = (dQ/dP) dP 
= (dQ/dP)(P/T) dT, so that the rate of increase of heat energy 
due to rise of temperature at constant volume is (dQ/dT)rconst = 
(dQ/dP)(P/T). This rate of increase is called the specific heat of the 
gas at constant volume and is denoted by CV. Therefore, 

dP"P Cv 

The temperature of the gas can also be raised while holding its 
pressure constant, by increasing its volume by just the right amount 
during the heating. In this case dP = 0, dV = (V/T) dT, and the 
rate of increase of heating energy due to rise of temperature at con- 
stant pressure is (dQ/dT) Pcoast = (dQ/dV)(V/T). This rate of increase 
is called the specific heat at constant pressure and is denoted by C P . 
Therefore, 

dV'V Cp 

If, now, we allow the gas to change both volume and pressure, the 
change in heat energy will be the rate of change of Q with respect to 
V times dV, plus the rate of change with P times dP: 

from the equations for CV and C P . 

If the gas is expanded slowly enough to enable the temperature to 
remain constant, then dT will be zero in (22.4), and the relation between 



VI.22] THE EQUATION OF MOTION 221 

the change in pressure and the change in volume is (dP/P) = — (dV/ V) . 
However, the expansions and contractions in the sound wave are 
entirely too rapid for the temperature of the gas to remain constant. 
The alternations of pressure and density are so rapid that it is much 
more nearly correct to say that no heat energy has time to flow away 
from the compressed part of the gas before this part is no longer com- 
pressed. In this case, where the gas temperature changes but its 
heat energy does not change, the compression is said to be adiabatic. 
To find the relation between the change in pressure and the change in 
volume we must use the equation for dQ, with dQ set equal to zero: 
C P (dV/V) - =C v (dP/P). This is the equation that will be used to 
find the elasticity of the gas. 

To return to the study of the gas in the tube contained between 
the planes originally at af and at x + dx, the volume of the gas is S dx, 
and the change in volume is S dx(d£/dx) ; the pressure is P , and the 
change in pressure is p. The preceding equation relating change of 
pressure to change in volume in adiabatic compression takes on the 
form 

or, letting (C P /C V ) = y c , 

V = ~^ p of||j = ToPoS (22.5) 

from Eq. (22.2). The constant y c , the ratio of the specific heat at 
constant pressure to that at constant volume, has the value 1.40 for 
air at normal conditions. 

The Wave Equation. — From the requirement of conservation of 
matter we have obtained Eq. (22.2) relating the change in density to 
the rate of change of displacement; and from the thermodynamic gas 
laws we' have obtained Eq. (22.5), relating the change in pressure to 
the change in density. Each of the functions p, £, and 8 is a function 
of x and t. By using one more equation, that relating the acceleration 
of the gas between the planes to the net force on it, we shall have 
enough equations to solve for all three quantities. 

The force on the gas between the planes, due to the gas to the left, 
is [P + p(x)]S; and that due to the gas to the right of the planes is 
[Po + p(x + dx)]S = [P + p(x) + (dp/dx)dx]S. The difference be- 
tween these forces —Sdx(dp/dx) is the net force acting on the gas 
between the planes and must therefore be equal to the mass of the 
gas P S dx times its acceleration (d 2 £/dt 2 ). This gives us the needed 
third equation: 



PLANE 


WAVES OF SOUND 


a 2 i 
9 at 2 


du 


dp 
dx 



222 PLANE WAVES OF SOUND [VI.22 

(22.6) 

By combining Eqs. (22.2), (22.4), (22.5), and (22.6) we obtain 
formulas for £, p, and 5 and the change in temperature AT, in a plane 
sound wave: 



d 2 £ 
dx 2 


1 d 2 £ d 2 p _ 1 d 2 p d 2 8 _ 1 d 2 8 
~c~ 2 W dx 2 ~c 2 dt 2 ' dx 2 ~ ~c 2 W 


c 
V 


-iW- -pc.fi 


AT 


-0-^--^- 1 )M' 



(22.7) 



The displacement, density, and pressure all obey the equation for 
propagation of waves with a velocity c. For the simple case of plane 
sound waves of not too large amplitude, these waves will be propa- 
gated without change of shape and with a velocity c independent of 
shape. The three waves are not independent, being related by some 
of the equations above; once the wave of particle displacement is 
known (for instance), the waves of density and pressure are deter- 
mined. Where the pressure is greatest in magnitude, there the rate 
of change of displacement with x is largest; and where the pressure 
is zero, there the displacement is greatest (or least). Where pressure 
is greatest, there density is greatest, and temperature is greatest, and 
so on. 

It is a very interesting and important result that plane waves of 
sound, longitudinal waves, obey the same wave equation as do the 
transverse waves on a string. All the results that we have worked 
out for waves on a string can be taken over for plane sound waves, 
except that the meaning of some of the terms — displacement, shape 
of the wave, etc. — must be changed somewhat. 

The velocity of sound in air at standard conditions (20°C, 760 
mm of mercury) and the related constants are 

c = 34,400 cm/sec 
pc = 42 g/cm 2 sec 

p = 0.00121 g/cm 3 , Tc = 1-40 ) (22.8) 

pC 2 = Tc p = 1.42 X 10 6 ergs/cm 3 
P = 1.013 X 10 6 dynes/cm 2 



V1.22J 



THE EQUATION OF MOTION 



223 




The constant pc is called the characteristic acoustic resistance of the 
air, for reasons to be given in the next section. 

Energy in a Plane Wave. — We can now obtain an expression for 
the energy involved in a sound wave. The kinetic energy of any ele- 
ment of volume of the gas is ip(d£/d£) 2 
dx dy dz. The potential energy of the 
element is the amount of work required 
to compress the element from its equilib- 
rium volume dx dy dz to its new volume 
[1 + (d£/dx)] dxdydz = (1 — 8) dxdydz. 
Figure 51 shows a graph of the volume 
as a function of the pressure. The 
work done in compressing the gas jPdV 
turns out to equal the area of the four- 
sided figure (V, V , b, a). When p is 
small the side (a, b) can be considered 
to be ■ a straight line, and the area is 
equal to (P + %p)(V Q — V). If the wave whose energy we are mea- 
suring is a simple harmonic one or one made up of a number of simple 
harmonic components, the average value of the term Po(V — V) 
will be zero. Since we are interested only in average values of the 
energy, this term can be omitted. By the use of Eqs. (22.7) the re- 
maining term ip(V Q — V) becomes $pc 2 (d%/dx) 2 dx dy dz. The aver- 
age energy of a volume of gas due to the passage of a plane wave is 
therefore 

Average energy = \p I I |(^)+ c2 (^r) dxdydz ergs (22.9) 

where u = (di-/di). For a simple harmonic wave of frequency v = 
(w/27r), this can be expressed in terms of p alone: 



v v v— 

Fig. 51. — Relation between 
pressure and volume in a gas. 
The work done compressing the 
gas from volume Vo to volume V is 
equal to the area enclosed by the 
figure (Vo, a, b, Vo). 



Average energy = 



1 

2 P c' 



///[©'©■ 



+ p 2 



dx dy dz 



These equations correspond to Eq. (9.12) for the average energy of a 
string. The average energy per cubic centimeter will be denoted by 
W. 

Intensity. — The rate at which the energy is being transmitted 
along the wave, per square centimeter of wave front, is called the 
intensity of the sound wave and will be denoted by T. This will equal 



224 PLANE WAVES OF SOUND [VI.22 

the excess pressure p on the square centimeter, multiplied by the 
velocity of the gas particles : 

v (!) = " pc * JJ Tt er ^ seG ^ cm < 22 - 10 > 

The average value of this is the intensity T. 

If the wave is a simple harmonic one, with frequency v, the expres- 
sions for pressure, energy, and intensity are fairly simple. For a wave 
going to the right, having a maximum pressure P+, 

p+ = P +e *(*-«), £ + = A + e ik ^~ ct \ f^tj = EZ+e*'*^') 

P+ = pc(^) } (22.11) 



»-c 



m 



T + = ZirWpcA*. = (||) = ipcET* 

To obtain the values of W and T given here it is necessary first to take 
the real parts of the expressions for p and £ before substituting in 
formulas (22.9) and (22.10) and averaging over time. 

For a wave going to the left with maximum pressure P_, 

rp_ — p_ e -ik(x+ct) £__ _ A_ e -ik(.x+ct) (_lr- J = JJ_ e -ik(x+ct) 

A --(dfe) u ~ = -(^} p - = ~ pc (i) 

T_ = -2w 2 v 2 pcAl = - (pA = -ycUl 

The intensity in this case is negative, since the wave is going in the 
direction of the negative z-axis. 

The expression used in Sees. 20 and 21 for the pressure on a con- 
denser microphone diaphragm is obtained from Eq. (22.11), for if the 
wave is all going toward the diaphragm, the total intensity will equal 
the square of the maximum pressure, divided by 2pc. - Since pc = 42, 



(22.12) 



VI.22] THE EQUATION OF MOTION 225 

the maximum pressure will be V84T = 9.2 V T, which was the expres- 
sion used. This is not the correct expression if the microphone is large 
enough to distort the wave as it passes, as we shall see in Chap. VIII. 

The quantity (c/v) is called the wavelength X of the waves, and 
the quantity (k/2w) = (v/c) = (1/X) is called the wave number. The 
wavelength corresponding to y = 500 is 69 cm, or about 2 ft; that 
corresponding top = 4,000 is 8.6 cm, or about 3 in. 

The Decibel Scale. — While physical instruments can be designed 
to measure intensity in ergs per sec per square centimeter or in micro- 
watts per square centimer (1 microwatt per sq cm = 10 ergs per sec 
per sq cm = 10 -6 watt per sq cm), the ear does not respond in a man- 
ner proportional to intensity or to amplitude. It is outside the scope 
of this book to go into detail concerning physiological acoustics, but it 
is necessary to include a brief outline of the subject in order to have an 
idea of the ranges of values of intensity and frequency encountered 
in acoustical work. 1 

The human ear is a remarkably rugged yet sensitive organ. It 
responds to a frequency range of about ten octaves, whereas the eye 
responds to less than one. It responds to air vibrations whose ampli- 
tude is hardly more than molecular size; it. also responds without 
damage to sounds of intensity 10 million million times greater (sound 
much louder than this becomes painful) . 

The response of the ear is not proportional to the intensity, how- 
ever; it is much more nearly proportional to the logarithm of the 
intensity. If we increase the intensity of a sound in steps of what 
seem to be equal increments of loudness, we find that the intensities 
form a sequence of the sort 1, 2, 4, 8, 16 • • • or 1, 10, 100, 1000 • • • 
and not of the sort 1, 2, 3, 4 • • • or 1, 10, 19, 28 • • • • Consequently, 
a logarithmic scale is often chosen in which to express acoustic ener- 
gies and intensities. The unit is the decibel (abbreviated db). 

A sound is said to be a decibel higher in level (or one decibel more) 
than another sound if its intensity is 1.259 (= \/l0) times the inten- 
sity of the other. If the ratio of intensities is 10, the difference in 
level is said to be 10 decibels (or one bel) ; if the ratio is 1,000, the differ- 
ence is 30 db, and so on. The difference in level of two sounds in 
decibels is equal to ten times the logarithm to the base 10 of the ratio 
between the intensities: 

1 Many of the data quoted in this section have been obtained from H. Fletcher, 
"Speech and Hearing," D. Van Nostrand Company, Inc., New York, 1929. The 
xeader is referred to this book or to Stevens and Davis, "Hearing, Its Psychology 
and Physiology," John Wiley & Sons, Inc., 1938, for further details. 



226 PLANE WAVES OF SOUND [VL22 

Level difference = 10 log 10 ( yM (22.13) 

Intensity and Pressure Level. — At times it is convenient to deter- 
mine the absolute value of the intensity of some sound in the decibel 
scale. This is done by giving its level above or below 10 -10 microwatt 
per sq cm (= 10 -9 erg per sq cm per sec). Thus the intensity level 
of the sound is 

Intensity level = 10 log 10 (10 9 T) =90+10 logic (T) (22.14) 

This reference level is chosen because it is approximately the minimum 
audible intensity at 1,000 cps. 

Unfortunately (or fortunately, perhaps) we seldom measure sound 
intensity; what is measured is usually root-mean-square pressure 
amplitude p ims . If the sound is a sine wave, traveling in one direc- 
tion, then Eq. (22.11) shows the relation between p r m S and T; T = 
(prms) 2 /42, for air at standard conditions, since (twO 2 = i|p| 2 . How- 
ever, if the wave is not a plane traveling wave, the relation between 
T and p is not at all as simple. Indeed there are many cases when we 
do not know enough about the sound field to be able to compute the 
intensity once we have- measured the pressure ; so that we cannot give 
the value of the intensity level if we have measured only pressure. 

To obviate this difficulty, we can define a level in terms of pressure 
squared, the reference level being 0.000200 dyne per sq cm which cor- 
responds (within 2 per cent) to the pressure amplitude in a plane wave 
of zero intensity level. The pressure level of a sound is 

Pressure level = 20 logio L ^7o-J = 20 r °g»W + 74 < 22 - 15 ) 

When the sound is a plane wave going in a single direction, then its 
pressure level is only 0.1 db higher than its intensity level; when the 
sound has a more complex distribution, the pressure level, may differ 
from the intensity level by many decibels ; but in any case if we change 
the amplitude of a sound but not its distribution in space the decibel 
difference between pressure and intensity level will remain constant, 
the value of the difference being characteristic of the space distribution 
chosen. 

The response of the ear is not exactly proportional to the decibel 
scale, though it corresponds much more closely than it would to the 
intensity scale. Corresponding to the physical quantities intensity 
and frequency are the physiological (or, rather, psychophysiological) 
quantities loudness and pitch. The loudness of the sound depends on 



VI.22] 



THE EQUATION OF MOTION 



227 



both its intensity level and its frequency; its pitch depends chiefly 
on frequency, but to seme extent on intensity. Contours of equal 
loudness for the average person are plotted in Fig. 52 in terms of 
intensity level and frequency. 

The bottom curve, for zero loudness, is the threshold of hearing, 
below which a sound of that frequency is inaudible to the average 
person. The upper contour is the threshold of pain, above which 
the sensation is more of pain than of sound (and the result is more or 



150| ,--.:,.,.W, 




20 



50 100 



5000 10,000 20,000 



200 500 1000 2000 

Frequency, cycles per second 

Fig. 52. — Contours of equal loudness, plotted against intensity level and frequency 
for the average ear. Contours are numbered by correspondence with intensity leve 
at 1000 cps. Unshaded portion, the auditory area, shows range over which sound can 
be heard. 



less damaging to the ear). About 5 db below this upper threshold 
the sound begins to "feel uncomfortable"; at the threshold plotted 
the sensation is perhaps best described as a "tickling sensation." 
About 5 db above this, the sensation is "pure pain." 

The area enclosed within the two thresholds represents the range 
of audible sound in frequency and intensity. We see that the ear can 
hear sounds of frequency as low as 20 cps and as high as 20,000, 
although at these extreme limits the intensity range perceptible as 
sound is very small. On the other hand, the range of perceptible 
intensity at 1,000 cycles is as much as 125 db. 



228 PLANE WAVES OF SOUND [VI.22 

This area covers the range of frequency and intensity that sound 
reproduction equipment must cover for perfect performance. The 
ear is most sensitive at about 2,000 to 4,000 cps. If our abscissa 
were pressure level at the ear diaphragm rather than intensity level 
just outside the ear, the wavy appearance of the loudness contours 
between 2,000 and 10,000 cps would not occur, for these fluctuations 
are due to "cavity resonance" in the outer ear canal, increasing sensi- 
tivity at about 4,000 cps and decreasing it at about 8,000 cps. In 
terms of pressure level at the ear diaphragm, the ear is most sensitive 
at about 1,000 cps. Cavity resonance will be discussed at the end of 
Sec. 23. 

An interesting result of the peculiar response of the ear, evidenced 
by the fact that loudness-level contours, as shown in Fig. 52, are 
parallel neither to the intensity-level contours nor to each other, is that 
a complex sound changes its quality when its over-all intensity is 
changed. Suppose that one hears a sound consisting of many different 
components, some at frequencies around 100 cps and some around 
1,000 cps, first when most of the components have individual intensity 
levels of about 100 db and then when their intensity levels are about 
20 db (owing either to the fact that the source is farther away or that 
the sound is recorded and played back at a lower level). In the first 
case, when the components are at about 100 db, all components are 
of approximately equal loudness (since the 100-db loudness level 
contour is roughly horizontal). In the second case, with the intensity 
level of the components about 20 db, the 100-cps components would 
not be heard at all (for this level is below the threshold for 100 cps), 
whereas the 1,000-cps components would still be heard. Therefore 
in order to change the loudness of a complex sound without changing 
its quality, the different frequency components should be attenuated 
by different amounts, according .to the loudness-level contours of 
Fig. 52. 

Sound Power. — The amount of power produced by sound gen- 
erators is very small compared with usual electrical powers. The 
average power produced by a person talking in an ordinary conver- 
sational tone is about 10 -5 watt, or 100 ergs per sec; although the 
range of power that can be produced by the voice varies from about 
1 erg per sec for very weak speech to about 10 4 ergs per sec for loud 
speech. Greater powers can be produced by singing voices, the range 
from pp to ff corresponding approximately to the variation from 10 3 
to 3 X 10 6 ergs per sec. 



VI.22] THE EQUATION OF MOTION 229 

Some musical instruments can produce greater power than the 
human voice. The violin cannot, but the trombone and cornet can 
produce about 4 X 10 5 and the bass drum about 6 X 10 5 ergs per sec. 
The ratio of maximum to minimum power produced by a band or 
orchestra while playing may be as great as 10 5 . The important range 
of frequency is from 30 to 8,000 cps. 

If we assume that the acoustic power from any of these sources 
radiates out equally in all directions (which it certainly does not do 
in many cases), then the relation between intensity and power output 
II would be T = Jl/^irr 2 . To this approximation, the intensity level 
due to a source of power output II ergs per sec at a distance of 3 ft 
(100 cm) would be 10 logio(n) + 39. The intensity level of a person 
talking normally 3 ft away is therefore approximately +60, corre- 
sponding roughly to a loudness of 60, according to Fig. 52 (since most 
components of spoken sounds are between 500 and 5,000 cps). The 
relation between power output and sound-intensity level farther from 
the source, particularly in rooms, will be discussed in Chap. VIII. 

Frequency Distribution of Sounds. — Since the ear does distinguish 
between sounds of different frequencies, it is often important to 
analyze the sound into its frequency components. This is, of course, 
what we have been doing throughout the book, even when we have 
discussed transient sounds. Referring to Eq. (2.19); we see that we 
can express any dependence of pressure on time by the formula 

p(t) = f "^ P{v)e-^ ivt dv 

where we can determine the function P(v) by the reciprocal relation 

P(„) = J_ f p(*)e 2 ""d< (22.16) 

The value of P (v) = \2P(v)\ or \2P{-v)\ is the pressure ampli- 
tude of that part of the sound in a unit frequency band around v 
(i.e., between v — \ and v + \ cps). If the sound is a plane wave, 
the square of P , divided by 2pc, gives the intensity of this part of 
the sound, T(v) = Pl(v)/2pc. The frequency distribution of a sound 
is often given in a decibel scale, in terms of the so-called spectrum 
level, which is the intensity level of that part of the sound in a unit 
frequency band at frequency v, 

Spectrum level = 10 logi [T(^)] + 90 (22.17) 

~20 1ogio[Po(»')] + 71db 



230 PLANE WAVES OF SOUND [VI.22 

where the second line is valid only when the sound is approximately 
a plane wave and when pc = 42. 

It should be pointed out that a measurement of spectrum level 
(which can be obtained from a sound analyzer) is not sufficient to 
give the actual shape of the pressure fluctuation, for it gives values 
of Po(v), which is proportional to the amplitude only of P{v) and of 
P( — v). It follows that the phase angle of the frequency component 
of pressure P(v) is not determined by the sound analyzer; there- 
fore Eq. (22.15) cannot be utilized to compute p(t), if we have only 
sound-analyzer data. This usually does not matter, for the ear 
seems to be less sensitive to the phase of P{v) than it is to its mag- 
nitude. This is, of course, not true of very short pulses, for here the 
phases must be related in just the right manner to produce the pulse. 
Consequently, specifying the spectrum-level curve of a pulse does not 
specify it uniquely. 

A "pure" noise corresponds to a constant value of P over a large 
range of frequency, whereas a note of one frequency has a P which 
is zero everywhere except at just one frequency, where it is very large. 
Musical sounds correspond to a series of sharp peaks in the curve for 
P , equally spaced in frequency, the peak of lowest frequency corre- 
sponding to the fundamental, the next peak, at twice the frequency, 
corresponding to the second harmonic, and so on. The area under 
each peak corresponds to the amplitude of that harmonic. 

The peaks for a musical sound are never in practice infinitesimally 
narrow, corresponding to exactly one frequency, for there are always 
slight variations in pitch and intensity during the production of the 
sound to give it a "spread" in frequency. As a matter of fact, only 
a sine wave that continues forever with undiminished amplitude and 
unchanged frequency can have a peak of zero width. Any starting or 
stopping of the vibration will spread out the peak. For instance, 
we can utilize Eq. (22.16) to show that a sinusoidal oscillation of 
finite length 

(0 (t < - £A<) 

p(t) = <P m cos (2rv t) (- $At < t < iM) 

{ (t> iAt) 

has a distribution in frequency corresponding approximately to 

r> / \ f P A sinh-^o - v)M] 



VI.22] 



THE EQUATION OF MOTION 



231 



(where we have neglected a term involving (v + v)~\ since it is 
usually much smaller, at least when v is approximately equal to v ). 

The approximation is valid if v is larger than (1/At), i.e., if the 
pulse contains several cycles. The quantity P (v) has a peak at 
v = vo, of height (P m At/2T), it falls to about half this value at 
v = v ± (l/2At), and is very small for (v - v) large. Figure 53 




Fig. 53.— Two pressure pulses, whose shape is plotted as a function of time, and their 
corresponding distribution-infrequency for intensity. 

shows curves of the distribution in frequency of the intensity [P§0)/2pc] 
as functions of frequency. In order to build a pulse of finite duration 
we must use waves of different frequency, so arranged that they all 
cancel out except for the duration of the pulse. The shorter the pulse 
the more different frequencies must be used to form it. A measure 
of the "spread" of frequency needed for a pulse is the half width Av 
of the peak in the distribution in frequency, the frequency difference 
between the two points on the curve that have half the maximum 
height. This half width is related to the length of the pulse b V the 
relation 



AvAt^l 



(22.18) 



This relationship is approximately true even if the pulse is not started 
and stopped abruptly, as long as A* is taken as the half duration of the 



232 PLANE WAVES OF SOUND [VI.22 

pulse, the length of time between two parts of the pulse where the 
amplitude is half the maximum amplitude. 

Thus, the peaks in the curve of distribution in frequency of music 
or speech have a width that is at least as wide as the reciprocal of the 
length of time during which the note is maintained unchanged in 
amplitude and frequency. In the case of music, the duration of the 
tone is usually longer than ^ sec, and the half width is quite small, 
the actual width of the peak being chiefly due to the vibrato effects 
of the musician, who usually makes the frequency fluctuate slightly 
during the production of the note. It may well be that this increased 
spread in frequency is the reason that the vibrato note sounds better 
than a pure tone; for we shall see in the last chapter that the wider the 
spread in frequency a note has, the less marked are its resonance 
effects in a room. 

In the case of speech, however, the duration of any particular sound 
is quite short, in the case of some consonants being as short as ^V. sec. 
In this case the half width of the peaks is large, and the feeling that 
the sound has a definite pitch is lost. 

The Vowel Sounds. — The vowel and semivowel sounds are more or 
less musical in nature, however; their distribution in frequency being 
a sequence of many separate peaks, corresponding to the first 10 or 
more harmonics of the natural frequency of the vocal cords, the width 
of each peak corresponding to the duration of the sound. A bass 
voice will have its fundamental at 100 cycles (or thereabouts) and will 
have 10 peaks below 1,000 cps. A soprano voice with its fundamental 
at 250 cycles will have only four peaks below 1,000 cycles. Ordinarily, 
most of the peaks are small in height, only two or three of them being 
large. The position of the large peaks in the frequency scale deter- 
mines the sort of vowel that is spoken. For instance, the vowel " ah " 
has the peaks near 900 cycles enhanced, whether they correspond to 
the eighth, ninth, and tenth harmonics for the bass voice with a funda- 
mental at 100 cycles or to the third and fourth harmonics for the 
soprano voice of fundamental frequency 250 cycles. What seems to 
be happening is that the oral cavity for all individuals shapes itself 
for the vowel "ah" so that it resonates to a frequency of about 900, 
enhancing whatever part of the sound, coming to it from the vocal 
cords, is near this frequency. 

The vocal cords differ from person to person, some sending to the 
mouth cavity pulsations that can be analyzed into a Fourier series 
with a fundamental frequency as low as 100 cycles, some with a funda- 
mental as high as 250 cycles, etc. The mouth, then, is shaped to pick 



VI.23] PROPAGATION OF SOUND IN TUBES 233 

from these frequencies the ones that correspond to the characteristic 
frequencies of the vowel spoken; or, from another point of view, the 
mouth is shaped so that the air in it is periodically set into vibration 
at the characteristic frequencies of the mouth by the pulsations sent to 
it from the vocal cords. The two points of view are essentially the 
same, one corresponding to an analysis of the motion in time and the 
other to an analysis in frequency. 

The characteristic frequency of long "oo" is 400, that of long "o" 
is 500, that of " ah " is 900. Some of the other vowels have two charac- 
teristic frequencies: Those of the "a" in "tan" are 750 and 1,600, 
those of long "a" as in "tame" are 550 and 2,100, those of long "ee" 
are 375 and 2,400, and so on. 

23. PROPAGATION OF SOUND IN TUBES 

A large number of sound-generating devices are tubular in shape, 
sound waves of large amplitude being set up inside the tube, and some 
of this stored-up energy being radiated out into the open. Organ 
pipes, wood-wind and brass musical instruments, and horn loud- 
speakers are sound generators of this type. Inside the tube the sound 
waves are approximately plane waves and can be treated by the 
methods of the preceding section. We shall discuss the behavior of 
sound in tubes of uniform cross section in the present section and deal 
with waves in tubes of changing cross section in the next section, 
leaving the more complicated radiation in the open to be treated in 
the next chapter. 

In many cases the various parts of the tube (valves, constrictions, 
etc.) are small in length compared with a wavelength of sound. When 
this is the case the behavior of sound in the tube is analogous to the 
behavior of electric current in a circuit with lumped circuit elements 
of inductance, capacitance, etc. We shall take up this case first and 
work out { he analogous electric circuits for a few examples. When the 
tube elements are longer than a wavelength of sound, the electric 
analogue is the transmission line. This more complicated case will 
be taken up later in the section. 

Analogous Circuit Elements. — A tube of variable cross section, 
where the length of the elements are shorter than a half wavelength 
of the transmitted sound, is analogous to an electric filter circuit. 
The analogue of the pressure in a tube element is the voltage across 
the corresponding part of the filter circuit, and the analogue of the 
current at some point in the circuit is the total flow of air, the particle 



234 PLANE WAVES OF SOUND [VI.23 

velocity times the area of cross section 8 of the tube at its correspond- 
ing part. 

The analogous impedance is thus not the specific acoustic imped- 
ance z (which is defined as pressure over velocity) but z divided by S. 
It must be emphasized that the electric circuit is merely an analogue, 
to help our analysis (since electric filters are more familiar to most of 
us than acoustical filters). The analogous current comes out in cubic 
centimeters per second, and the voltage is in dynes per square centi- 
meters. The capacitances and inductances are not in farads and 
henrys but in the analogous acoustical units, and the power trans- 
mitted comes out, not in watts, but in ergs per second. 

Tube elements are usually of two kinds: pieces of small cross- 
sectional area (opening at both ends into larger chambers), which we 
shall call constrictions; and chambers of larger cross section (feeding 
into constrictions at both ends), which we shall call tanks. The air 
in the constrictions is mass controlled, and therefore the analogous 
circuit element is an inductance; the air in a tank is stiffness controlled, 
so the analogous element is a capacitance. The physical reasons 
underlying these statements, and their proof, will have to be post- 
poned until Chap. VII, when we shall work out the necessary mathe- 
matics. In the meantime some very crude analysis will have to 
suffice to compute the values of the analogous elements. 

Constriction. — The air in a constriction has a total mass of pSl e g, 
where S is the cross-sectional area and l e the effective length of the 
tube. We use an effective length because it turns out that some air 
beyond the ends of the constriction moves along with the air in the 
constriction, and this must be included to obtain the effective mass. 
It will be shown in Sec. 28 that the effective length l e is related to the 
actual length I by the formula 

Z e ~ Z + 0.8 V£ (23.1) 

Therefore even a hole in a thin plate has a nonvanishing effective 
length, proportional to the square root of the area of the hole. 

The force accelerating the mass pSl e is the difference in pressure 
at the two ends of the constriction, multiplied by the area S. The 
acceleration -iwt* is therefore equal to S times the pressure drop, 
divided by ( P Sl e ); and the analogous impedance, the pressure drop 
divided by Su, is -ioWS). A constriction is thus analogous to 
an inductance of value 



(f)g/cm 4 



(23.2) 



VI.23] PROPAGATION OF SOUND IN TUBES 235 

where £ is the cross-sectional area of the constriction, I its length, and 
h is given by Eq. (23.1). 

Tank.— If total flow of air Su is analogous to current, the analogue 
of charge is S times the displacement £. If a volume of air S% flows 
in through one opening of a tank, the increase in pressure in the tank 
is obtained by using a variant of Eq. (22.5), p = -pc\dV/V), where 
V is the volume of the tank and -dV is the volume of air introduced, 
St The analogous capacitance is the ratio between analogous charge 
(££) and analogous voltage p 

Ca ~ (^) cm4 sec2 /g (23.3) 

It will be shown later [see Eqs. (23.14) et seq.] that, when the output 
end of a constriction opens into free space, the radiation load on the 
opening is represented in the equivalent circuit by an analogous terminal 
resistance 



R T ^ 



fe) 



(23.4) 



which is shunted across the termination of the circuit, in series with 
the analogous inductance L a of the output constriction. This resist- 
ance becomes very small for low frequencies, since a small opening 
radiates very poorly at low frequencies. The power dissipated in this 
resistance is the power radiated out of the acoustical net. 

Examples.— By use of these formulas in the frequency range where 
they are valid, we can build up various acoustic circuits and study their 
properties, by studying the behavior of the analogous electric circuits 
A few examples are shown in Fig. 54. 

Case a is the simple Helmholtz resonator, analogous to the simple 
L-C series circuit. The analogous impedance at A is 

and the resonance frequency is 



v = 



2tt VL a C a 



cps (23.5) 



Case b is a simple low-pass filter. Any standard text on filter 
circuits will prove that the cutoff frequency, above which the sound 
will not be transmitted, is at 



236 



PLANE WAVES OF SOUND 



[VI.23 



1 _1 lc*S 



Case c is a simple high-pass filter, where frequencies above this value 
vo will be transmitted. Other, more complex, circuits may be built 
up and analyzed in a similar manner. 



l]~ | ■*■!- Area S 





(«) 




-a "-a 






|..J ^ AreoS 



Area 5- 





(c) 



Fig 54 —Acoustical circuits on left and analogous electrical circuits on right. 
Constrictions of length I and cross-sectional area 8 are analogous to inductances La ; 
tanks of volume V are analogous to capacitances C«; as long as all dimensions oi indi- 
vidual elements are less than a half-wavelength of the transmitted sound. Analogous 
impedance at A equals ratio of pressure to S times particle velocity at A. 

The lumped circuit approximation is not so good a one for sound 
as it is for electric currents, because of the considerably smaller speed 
of sound and the corresponding shortness of wavelength for a given 
frequency. JJsually the formulas given above cease to have any 
validity for frequencies above 1,000 cps or so (for when an acoustical 
system is longer than a half-wavelength it behaves like a transmission 
line, and the wave formulas must be used). 

We have now encountered three kinds of acoustic impedance, a 
confusing redundancy only partly excused by the fact that the different 
impedances are useful in different kinds of calculations: the analogous 
impedance when we deal with lumped-circuit elements at low fre- 
quencies, the specific acoustic impedance when we deal with transmis- 



VI.23] PROPAGATION OF SOUND IN TUBES 237 

sion-line calculations, and the usual mechanical radiation impedance 
when we calculate the coupling between the waves in the tube and the 
driving piston or the load at the output end. We list the three 
impedances here for comparison; their ratio turns out to be the cross- 
sectional area S of the tube. 

Analogous impedance Z a — (pressure/volume flow) 
(Greek subscripts, see page 234). Used in analogous circuits. 

Specific acoustic impedance z = (pressure/velocity) = SZ a 
(Lower case z, see page 239.) Mechanical impedance per unit area. 

Radiation impedance Z a = (force/velocity) = Sz = S 2 Z a 
(Lower case subscripts, see page 238.) The part of the mechanical 
impedance due to the sound field. 

There is also the acoustic impedance ratio f = — ix = z/pc, the ratio 
of the specific acoustic impedance to that of the medium, as indicated 
on the next page. 

In each of these cases we can draw equivalent electric circuits to 
help us understand the behavior of the acoustical system. These 
circuits are only analogous, however, for the impedances do not have 
the dimensions of ohms; instead of wires and amperes and volts we 
are dealing with tubes and air flow and pressure. Since we are more 
familiar with electric circuits than with acoustic ones, the equivalent 
circuits are often useful. The circuits equivalent to the analogous 
impedances are usually most useful, for there is a direct relation 
between currents and voltages in each part of the equivalent circuit 
and the pressure and volume flow in the corresponding part of the 
acoustical circuit. Another way of stating the same property is that 
the analogous impedance does not change markedly with distances 
along the tube of less than a half wavelength, even though the tube 
cross section changes markedly. Equivalent circuits for z and Z a are 
more difficult to interpret, but are sometimes helpful. 

Characteristic Acoustic Resistance. — We are now in a position to 
study the behavior of plane waves of sound in tubes of uniform cross 
section, which are longer than a half wavelength. The treatment 
will be quite analogous to that of Sec. 10 for wave motion on a string. 
We first take the case of a tube of infinite length, with no reflected 
wave, to get the simplest sort of air reaction. We suppose that we 
have a tube of uniform cross-sectional area S, starting at the origin 
and extending to the right along the #-axis for such a great distance 
that no wave is reflected back to the origin from its far end. Suppose 
that we fit the end at x = with a flat-topped piston which can vibrate 
and generate waves, so that its displacement, as a function of time, 



238 PLANE WAVES OF SOUND (VI.23 

is E(£)- The wave produced in the tube must be such that the dis- 
placement of the air next to the piston (i.e., at x = 0) is equal to 
S(0- Such a wave would be 



-0-5) 



which travels to the right in the tube with velocity c. 

The reaction F of the air back on the piston will be equal to the 
area of cross section of the tube S multiplied by the excess pressure p 
due to the wave. Using Eqs. (22.7) for the pressure, we have 



F = Sp = -Spc 



(©--"*[*(' -31 



<, (dZ\ . ( . _ dZ(z) 



(23.6) 



The reaction on the piston is therefore proportional to the velocity 
of the piston, a purely frictional reaction. The proportionality 
constant (Spc) is called the radiation resistance of the air on the piston. 
The resistance per unit area of the piston R = pc is called the charac- 
teristic acoustic resistance. Its value for air at normal conditions is 
42, as we showed on page 222. This is the expression used on page 
199 for the reaction on the diaphragm of a condenser microphone. 

We have just seen that when the tube is so long that there is no 
reflected wave the acoustic resistance on the piston is independent 
of the type of motion of the piston. When the tube is not infinitely 
long, however, there will be some wave motion reflected back from 
the far end which will also affect the piston. When this happens, the 
reaction on the piston will be different for different motions of the 
piston. It will then be necessary to analyze this motion into its 
simple harmonic components and to treat each component separately. 

Incident and Reflected Waves. — When the sound wave is not all 
radiated out of the far end of the tube, both incident and reflected 
waves will be present. By referring to Eqs. (22.11) and (22.12), we 
see that the pressure, particle velocity, energy density, and intensity 
of a plane wave in a uniform tube with both waves present are 

~y _ p gik(x-ct) _J_ p_g—ik(x+ct) 
( zl \ — Jl (P e iHx-ct) _ p _ e -ik{x+ct)-y 

J_ ,™ , ™s > (23.7) 

2pc 2 

J_ 
2pc 



W-^Pl+Pl) 



t = ^r (P+ - p -) 



VL23] PROPAGATION OF SOUND IN TUBES 239 

To simplify the equations we shall express the ratio of the ampli- 
tudes of the waves in the two directions by means of an exponential: 
-(P-/P+) = (A-/A+) = (U-/U+) = e-w, where $ is, in general, a 
complex number; in other words, $ = irao — iirfio. Twice the imagi- 
nary part of yf/, —2x^ , gives the phase angle between the two waves 
at x = 0; and e~ 2Re ^ ) [where Re{<f) = ira is the real part of ^] is the 
ratio between the amplitudes of the two waves, without regard to 
phase. The ratio between the numerical magnitudes of the intensities 
of the two component waves is e~* ReW = e~~ 4wa . The expressions for 
the pressure and particle velocity, and their ratio, then become 



= 2P + e-+- 2 * i > t sinhU + ^llA 

\Wai) = 



<x) = (?P±) e -*-^« ..... L . *•■«* > (23- 8 ) 




z ( x ) = \ a // aj ) = pc tanh 



Specific Acoustic Impedance. — The quantity z(x) in Eq. (23.8), 
the ratio between the pressure and particle velocity, is called the 
specific acoustic impedance at the point x for the frequency v. It is a 
most useful quantity, for, once it is known, the reaction of the air 
on a vibrating system can be determined. As an example of its 
utility, consider a uniform tube closed at one end by a cap that yields 
somewhat to pressure and having a flat-topped driving piston at the 
other end. If we know the mechanical impedance of the cap, the 
ratio of driving force to cap velocity, we can determine the specific 
acoustic impedance of the air in the tube next to the cap's inner 
surface. For the velocity of this air must equal the velocity of the 
cap, and the pressure times the area of cross section of the tube must 
equal the force on the cap. Knowing z at the cap, we can find ^; and 
knowing \j/, we can find the value of z at the piston end of the tube. 
If this value is known, we can find the ratio between the driving force 
on the piston and its velocity, which is the effective mechanical 
impedance Z m of the piston for the frequency v. If the piston is driven 
by a force of magnitude i^o and frequency v, the velocity amplitude 
of the piston will be F divided by the value of Z m for the frequency p. 
Similarly, if the driving force is a combination of several components 
of different frequencies, the piston velocity is a similar combination 
of velocities of different frequencies, each with amplitude equal to 
the ratio between the magnitude of the component force and the 
impedance for the corresponding frequency. Thus a consideration 



240 PLANE WAVES OF SOUND [VI.23 

of the behavior of the specific acoustic impedance of a plane wave will 
enable us to work out the details of its interaction with various mechan- 
ical systems. 

In a great many cases it is most useful to express the acoustic 
impedance in units of the characteristic impedance of the medium, 
pc. The dimensionless quantity f = (z/pc) = 6 — ix is called the 
acoustic impedance ratio, and its reciprocal 77 = (1/f) = k — ia the 
acoustic admittance ratio. Its value indicates the amount of impedance 
mismatch at any change of medium or change of cross section. Where 
sound strikes the surface of a medium other than air (as, for instance, 
the bounding surface of a tube or the wails of a room) then the value 
of the admittance ratio 77 for the surface is a measure of the effect the 
surface will have on the wave motion striking the surface, as will be 
shown later. 

Standing Waves. — Equations (23.7) and (23.8) are duplicates of 
those discussed in Sec. 13 for waves in strings. The properties of the 
hyperbolic functions were given in Eq. (13.7); and the relationship 
between the behavior of the waves, which was discussed there, can 
be applied to the present case, though the motion here is longitudinal 
instead of transverse. The air displacement corresponds to the string 
displacement and the pressure to the transverse force. We note that 
the measurable quantity here is usually the pressure, whereas the 
measurable quantity with the string is the displacement. 

We shall need to distinguish between the real and imaginary parts 
of the quantity in the brackets in Eqs. (22.14), as before, by setting 

* + fe^*\ = x(a _ ij3) 

a = (l/ir)ifc#); = -(lA)/m(^) - (2x/X) 

where Re(\p) is the real part and Im(4>) the imaginary part of rp, and 
X = (c/v) is the wavelength of the sound wave. In terms of these 
quantities the acoustic impedance ratio, the amplitude of the pressure 
fluctuations, the amplitude of the air velocity, the energy density and 
the sound intensity at the point x are, respectively, 

I — J = f = - ix = tanh[x(« - #)] 
|pj = 2P+e-* a Vcosh 2 ^) - cos 2 (7r/3) 
\ u \ = (— ) e~* a Vcosh 2 (ra) - sin 2 (x/3) ^ 



VI.23J PROPAGATION OF SOUND IN TUBES 241 

The minima of p are at the same places the maxima of string dis- 
placement were in the equations of page 141. This is because p corre- 
sponds to (dy/dx) for the string, whereas £ corresponds to y, so that 
the quantities observed are opposite in behavior with respect to ft 
Plates I, II, and III at the back of the book can be used in the present 
case as well as for the string, however, as long as this difference is 
kept in mind. 

Suppose we consider the case where the "load" is at x = I and the 
driving piston is at x = 0. The average impedance ratio of the load 
is (Z m /pcS) = izi/pc) = 6i — ixi, which is related to the corre- 
sponding phase parameters for the load by the equation di — ixi = 
tanh[7r(a!j — ipi)]. We are neglecting attenuation along the tube 
so that a = ai, a constant. However, = /3o — (2x/\) = ft + 
(2/X)(Z — x), so that /3 changes as we go along the tube, increasing 
as we move from load to source, and decreasing as we go from source 
to load. Therefore as we go from load to source the point correspond- 
ing to the impedance ratio moves from (aj,ft) on Plate I or II along 
a curve of constant a, in the direction of increasing ft The point 
moves in a clockwise direction by an amount of complete circuits equal 
to the number of half wavelengths in the tube length. This motion 
eventually brings us to the point (a ,ft>), where a = m, 0o = ft + 
(2Z/X). From this we can compute the load on the driving piston due 
to the air column plus load Z r = pcS tanh {ir[ai — ft — (2il/\)]}, by 
means of Plates I or II. 

The amplitude of the pressure fluctuation is a maximum where j3 
is a half -integer (•••—£,+ £,+ $•••). At these points the 
amplitude of the particle velocity is a minimum. Where /3 is an 
integer ( • • • — 1,0,1,2 ••• ) the pressure amplitude is a minimum 
and the amplitude of the particle velocity is a maximum. Values 
of pressure and velocity amplitudes and phase angles can be read 
from Plate I in the back of the book. 

The values of the real and imaginary parts of (z/pc), 0, and x, 
corresponding to given values of a and ft are given in Plate II and 
in Tables III and IV at the back of the book. Alternatively, the 
magnitude of f and its phase angle can be obtained from Plate III 
and Table III. Examples of the use of the figures for calculation were 
given on page 138. 

The usefulness of the foregoing analysis will become more apparent 
later in the chapter. At present, we can see from Eqs. (23.8) that 
in wave motion composed of two plane simple harmonic waves travel- 
ing in opposite directions, if the specific acoustic impedance at some 
point x can be represented by the values R and X or by the con- 



242 PLANE WAVES OF SOUND [VI.23 

stants a and /?o, then the impedance at some other point x is given 
by the values of X and R corresponding to a = « and 18 = /3 — 
(2/\)(x — x ). The different impedances for different points along 
the wave correspond to the points on a circle of constant a in the 
figures, with /3 equal to the distance along the wave in units of half 
wavelength. Both R and X repeat themselves each half wavelength 
increase of x. 

The expression for the intensity in Eqs. (23.9) is obtained from 
those for the maximum pressure and particle velocity by multiplying 
the product of yw a nd (dg/da pmax by the cosine of the phase angle 
between them, (R/\/R 2 + X 2 ). This gives the maximum intensity, 
and the average intensity T is one half of this. 

This intensity is independent of (3 and is therefore independent 
of x. Although the pressure, the particle velocity, and the phase 
angle between them all change with x, they change in such a manner 
that the flow of energy along the wave is everywhere the same. This 
must be, for otherwise the energy would pile up at certain points and 
would continually diminish at others. 

We have assumed heretofore that a is independent of x, as indeed 
it is in a true plane wave, when there is no energy absorption during 
the progress of the wave. In some cases in practice, however, energy 
is absorbed by the walls of the tube (or in the air itself) as the wave 
travels along, and a will also depend on x. This case will be discussed 
later in this section. 

Measurement of Acoustic Impedance. — We can use the results 
of Eqs. (23.9) to devise a method for measuring specific acoustic 
impedance. We fit on the end of the tube, at x = I, the material or 
device whose impedance is to be measured; and a source of plane waves 
is placed at the input end, x = 0. The mechanical impedance of the 
device, Z m ( = force divided by velocity), divided by the cross-sectional 
area of the tube, S, is equal to the average specific acoustic impedance 
z ( = pressure divided by velocity) at the end of the tube, x = I. This 
terminal impedance is related to the quantities a and of Eq. (23.9) 
by the usual formula 



(s)- 



= 0i — ixi = tanh[7r(orz — ift)] 



The dimensionless quantity f = (z/pc) is the acoustic impedance ratio 
of the load, and and x are called the acoustic resistance and react- 
ance ratios respectively. 



VI.23] PROPAGATION OF SOUND IN TUBES 243 

The impedance, pressure, and particle velocity at any point of the 
tube are then given by the equations 



|p| = A -\/cosh 2 (xa) — cos 2 (7rj8) 

Id = (— I Vcosh 2 (7ro:) — sin 2 (x/?) 
\pcj 



m 



where 

a = oc i; j8 = ft + (2/X)(Z - a) 
As we have mentioned in our discussion of Eqs. (23.9), the pressure 
amplitude \p\ is minimum where is an integer, or (2/X)(Z — z) = 
n — jSi, and the value of \p\ at these points is A sinh (rat). Conversely, 
the pressure amplitude is a maximum where /3 is a half-integer, or 
(2/X)(£ — x) = n + i — ft, and its value there is A cosh (raj). 

Therefore we move a microphone (a small one to avoid distorting 
the wave) along the tube, measuring the distances from the output 
end of the tube (where the unknown impedance is placed) to the points 
where the pressure amplitude is a maximum or minimum, and meas- 
uring the values of \p\ at these points. The distance between minima 
(or between maxima) is a half wavelength (X/2). The distance 
between the output end and the nearest minimum, divided by (X/2), 
is 1 — ft. And the ratio between the value of |p| at its minimum to 
its value at its maximum is tanh(7ra!z)- Therefore both oti and 8 t 
can be determined, and the value of the mechanical impedance of the 
device, Spc(di — ixi), can be computed from Table III at the back of 
the book. The method is particularly accurate for values of imped- 
ance corresponding to values of a larger than 0.02 and smaller than 0.5. 

Damped Waves. — In actuality sound waves cannot transverse a 
tube without losing at least a small part of their energy, either by 
absorption at the walls of the tube or by the viscous friction impeding 
the air nearest the wall, or else by absorption in the air throughout 
the tube (or for all three reasons). Some of these effects will be 
discussed in more detail in the next chapter; here we are interested 
only in the result, which is that the amplitude of each wave is damped 
out as it progresses along the tube. The incident wave has a factor 
e -Kx/c anc j tn e reflected wave a factor e KX/c , where k is the damping 
constant for the wave. Therefore the correct expression for the pres- 
sure is 

p(x) = 2P + e-' a »+* T ^- iw ' sinh x I a — — — tjSo + i -^J 



244 PLANE WAVES OF SOUND [VI.23 

This means that in actual cases a, as well as /3, depends on x. The 
dependence is not usually so pronounced as with /3, for (k/c) is usually 
much smaller than 2w/\). However, when a depends on x we cannot 
simply take the ratio of the minimum value of \p\ along the tube to 
its maximum value to obtain the value of tanh (7ra) ; for each maximum 
and each minimum will have a different value. What is necessary is 
to plot the values of the maxima (against the corresponding values of 
x where they occur) on semilogarithmic paper and join the points by 
a smooth curve, also doing the same for the minima. The difference 
between the two curves, extrapolated to x = I, will then give the value 
of tanh^aj), which is used (via Table III) to obtain the acoustic 
impedance at the termination. 

The rest of the calculations of this section will be made tacitly 
assuming that a is independent of x. In general, and for most pur- 
poses, this assumption is a good enough approximation so that the 
additional calculations incident to the inclusion of damping are not 
worth while. If damping must be taken into account, however, the 
effects can usually be computed by considering that a varies linearly 
with x, inserting a = ao — (kx/tc) into the formulas that are to be 
developed. 

Closed Tube. — The simplest case for calculation is that of a uniform 
tube closed at x = I by a rigid plate. In this case the impedance at 
x = I is infinite, corresponding to ai = 0; ft = -g-. The specific 
acoustic impedance at x = is therefore 

' , [ (l . 2l\~] . r/27rZ\ . tt] . /coZ\ 

pc tanh \ —tti I H + t"/ = —ipciem l-yj + o =*pccot(— J 

If a piston were placed at the input end of the tube (x = 0), it would 
experience, because of the air in the closed tube, a mechanical imped- 
ance 

Z r = iSpc cot (j) = iSpc cot (?~j (23. 10) 

This is a pure reactance (although there would be a small resistive 
term if we took damping into account). 

At low frequencies (wl/c) becomes small and we can use the first 
term in the series expansion for the cotangent 



^6)W 



[Z«(c/co)] (23.11) 



VI.23] PROPAGATION OF SOUND IN TUBES 245 

which is a stiffness reactance, as would be expected of air in a closed 
tube. As noted earlier, the analogous impedance is equal to the corre- 
sponding mechanical impedance divided by S 2 , so that the analogous 
impedance for a closed tube (for I smaller than \/%r) is 



-my 



This justifies Eq. (23.3). 

A closed organ pipe is a closed tube driven at the "open" end 
x = by a jet of air blown across the opening. The jet is so adjusted 
that it is very sensitive to flow of air into and out of the tube, so that 
a small change in its direction makes a change in the pressure at the 
open end. In addition, the jet of air is adjusted so that it strikes a 
sharp-edged "lip" at the opposite side of the opening, which tends to 
set up a periodic motion perpendicular to the boundaries of the jet 
(sometimes with periodic vortex formation), the period depending on 
the speed of the air in the jet and on the distance between jet and 
"lip." This motion (called an edge tone) produces a net pressure 
fluctuation on the air in the closed tube. If the fluctuations are at 
the frequencies for which the impedance near the mouth of the tube 
is small, the tube response is large and, by reacting back on the jet, 
the oscillations will settle down into a vibration that is nearly simple 
harmonic. 

The frequencies to which the closed tube most strongly responds 
are those for which the Z r of Eq. (23.10) is smallest, i.e., for (wZ/c) = 
(n 4- £V, or 

Xn =~; v« = xi n (n = 1,3 » 5 ' ' ' ) (23>12) 

If now the closed organ pipe is "blown" at just the right air speed, 
the frequency of transverse vibration of the jet will equal the funda- 
mental frequency of the tube vi = (c/4Z), and the sound will consist 
almost entirely of the fundamental; the odd overtones will be present 
to some extent because the jet oscillations are not purely sinusoidal 
even when aided by the tube resonance. 

When the pipe is blown more strongly, the jet frequency is first 
held near the fundamental free frequency of the tube by the strength 
of coupling, but when the edge tone by itself would exhibit a frequency 
close to the third harmonic of the pipe, the note suddenly changes to 
this overtone and "locks in" at the new frequency. The pipe is then 
said to be "overblown." As indicated in Eq. (23.12), only the odd 



246 PLANE WAVES OF SOUND [V1.23 

harmonics are present to any extent in the sound from a closed pipe; 
the dependence of the amplitudes of the higher harmonics on the 
dimensions of the driving jet of air and on the location and shape of 
the "lip" regulating the edge tone is too complex to analyze in detail 
in this book. 

Open Tube. — The other case of particular interest is that of a 
uniform tube (of circular cross section of radius a) which is open at 
the end x = I. In Ghap. VII [Eq. (28.6)] we shall show that if the 
open end is fitted with a flange that is wide compared with a wave- 
length (so that the open end is effectively a round hole in an infinite 
plane wall) then the acoustic impedance at the open end, looking out- 
ward, is 



Ji(w)] 



zi ~ pc 1 - 2 ^Jp I - ipcM{w) = pc(0 o - *xo) (23.13) 



where w = (2<oa/c) = (47ra/X), where a is the radius of the tube and 
of the open end and J\ is the Bessel function given in Eq. (2.5). 

Values of the acoustic resistance ratio O = 1 — [2Ji(w)/w] and of 
the acoustic reactance ratio M(w), for an open end, are given in 
Table VIII at the back of the book. Their behavior for limiting 
values of w is 



If 

Xo = M(w) 



9o = 1 _ 2 ZlW^ («V8) <»<0.6) 

w [ 1 (w > 5) 

(4w/3t) (w < 1) 

(4/ttw) (w > 12) 



(23.14) 



The approximate formulas differ from the correct values by less than 
10 per cent within the designated ranges of w. 

Consequently, the limiting values of specific acoustic impedance 
at the open end are 

- (a - • \ I (p« 2 ^ 2 /2c) - io)(8 P a/Sir) (X > &ro) 
zi - P c{d t X o) — \ pc _ (i/ w )(2pc 2 Aa) [X < fcra/3)] 

At low frequencies the resistive term is quite small, so that very 
little energy is lost from the open end. Open tubes having cross- 
sectional perimeter much smaller than the sound wavelength are 
therefore nearly as good hoarders of energy as are closed tubes, for 
only a small percentage of the stored energy can be radiated away in 
any cycle. The reactive term is a mass load, equal to a mass of air 
(8pa/37r)g per sq cm of opening. When there is no flange on the end 
of the tube, or when the flange is less than a wavelength in size, the 



VI.23] PROPAGATION OF SOUND IN TUBES 247 

reactance is reduced somewhat in magnitude, changing from (8pa/3ir) 
to approximately 0.6 pa. Equation (23.1) for the effective length of 
a tube with two open ends, used an approximate correction inter- 
mediate between these limits. 

For short wavelengths, the impedance is almost entirely resistive, 
approaching in value the characteristic acoustic resistance pc. In 
this case there is practically no reflected wave; the incident wave 
radiates out of the open end with little subsequent spreading, as 
though the tube were still guiding it. The sound wave, as it emerges 
into the open, does not realize that it has left the tube and so sees no 
need for sending back a reflected wave. 

The values of a and corresponding to the impedance Zi for the 
open end will be labeled a p and /3 P . Their values as functions of 
w = (^irva/c) are given in Table VIII at the back of the book. For 
very large or very small values of w they become 

I 27r(a/X) 2 = (co 2 /8x) (X > &ra) 

ap — \ (l/2ir) ln(artyX) = (l/2r) ln(«ti>/2) [X < (ra/3)] . 
(16a/&rX) = (4w/3tt 2 ) (X > 8xa) 
(i) [X < (W3)] 

According to Eqs. (23.9), the specific acoustic impedance at the 
driving end of the tube, x = 0, is 

pc tanh LY a p - ip p - i — J (23.16) 

. tan(27rZ p /X) +z27r 2 (a/X) 2 ,. ^ , 

~ lpC l-^ 2 (a/X) 2 tan(2x^/X) (X > Swa) 

pc-i^- e** il/ * [X < (W3)] 

Twa 

where l p = I + (8a/3x). 

Small-diameter Open Tube. — For wavelengths long compared with 
the tube perimeter 2xa, the input impedance is almost purely reactive, 
except at the resonances and antiresonances [where tan(27rZ p /X) is 
zero, or where it is infinity]. When the frequency is low enough (so 
that X > 2l p ) , then the input impedance becomes 

z — —ipcy^- 2 ) = —iuplp = —iupll + ^-)> X > 2l p ^> 4ra 



z = 






X 



In this case the impedance is a mass load equal to a column of air of 
length l p = I + (8a/37r) ; the correction for the effective length of the 



"248 PLANE WAVES OF SOUND [VI.23 

column being proportional to the tube radius a (if the flange is not 
large, the correction is approximately 0.6a). If the tube were open 
at x = also, the correction would be twice this: l e = I + (16a/37r). 
The effective length of the inductive element given in Eq. (23.1) is 
intermediate between 1.7a and 1.2a (where S is set equal to xa 2 ). 

As* long as the wavelength is long compared with 4ira, resonance 
occurs (zq minimum) when tan (2tI p /\) is zero, i.e., when 

27 r 

K = =g; v n = ^-n (n = 1,2,3 • • • ) (23.17) 

where l p = I + (8a/37r) for a large flange, ~ I + 0.6a for a small 
flange. The impedance at the input end for these frequencies is 

a pure resistance which is lowest for the fundamental frequency and 
increases as n 2 for the higher harmonics. 

An open organ pipe is driven by a jet of air similar to that for a 
closed organ pipe. When it is blown at the right jet speed, most of 
the energy is in the fundamental, but there are small amounts of all 
the upper harmonics present. With increasing jet speed the second 
or higher harmonics can be emphasized. 

A flute is also an open tube, set into oscillation by a jet of air, 
which the player blows across the end at x = 0. The various musical 
notes are produced by varying the effective length of the tube (by 
opening one or more holes in the side of the tube) and by varying the 
strength of the jet, so as to emphasize one or the other of the har- 
monics. The inner bore of the actual flute is not a uniform tube, but 
is conical in shape; the effect of the conical shape will be discussed 
later in this chapter. 

Reed Instruments. — Clarinets are tubes, fairly uniform in diameter, 
open at the output end, which are driven by a reed set into vibration 
by a blast of air into the tube. Coupling with the air column in the 
tube sets the reed into periodic oscillation, the air blast through the 
reed being modulated by the motion of the reed. The driving system 
here is more analogous to a constant-current generator than to a 
constant-voltage one; the greatest response is when the specific 
acoustic admittance (1/zo) is smallest, or z largest. The natural 
frequencies are when tan (2rrl P /\) is infinite, or when 

X m = ^; v m = 4- m (m = 1,3,5 • • • ) (23.19) 

in 4tj> 



VI.23] PROPAGATION OF SOUND IN TUBES 249 

In other words the natural frequencies of a "closed pipe" (i.e., 
open at only one end) excited at the "closed" end by a reed are the 
same as those of a "closed pipe" excited at the "open" end by a jet 
of air, as given in Eq. (23.12). 

The impedance at the driving end of the tube at the antiresonance 
frequencies given in Eq. (23.19) is 

as long as X TO is considerably larger than 4ira. 

The air jet through the reed or reeds is modulated by the reed 
vibration, and the reed vibration is maintained by the pressure fluctu- 
ations caused by the modulation, so the complete analysis of the 
motion would be quite intricate. A crude approximation to the 
actual state of affairs for a clarinet can be obtained, however, by 
assuming that the force driving the reed or reeds is an impulsive one, 
with the fundamental frequency of the tube i 



''-'•S-I- 15 ^} 



= gfo[ • • • - Kt + T) + 5(0 - 8(t - T) + 8(t - 2T) - • • • ] 

(where T = 2l p /c), corresponding to a positive impulse (tF /2) at the 
beginning of each cycle and a negative one — (wF Q /2) a half -period T 
later. This may be caused by the production of vortices at the tip 
of the reed; at any rate it is governed by the air vibration in the tube 
and possibly by the player's tongue. 

Motion of the Reed. — The reed has its own resonance frequency 
(a)r/2r), which is usually several times larger than vi. The reed 
impedance is therefore (m r / — io))(a)l — co 2 ) where m r is the equivalent 
mass of the reed, and the displacement of the reed from equilibrium is 



^l (F /m r ) r , . , _ ire 

Ji = 

(23.21) 

The air flow through the reed is proportional to the difference 
between the pressure P maintained in the mouth of the player and the 
pressure p at the input end of the tube and to the width (y + y ) of 
the opening at the reed, where y is the equilibrium width. The 
particle velocity at the input end of the tube is therefore 



250 PLANE WAVES OF SOUND [VL23 

u^o = G(P - p)(y + y) 

where G is the "conductivity constant "-of the mouth piece. Jf u is 
expressed in terms of a Fourier series in odd harmonics, 

00 

w*-o = U + 2 #»e~~* ,,l(2n+1)< (23.22) 

w = 

then the pressure at x = is obtained by multiplying each term in the 
series for the velocity by the corresponding specific acoustic resistance, 
obtained from Eq. (23.20), 

°° |- -, 

P-o = fii 2 [ (2» + l).» J e_l ' wl(2n+1)< J «i = (8pcZ p 2 A 2 « 2 ) 

This is valid as long as [2l p /ira{2n + 1)] is larger than unity; otherwise 
our approximate formula for the impedance of the open end [first 
formula of Eq. (23.16)] is not valid. Ordinarily l p is approximately 
50a, so that the series is valid for n smaller than 10 or 15 (2n + 1 
smaller than 20 or 30) The resistance R ± is approximately equal to 
10 5 , indicating that the pressure fluctuation at the mouthpiece is much 
larger than the velocity fluctuation (which corresponds to the fact 
that the mouthpiece is at a velocity node). 

We next insert the series for y [Eq. (23.21)] and p in the expression 
for u and solve to obtain the values of the constants U n . At first we 
shall neglect the small term Gyp and set 

u = Uo + 2 u » e ~™ l(2n+1)t = GP Vo + GPy ~ Gy p 

^Li 1 — {a)i/ur) 2 {2n + 1) 



e — iui(2n+l)« 



— Gv a R\ ^? — e -wi(2«+i)i 

Equating coefficients of the different exponentials, we have 

Uo = GPy ) 

v = B{2n + l) 2 } (23.23) 

^ n [(2w + l) 2 + M 2 ][iV 2 - (2» + l) 2 ] ) 

where 5 = (GPF /m»«!), M 2 = GRiy ; and where iV = (« r /«i) is the 
ratio between the reed frequency and the fundamental frequency of 
the tube (N ~ 10 for the clarinet). 

Pressure and Velocity at the Reed. — The pressure in the tube, a 
distance x from the mouthpiece is, therefore, to the first approximation, 



VI.23] 



n = 



PROPAGATION OF SOUND IN TUBES 

(_l)ng-io> l (2n+l)t 

[(2» + l) 2 + M*][N 2 - (2n + l) 2 ] ' 



251 



• sin \~ (2n + l)(l p -x)+i g (2n + l) 2 ] 

The expression for the fluctuations in the velocity can be obtained 
from Eq. (22.6), so that 



uc~ Uo + 



RiB 
pc 



j(— \\n e iat(2n+l)t 



» = 



[(2» + l) 2 + M 2 ][N 2 - (2ft + l) 2 ] 



Ul 



pc 



cos I ~ (2ft + l)(l p - *)> i j£ (2n + l) 2 



and the pressure and velocity at the effective output end of the tube 
u One Period >| 




Time 



Fig. 55. — Motion of reed, wave pressure and air flow at mouthpiece for a simplified 
model of clarinet. Resonance frequency of reed is ten times the fundamental tube 
frequency. 

[x = l p = I -}- (8a/37r)] are 

Pwmh „ pcB *y __._(»» + l)HX-l)v*"<*+»« 

U + 



Ux—lv 



[(2n + l) 2 + M 2 ][N 2 - (2ft + l) 2 ] 

RlB ^ t '(_l)ng-i«i(2n+l)t 

pc 



[(2ft + l) 2 + M 2 ][N 2 - (2ft + l) 2 ] 



Curves for pressure and velocity at the mouthpiece are shown in Fig. 
55. We confirm that at the beginning of each half-period there is a 



252 PLANE WAVES OF SOUND [VI.23 

peak of back pressure from the tube at the mouthpiece, and a corre- 
sponding reversal in the motion of the reed. During the rest of the 
cycle the oscillations correspond closely to those for the free vibration 
of the reed, which has a natural frequency ten times the tube funda- 
mental (N = 10) for the case shown. This justifies (approximately) 
the assumption made to obtain Eq. (23.21) for the motion of the reed. 

We notice several interesting facts concerning the air vibrations 
in this approximate solution for a clarinet. Here, in distinction from 
the flute and open organ pipe, the odd harmonics are excited and the 
driving end is at a minimum of the a-c component of particle velocity, 
rather than a minimum of pressure. At the open end the a-c com- 
ponent of velocity is large (because Ri is large) and the pressure is 
small, as it must be at an open end of dimensions smaller than a 
wavelength. 

The frequencies in the sound which are emphasized are the funda- 
mental of the tube and those overtones which are near the resonance 
frequency of the reed, which is near the ninth or eleventh harmonic 
for tones in the middle register of the clarinet. The total output 
power can be computed from the pressure and a-c component of 
velocity at either the mouthpiece or the output end, by multiplying 
the a-c velocity and pressure, term by term, averaging and adding 
and finally multiplying by the tube area, S = ira 2 : 



n ~ — 

— 2 



RlB2 2{[(2n + l) 2 + MW- (2» + 1)»]} (23 ' 24) 



where the expressions for the constants R lf B, M, and N are given on 
the preceding pages. The individual terms in the series equal the 
power output in the different harmonics, and the square root of each 
term is proportional to the pressure component for that frequency 
in the sound which is heard. 

Even Harmonics. — In the actual sound from a clarinet there are 
present a few components of even harmonics, which are not included 
in the series given above. This is because the modulation of the air 
flow by the reed is not purely linear : in other words we have neglected 
the term Gyp in the expression for u. This is small, but it does bring 
in even harmonics. The largest even terms are due, of course, to 
the product of the largest terms in the series for y, which is for (2w + 1) 
ca N — (a>r/coi), times the largest term in p, which is for n = 0. 
Therefore the largest even-harmonic terms in the series for the velocity 
at x = are (using the formula for the product of two cosines) 



VI.23] PROPAGATION OF SOUND IN TUBES 253 

{RyB 2 /2P) 



(1 + M 2 )(N 2 - 1)[N 2 - (2n r + l) 2 ] 



[cos (2n r o)it) — cos[(2n r + 2)w^]} 



where n r is the integer for which (2n r + 1) is closest to N (for the clari- 
net n r is 4 or 5). We have neg ected the small term (pc/Ri)(2n -f- l) 2 . 
We see that the most important even harmonics are those just above 
and just below the odd harmonic which is nearest the reed resonance. 
The pressure at x = for these frequencies is obtained by using 
Eq. (23.18) for the impedance for even harmonics. It is equal to 

(8 P 2 c 2 B 2 /P) 

(1 + M 2 )(N 2 - 1)[N 2 - (2n r + l) 2 ] ' 

• {n 2 cos[2n r coid — (n r + l) 2 cos[(2w r + 2)wifl} 

Other even harmonics can be computed by using the less important 
terms in the series for x and p, but their amplitude is too small to 
make it worth while including them. Other corrections would also 
come in, owing to the fact that the actual tube may be slightly conical 
rather than of uniform cross section. In the case of the oboe, the 
conical shape is so pronounced as to modify considerably the empha- 
sized frequencies, as will be indicated later. 

The behavior discussed in the example we have treated in such 
detail is typical of most wind instruments. The air in a tube, open 
or closed, is excited by the oscillations of a reed, a jet of air, the horn 
player's lips, the vocal cords, or some similar mechanism. The 
oscillations in the tube react back on the driving oscillator to modify 
its motion. Often the coupling is sufficiently close for the driving 
oscillator to move with the frequencies characteristic of the tube, as 
is the case of the reed instruments (although the reed instruments can 
also oscillate with the frequency of the reed when played by an 
unskilled player). In such instruments the natural frequency of the 
driving oscillator is usually higher than the tube fundamental. 

Other Wind Instruments. — In other instruments the frequency of 
the driving oscillator is adjusted to the fundamental or to some har- 
monic of the tube. This is the case of the flute, the organ pipe, and 
the horns; in such cases the resulting note depends on the tuning of 
the driver (which tube harmonic is picked out) as well as on the 
effective length of the tube. In the case of the singing voice, the 
fundamental frequency of the driver, the vocal cords, is lower than 
that of the cavity, and the coupling is weaker than in the other 
examples. Here the fundamental is determined primarily by the 
driver, and the resonating "pipe," the throat and mouth, are (more 
or less) adjusted to one of the harmonics of the driver. 



254 PLANE WAVES OF SOUND [VI.23 

In all these instruments the coupling between driver and tube is 
nonlinear, which makes it possible to extract energy from a steady 
blast of air, but which makes the analysis of the motion quite difficult. 
Once the oscillations start, they build up at the frequencies that allow 
the system to extract a maximum amount of energy from the blast. 
For instance, in the case of the driving jet across the tube end (flute, 
organ pipe) a vibration of the jet causes a pressure fluctuation, and 
the frequencies emphasized are those with the smallest input imped- 
ance (to make the velocity large), so that the driving end is at a pres- 
sure node and at a velocity maximum. When a reed is the driver, a 
vibration of the reed causes a velocity fluctuation; therefore the resonant 
frequencies are those with the largest input impedance (to make the 
pressure large) so that the driving end is at a velocity node and at a 
pressure maximum. 

In some cases, there are alternative choices of frequencies, either 
choice giving large energy, as for instance the low and middle registers 
in reed instruments. Once started in one or the other of these choices 
the energy builds up, and the air blast must be nearly stopped to 
change to the other type. • Part of the skill of the player comes in 
avoiding undesirable oscillations of the instrument. 

A great deal of experimental and theoretical work is needed before 
we can say we understand thoroughly the behavior of any of the wind 
instruments. 

Tube as an Analogous Transmission Line. — From our discussion 
at the beginning of this section we see that we can consider any tube 
of uniform cross-sectional area *S to be analogous to an electric trans- 
mission line, with distributed series inductance L T = (p/S) per unit 
length and distributed shunt capacitance C T = (S/pc 2 ) per uni t leng th. 
The analogous characteristic impedance of the line is Z a = \Z L T /C T = 
(pc/S), and the analogous propagation constant is —ia> \/L T C r = 
— ioi/c, as long as we neglect the energy loss of the wave as it travels 
along the tube. 

The analogous impedance of a round open end, of radius a, fitted 
with a large flange is 



Z„ — „ 

1 & 



pc_ 
ira 2 



L (2coa/c) J ira 2 \ c / 



( (jk>*/2kc) - ^(8p/37r 2 a) [« « (c/2o)] m „_. 

\ (pc/Tra 2 ) - (;/a,)(2pc 2 Ar 2 a 3 ) (« » (c/2a)] V ' ' 



shunted across the end of the transmission line. 



VI.23] PROPAGATION OF SOUND IN TUBES 255 

A change of cross-sectional area corresponds to a change of dis- 
tributed constants for the line; an increase of area corresponding to 
an increase in the distributed capacitance and to a decrease in the 
distributed inductance; and vice versa. Therefore a narrow portion 
of the tube corresponds more nearly to a pure inductance and a wide 
portion to a pure capacitance, as we assumed at the beginning of this 
section. 

A hole in the side of the tube will correspond to an inductance and 
resistance shunted across the line at the proper point. According to 
Eq. (23.25) the resistance should be approximately (p« 2 /27rc) and 
the inductance should be approximately (0.5p/b) where b is the radius 
of the open hole, and where we have taken into account the fact that 
the effective mass of air in the hole extends somewhat inside the hole, 
as well as outside, as was done in deriving Eq. (23.1). 

After the analogous impedance at the input end has been com- 
puted, the specific acoustic impedance can be obtained by multiplying 
by Si, and the mechanical impedance of the air column can be obtained 
by multiplying the analogous impedance by Sf, where Si is the area 
of the input end. 

Open Tube, Any Diameter. — We now return to the general problem 
of the piston and open tube and treat the intermediate case where the 
diameter of the tube is neither large nor small compared with the 
wave length. As we shall see in the next chapter, the acoustic imped- 
ance ratio at the open end of a cylindrical tube equipped with a baffle 
flange is 



?-"»,- 



M - 2 *^M _ iM(w)\ (23.26) 



where w = (Anrva/c) = (4ra/X) and a is the radius of the tube. Values 
of the functions 1 — [2Ji(w)/w] and M(w) are given in Table VIII 
at the back of the book. Equation (23.26) is valid for all wavelengths; 
the two limiting cases, corresponding to w — » and w — > °° , have been 
discussed previously. 

According to Eqs. (23.9), the direct and reflected waves inside the 
tube must adjust themselves so that 



tanh $ + 



r-v»)l 



f%nP 



is equal to the value of f (l) given above. To facilitate this calculation, 
solutions of the equation 

tanh [ir(a p - #„)] = 1-2 ^^ - iM(w) 



256 PLANE WAVES OF SOUND [VI.28 

are also given in Table VIII. Once the values of a p and P are found 
for a given value of w = (47ra/X), calculation of the properties of the 
piston-tube system is a simple matter. The ratio of reflected to 
incident amplitude is e~ 2 * a . The specific acoustic impedance at the 

piston is pc tanh Ur ( a p — i$ p — i — ) \> and the radiation impedance of 
the piston itself is 

%r = pc (-q) tanh L- f a p — i$ p — i— J = R r — iX r 

R r = pee (fjf), x r = pcx (§) 

Values of e and x can be obtained from Plates I and II, or from Table 
III at the back of the book. 

If the piston has a mechanical impedance Z p = R p — iX p , then 
the total mechanical impedance of the piston is Z m = R m — iX m , 
where R m = R r + R P , X m — X r + X p . If the piston is driven by a 
force of amplitude F , then its velocity u and amplitude of motion 
So, the intensity T of the sound in the tube, and the power n radiated 
out of the end of the tube are given by the formulas 



Uq = 






(23.27) 



2(RI + XIY \SJ 2S{Rl + Xl) 



As an example, the acoustic resistance ratio e and reactance ratio x 
at the piston end are shown in Fig. 56 for a tube whose length I is ir 
times its diameter 2a [so that w = (Ara/X) = 2Z/X]. The quantities 
are plotted as functions of (2Z/X) = (2vl/c). For the low frequencies 
the impedance at the open end is nearly all reactive, so that little of 
the energy escapes, and large resonance peaks can occur. The points 
of minimum impedance, where resonance occurs, are separated by 
points of high impedance. As the frequency increases, more and more 
of the energy reaching the open end is radiated out, less and less is 
reflected back to help in the resonance, and the peaks and valleys in 
the impedance curve get less and less pronounced. At very high 
frequencies the reactance is zero, the resistance is pc, and no resonance 
occurs. 

The integral values of (2Z/X), shown by the dashed vertical lines, 
correspond to the frequencies of resonance of a tube with an open end 



VI.23] 



PROPAGATION OF SOUND IN TUBES 



257 



which allows no energy to escape (i.e., a "perfect" organ pipe).. We 
notice that the actual frequencies of resonance are slightly lower than 
this for low frequencies and are considerably lower at high frequencies, 
approaching the points midway between two lines. 

The other two curves in Fig. 56 show the behavior of the acoustic 
conductance and susceptance ratios, the real and imaginary parts of 
the admittance (pc/z), at the end x = 0. If the driving piston at 
x = has a constant driving force to apply to the air in the tube 
[which would be the case in Eq. (23.27) if Z p <£ Z r ], then the power 

3r 



to 



y 



y 




_J5 



I I I 



Zr 












2- 


\U 










1 - 
"I 

n _• i 




1 ,. ,1 1 1 




4 6 8 Z 4 

XllM (21 A) 

Fig. 56. — Acoustic resistance, reactance, conductance, and susceptance ratios for 
the driving end of a tube with other end open. The length of the tube is ir times its 
diameter. If the driving piston is a "constant velocity generator," the power radiated 
is proportional to the resistance; if the piston is a "constant pressure generator," the 
power radiated is proportional to the conductance. 

radiated would be proportional to the conductance ratio k — 
0/(0 2 + x 2 ). The curve shows that if the piston were driven by a 
".pure noise" force, having all frequencies present in equal magnitude, 
those frequencies corresponding (approximately) to the first three or 
four harmonics of the resonance fundamental (c/2l) will be strongly 
reinforced, but the higher harmonics will not be reinforced very much. 
If the piston impedance Z v is larger than Z r , the veloeity amplitude 
of the piston will be nearly independent of v (if Z v is nearly constant) 
and the power radiated will be proportional to 0, the acoustic resistance 



258 PLANE WAVES OF SOUND [VI.2S 

ratio. In this case, as the curves show, the first three or four of the 
odd harmonics of the antiresonance frequency (c/4Z) will be reinforced, 
and we have the analogue of the clarinet, as was discussed earlier in 
this section. 

In either case, however, the higher harmonics will not be reinforced 
very much, so that a tube of this shape (I = 2ra), driven by a "pure 
noise" force, will give out sound having only the first three or four 
harmonics present in any appreciable intensity. A pipe that has a 
larger diameter for its length than this will have even fewer high 
resonance peaks and will give out a tone even poorer in harmonics. 
One having a smaller diameter will have more peaks, and the tone 
given out will be rich in harmonics. 

Cavity Resonance. — Another problem that can be solved by the 
hyperbolic tangent method is that of a plane wave falling on the 
open end of a tube fitted with a flange and closed at the other end. 
Suppose that the plane wave travels along the a>axis, which is the 
axis of the tube and is normal to the flange, that the wave has fre- 
quency v and that far from the tube it has pressure amplitude P f and 
velocity amplitude Uf = P//pc. When it strikes the open end of the 
tube, it agitates the air in the open end, sending a wave down the tube 
to the closed end where it is reflected. The reflected wave returns to 
the open end, and part of it radiates out into the open again. 

This method of analysis considers the wave reflected from the 
tube-plus-flange system to be made up of two parts : the wave reflected 
from the flange, and the wave radiating from the open end. For wave- 
lengths long compared with the radius of the opening a, the wave 
radiated from the mouth will spread out in all directions (as we shall 
see in the next chapter), while the wave reflected from the baffle will 
return along the #-axis, if the baffle flange is large compared with 
the wavelength. If the wavelength is small compared with a, both 
waves will return along the negative z-axis without appreciable side- 
ward spreading, and there will be interference between the two waves. 
For very short waves, the phase difference between the two reflected 
waves is just (4xZ/X). 

If the open end of the tube is provided with a large baffle flange, 
the incoming plane wave will be reflected from it; and close to the 
flange (which will be taken as the plane x = 0) the pressure due to the 
plane wave has the value 2P f e~ 2 ' iyt , and the corresponding particle 
velocity is zero. The baffle flange can sustain this pressure without 
moving, but the air in the end of the tube cannot. To the degree of 
approximation considered in this chapter, the air in the tube mouth 



VI.23] PROPAGATION OF SOUND IN TUBES 259 

will act as though it were a plane piston without mass or thickness, 
driven by a pressure of amplitude Po = 2P f , sending waves down the 
tube and also out into the open. The specific acoustic impedance 
for each square centimeter of the piston is that due to a tube of length I 
closed at the other end plus that due to radiation from a piston set in a 
baffle. From Eqs. (23.10) and (23.13) the impedance ratio is 

To = [l - 2 1M] - i [M(w) - cot(^)]; w = 4WX 

The velocity of the air in the mouth of the tube is therefore (d%/dt) 
= (Po/pcf o^ -2 ™"', and the velocity and pressure inside the tube are 

u = _ Po ,{ P< L, sin \^-a-x)\ e- M * 



f o sin(27rZ/X) 

iPo 
fo sin(27rZ/A) 



sin y (Z - x)\ 



V = v ^/o_7/^ c os y (Z - x) 



by Eq. (23.8). The pressure just outside the open end is the driving 
pressure P e~ 2 * ivt given above, plus the pressure due to the outgoing 
wave caused by the vibration of the air in the mouth. 

The ratio of the pressure amplitude at the closed end to the pressure 
amplitude at the open end is obtained from the foregoing equations: 



®-{[i-»^r-'(?) 

+ [MMsin(^)-cos(^ 



-i 

(23.28) 



where w = (4ira/X). Tables of the functions involved are given at 
the back of the book. The phase difference between the pressure at 
x = I and what it would be at the same spot (x = I) if there had been 
no tube or flange present turns out to be 



q = cot" 1 



cot(2TZ/X) - M(w) 

1_2 Jl ^ 
w 



(t) 



When the wavelength is so long that w is very small, the pressure 

p 

at the far end of the tube becomes Pi -* " 7/ . . Resonance 

cos (2xi/X) 
occurs whenever the frequency equals one of the natural frequencies 
of vibration of the closed pipe v n = (nc/4Z), (n = 1, 3, 5 • • • )• What 
happens is that some of the wave enters the tube and is trapped 



260 



PLANE WAVES OF SOUND 



[VI.23 



there, because of the low radiating efficiency of the open end for 
long wavelengths. If the wavelength happens to be the proper size, 
resonance occurs. 

When the wavelength is so small that w is large, the pressure at 
x — I becomes Pi — » P , independent of v. The efficiency of the open 
end for radiating sound is so large that no wave can stay long enough 
in the tube to produce resonance. The ratio Pi/Po is plotted in Fig. 
57, for a tube whose length is t times its diameter (as in Fig. 56), as 



H> 




2l A 



Fig. 57. — The ratio of pressure at the closed end to pressure at the open end, 
for the tube shown in Fig. 56 when I = 2ira. The peaks are examples of cavity 
resonance. 



a function of (2Z/X) == (2lv/c). It shows the transition from strong 
resonance at low frequencies to a nearly uniform response at very 
high frequencies. 

We have assumed above that the driving pressure at the open end 
of the tube P is twice the pressure amplitude in free space P f . This 
is true as long as the baffle flange is large compared with the wave- 
length of the sound; but when the wavelength is very large compared 
with the size of the baffle, P becomes equal to P f . The curve for the 
ratio of Pi to Pf will therefore be similar to the curve shown in Fig. 57, 
over most of the range of (2Z/X), except that the vertical scale will be 
doubled. If the flange is not very large, the curve for (Pi/Pf) will 
be more like the dotted curve at small values of (2Z/X), reducing to 
unit value (on the doubled scale) as a lower limit, owing to the reduc- 
tion of the value of (Po/Pf) from two to unity for long wavelengths. 
This will be discussed in detail in the next chapter. 



VI.23] PROPAGATION OF SOUND IN TUBES 261 

Many condenser-microphone diaphragms are stretched by means 
of a ring clamp set in front of the metal membrane, forming a short 
tube with the diaphragm closing - one end. When the wavelength 
of the sound is equal to four times the length of the tube, resonance 
of the sort discussed above occurs and will be quite marked unless the 
radius of the tube is much larger than its length. Such resonance in 
condenser microphones is termed cavity resonance. 

It is interesting to note that the dips in the contours of loudness 
level, in Fig. 52, at 4,000 cps, are due to cavity resonance of the air 
in the outer ear. At this frequency the pressure at the ear diaphragm 
is several times the pressure just outside the ear; so that cavity 
resonance acts to make the ear as a whole more sensitive at 4,000 cps 
than at 1,000 cps or at 10,000 cps, although the inner ear is not more 
sensitive at 4,000 than it is at 1,000 cps. 

Transient Effects, Flutter Echo. — As an example of the application 
of the contour-integral methods of calculation to waves in tubes, we 
shall consider the case of a uniform tube of length I, fitted with a 
piston at the end x = 0, and fitted at x = I with a termination having 
a purely resistive mechanical impedance. The piston will be held 
at rest until time t = 0, when it will be suddenly moved inward a 
unit distance. This will cause a pulse to travel down the tube. The 
pulse reflected from the resistive termination will not be so intense 
as the incident pulse. By the time this reaches x = the piston is 
at rest, so that the pulse will be reflected back down the tube without 
loss of intensity. At every reflection from the end at x = I, how- 
ever, there will be a further reduction in amplitude. 

One can guess the general form the reechoing pulse wave must take. 
Presumably it can be expressed in terms of the following function: 

y-y [o<t<(i/ C )] 

_ e - 2 ™ 6 ^ _ 2 l^j [(l/c) < t < (2l/c)] 
Flu(x,t) = \ ^" «V --H IW C ) < « < <«/«>] (23.29) 



f-4±*) 



[(4Z/c) < t < (52/c)] 



etc. 



262 PLANE WAVES OF SOUND [VI.23 

This represents a pulse reflecting back and forth from one end 
of the tube to the other, reducing in amplitude by an amount e~ 2va 
every time it reflects from the end x = I. Such pulse echoes are 
called flutter echoes. They are encountered in rooms with plane 
parallel walls of high acoustic reflectivity. Any pulse wave generated 
between the walls will bounce back and forth many times before 
dying out. Proper auditorium design is aimed, in part, at the elimi- 
nation of flutter echoes, as will be explained in Chap. VIII. 

Of course we have not yet shown that Eq. (23.29) is the correct 
solution of the problem we outlined at the beginning of this section. 
Before we use the contour-integration method to show that this 
formula is the correct solution, it will be advisable to modify the form 
of the expression for the function Flu(x,t), so we can recognize our 
answer when it is obtained. A pleasant exercise for the reader would 
be to juggle the delta functions around to arrive at 

Flu(x,t) = <rW»<«<-> ^ « 6 " 2m ' c + * ) 

jgjA.a-LzJ^ (0<x<t) 



m = 
.g— Ora/0 (ct+z) 



The two sums represent periodic waves, the first going in the positive 
^-direction, the second in the negative ^-direction. Each sum is 
modulated by an exponential damping term which in effect reduces 
the amplitude of each succeeding pulse by a factor e~ 2 ™. 

The sums, being periodic, can be expressed in terms of a Fourier 
series. The first sum, for instance, can be expressed in terms of the 
series 

The coefficient A n can be obtained, as on page 87, by multiplying 
both series by cos [(WO (^ - x)] and integrating over [t - (x/c)] 
from -(l/c) to +(l/c). This results in the equations 

1 = (^)a ; l = f-Vn (n = 1,2,3 • • • ) 



= (?)-; -ev 



Therefore the flutter-echo function Flu(x,t) can be expressed in terms 
of the series 



VI.23] PROPAGATION OF SOUND IN TUBES 263 

Ftu M = ^<« {£ + \ J cos [(=-<) (* - f)]} 



_ ^,-(ira/I)(c<+x) 

)2Z ' I _ 



provided, of course, that we waive questions of convergence (as we 
mentioned on page 98). To complete our discussion we must show 
that this series is, in fact, obtained by contour integration of the 
transient function. 

From Eqs. (23.8) and (23.9) we see that the air velocity in a simple 
harmonic wave in a tube with terminal acoustic resistance R at x = lis 



u = Ae'™ 1 cosh \ira — -~ (J — x) 



f iia 
I ira 



= Ae~ iat sinh \ira (I — x) 

(tic \ 
ira — -~ J = (pc/R) = k. If the piston 

velocity is e-™', the velocity in the tube is 

_ _ faf sinhfra — (ia>/c)(l — x)] _ _ e'™ 1 sinhfra — (ua/c)(l - x) ] 
~ sinh [ia — (icol/c)] i sin [((ol/c) + *Va:] 

According to the discussion at the end of Sec. 6, if the piston 
velocity is 8(t), the air velocity in the tube is 

v ' Am J- a, sin[(wt/c) + lira] 

The poles of this integrand are at co = (c/l)(irn — iira), where n 
is any integer, positive or negative (or zero). Close to the nth pole 
the sine function is ( — l) n (l/c)[w — (irc/l){n — ia)] and the integral 
around this pole, for the first exponential, becomes [since ( — 1)" = e™ n ] 



_ _L_ ( - | (D e -i«>(t-x/c)+Ta-(iul/c)+iirn ' dC0 



- ia)\ 



4iri \lj J |_ w ~ (irc/l)(n 

— _£_ (,-(Ta/l)(ct-x)-(iTn/l)(.ct—x) 

There is another pole at the corresponding negative value of n, and 
the two exponentials will combine to form the term (c/Z)e~ (Ta/Z)(c ' -:r) 
cos\{irn/l)(ct — x)]. Similar calculations for the second exponential 



264 PLANE WAVES OF SOUND [VI.28 

give similar terms; finally, by putting together all the terms, including 
those for n = 0, we are able to show that 

u s (t) = Flu(x,t) (23.31) 

Returning to Eq. (23.29), we can use the relation p(du/dt) = — (dp/dx) 
to obtain the corresponding pressure wave p&(t) = pcFlp(x,t), where 

5 (t - fj [0< * < (l/c)] 

e -2*a 8 L _ ^-~j [(I/C) < t <(21/C)] 

Flp(x,t) = < / 21 4- x\ (23.32) 

e -4™ 5 h _ ^^) [(3Z/c) < i < (4Z/c)] 

etc. 

The pressure wave does not change sign on reflection, but it reduces 
in amplitude on reflection from the end x = I by the same factor 
e -2wa = (r _ P c)/(R + pc) as does the velocity wave (as, of course, 
it must). 

As in our earlier discussions of transient effects, the wave generated 
by moving the piston in some arbitrary way, 

{0 (t<0) 

g(t) (0<t<A) 
(t > A) 

where g(t) is any arbitrary function of t, can be expressed in terms of 
the pulse waves: 

u{t,x) = f g(T)Flu(x,t — t) dr 

p(t,x) = pc J[ A g(r)Flp(x,t - t) dr 

If g(t) is a sensible sort of function of t, the resulting series for u and p 
will converge satisfactorily, so that the question of the convergence of 
the Fourier series for Flu is only academic, as was pointed out in 
Sec. 10. 

We note that if the duration A of the motion of the piston is less 
than (21 /c), the reaction of the air during the motion is as though the 
tube were infinite in length (in other words it exhibits its characteristic 
resistance pc). We note also that, if the specific acoustic resistance R 
of the termination were to depend on co, the reflected wave would not 
in general be a simple pulse, but would be spread out more than the 



VI.23] PROPAGATION OF SOUND IN TUBES 265 

incident pulse. If the termination impedance is not pure resistive 
but has a reactive part that also depends on w, the reflected wave would 
also be "blurred out." 

24. PROPAGATION OF SOUND IN HORNS 

As we have noted before, and as we shall treat in detail in the next 
chapter, a tube whose open end has a diameter smaller than the wave- 
length of the sound sent out is a very inefficient radiator of sound. 
This inefficiency is a desideratum in organ pipes and other wind-instru- 
ments, for in these cases resonance is important, and most of the wave 
must be reflected back into the tube from the open end to have sharp 
resonance. 

The small radiating efficiency of a small opening (or a small dia- 
phragm) is a very great detriment in a loud-speaker, however, for a 
loud-speaker should have no marked resonance frequencies. Two 
general methods are in use for building loud-speakers so as to radiate 
sound efficiently. One method is to make the vibrating diaphragm 
large enough to radiate well (dynamic speakers are of this type). 
This type of design has the advantage of compactness, but it requires 
comparatively heavy moving parts with their consequent mechanical 
inefficiencies. The other method is to use a small diaphragm and to 
magnify its effective size by using a flaring tube, a horn. One purpose 
of the loud-speaker horn is to spread the concentrated waves coming 
from the diaphragm out over a large enough area so that they can 
continue out from the mouth of the horn with very little reflection 
back to the diaphragm. In this way a light diaphragm can be used, 
and acoustic efficiency can be maintained at the same time. Another 
occasionally important purpose of the horn is to concentrate the sound 
into a directed beam, so that most of the radiated energy is sent out 
in one direction. 

The horn must not flare too rapidly, for the sound waves will then 
not "cling" to the inner surface of the horn and spread out but will 
act as though they were already out in free space, radiating from a 
small diaphragm and exhibiting a correspondingly small efficiency. 
Therefore in order that the mouth of the horn may be large in diam- 
eter, the horn must be long. 

One-parameter Waves. — The analysis of wave motion in a horn is a 
very complicated matter, so complicated that it has been done in a 
rigorous manner only for conical and hyperbolic horns. If the horn 
does not flare too much, however, so that we can consider the wave 
as spreading out uniformly over a cross section of the horn as it travels 



266 PLANE WAVES OF SOUND [VI.24 

outward, then we can use an approximate method of calculation and 
obtain fairly satisfactory results. For in this case the displacement 
of the gas molecules all over a surface perpendicular to the axis of the 
horn will be the same, and the displacement, pressure, etc., will be 
functions only of t and of the distance x along the horn. 

The statements made in the last paragraph require further dis- 
cussion to bring out all their implications; although a detailed discus- 
sion would lead us into the far reaches of differential geometry and 
would require a more complete knowledge of the properties of the 
wave equation in three dimensions than we shall have room to cover 
in the next chapter.' Nevertheless, a few paragraphs of general 
discussion are needed here, to indicate the approximations and limita- 
tions of the assumption that a wave in a horn is a one-dimensional 
wave. 

The shape of the inner surface of the horn of course affects the 
shape of the wave, and since the horn cross section is not independent 
of x, we should expect the wave to depend on the coordinates per- 
pendicular to x as well as on x, and thus not be a plane wave. What 
we must discuss are the conditions under which the wave will approxi- 
mate the behavior of a plane wave, and what we should derive (but 
will not because we have not in this book the requisite mathematical 
tools) are the criteria showing when these conditions occur. 

A plane wave, traveling in the positive ^-direction, has a phase 
which is the same everywhere over the surface of a plane perpendicular 
to x. In the usual exponential notation, 

p = Ae**-™ 1 (24.1) 

The quantity A is the amplitude of the wave and can be taken to be 
real. The quantity 4>, also a real function, is the phase of the wave. 
In the case of the plane wave <f> = {a/c)(x - x ), so that over a plane 
perpendicular to the x-axis (x = a, for instance) <f> is constant. In a 
plane wave the surfaces of constant phase are planes, perpendicular 
to the x-axis, moving along x with a speed c. 

Other waves can be set up which have the same general form as 
Eq. (24.1), though their surfaces of constant phase are not planes, 
and their amplitude is not constant. In the next chapter we shall 
see that a spherical wave, radiating out from a point source, has the 
form of Eq. (24.1). The amplitude A for the spherical wave is 
inversely proportional to r, the distance from the source, and the 
phase <t> = (w/c)(r - r ) is proportional to r. In this case, therefore, 
the surfaces of constant phase are spheres, expanding outward from 



VI.24] 



PROPAGATION OF SOUND IN HORNS 



267 



the source. Just as with the plane wave, however, both amplitude 
and phase are functions of only one coordinate. 

In terms of these concepts we can restate our requirements for a 
"good" horn. We first set up a system of mutually perpendicular 




.-H=I0 



Fig. 58. — Coordinate surfaces corresponding to three types of horns. To the degree 
of approximation of Eq. (24.2), surfaces of constant phase coincide with the ju-surfaces 
and the particle velocity is parallel to the lines = constant. See Eq. (24.3). 

coordinates (/*,#) suitable for the horn, so that the inner surface of 
the horn corresponds to # = # s (a constant), the axis of the horn (the 
x-axis) corresponds to # = 0, and so that n = x when # = {i.e., the 
scale factor for fi is the same as for x along the axis of the horn). 
Samples of such coordinate systems are given in Fig. 58, for several 
types of horn shapes. 



268 PLANE WAVES OF SOUND [VI.24 

We then set up the wave equation in these coordinates (this is, of 
course, difficult and is part of the detailed discussion that must be 
omitted here) and see whether we can obtain a solution of the form 
given in Eq. (24.1), with amplitude and phase depending only on the 
coordinate /*. If this can be done accurately, the wave is not a plane 
wave, but it is a one-parameter wave, which can then be handled as 
we have been handling plane-wave expressions. In this case the 
problem is straightforward. 

It turns out that very few coordinate systems give rise to a wave 
equation simple enough to allow a solution that is a function of only 
one coordinate (only the first of the three systems shown in Fig. 58 
has this property). When the coordinates do not have this property, 
the particle velocity will not be parallel to the /z coordinate lines, and 
the wave will tend to reflect from the horn surface as it travels along, 
rather than moving parallel to it. As we have seen in the previous 
section, and shall see again in the next chapter, any reflection of the 
wave during its' progress along a tube reduces the amount of energy 
traveling out of the tube and tends to trap some of the energy inside, 
causing resonance for some frequencies and poor transmission for 
others. This condition is satisfactory for musical instruments, where 
we desire strong resonances, but it is not desirable for loud-speaker 
horns, where we desire a uniform transmission, independent of fre- 
quency. Consequently, we can say that one criterion for a good horn 
is that it should be possible to set up one-parameter waves inside it; 
that a possible solution of the Wave equation in coordinates suitable 
for the horn should be a function of the coordinate n alone. 

For some coordinate systems the correct solution is nearly a one- 
parameter function, and the properties of the corresponding horn, 
deduced by using an approximate one-parameter solution, will be 
close to the true behavior. It is always possible (though often diffi- 
cult) to check the degree of approximation involved for a given horn 
by setting up the exact wave equation in the suitable coordinates, 
substituting p = AGu)^ 00- * 61 ' in the equation and determining by 
what amount this function does not satisfy the equation. 

An Approximate Wave Equation. — If we find that a one-parameter 
solution is approximately correct for some horn shape, we should be 
able to set up a one-dimensional wave equation that is approximately 
correct, from which we can obtain the one-parameter approximate 
solution. This approximate equation is not so difficult to obtain or 
to work with as is the exact wave equation, and we are now in a 
position to derive it. 



VI.24] PROPAGATION OF SOUND IN HORNS 269 

We consider a thin shell of air between two neighboring surfaces 
of constant phase (between <t> and # + d<(>), rather than a plane sheet 
between x and x + dx, as we did for the plane waves. If a one- 
parameter wave is a good approximation to the correct solution, this 
is nearly equal to the air between the coordinate surfaces p and 
/x. + dft. The total area of that part of the coordinate surface p 
which is inside the horn is called S, which is a function of p only (if 
the horn is a "good" one, S turns out to be proportional to the area 
of the cross section of the horn which is tangential to the coordinate 
surface p) . Therefore the volume of the gas in the shell at equilibrium 
is S dp (or approximately S dx, since p = x along the horn axis). 

During the passage of the wave, the surface of air originally at p is 
displaced to p + £, and the new volume of the shell is 

giving a fractional change in volume of (1//S) ^- (S£), which corre- 
sponds to Eq. (22.3) and to the equation p = ~("o")r (^)> 

relating pressure and particle displacement. The net force on the 
shell of air is everywhere perpendicular to the ju-surf ace (if the approxi- 
mation is good), and therefore Eq. (22.6) still holds, since differenti- 
ation with respect to p is the same as differentiation with respect to 
x along the horn axis. We next differentiate twice with respect to 
time the equation for p given above, and substitute for (d 2 £/d£ 2 ) from 
Eq. (22.6), obtaining, finally 

l±(^dp\_L^p. §u _ldp ( 

Sdx\ dxj c 2 dt 2 ' dt P dx K } 

where, as before, c 2 = (Po7c/p) and where u is the magnitude of the 
particle' velocity (the direction is perpendicular to the ju-surfaces, 
parallel to the & coordinate). If the waves are simple harmonic, the 
relation between u and p is 

1 dp 

tup ox 

Possible Horn Shapes. — If approximate Eq. (24.2) is valid for a 
given horn shape, then we should expect, to the same approximation, 
that a wave traveling out of the horn would be represented by the 
form given in Eq. (24.1). Moreover, we should expect that the 



270 



PLANE WAVES OF SOUND 



[VI.24 



amplitude A will (to the same accuracy) be inversely proportional to 
the square root of S. For the intensity of sound in the wave should 
be proportional to A 2 , and the total energy flow out of the horn would 
therefore be A 2 S, which must be independent of x if energy is to be 
conserved. 

We set S(x) = ir[y{x)] 2 , where y is the effective radius of the 
cross section of the horn at a distance x from its small end. Inserting 
the expression p = {B/y)e i '^ iut into Eq. (24.2), we obtain 



(ejy iMA_M 2 

\dxj ^ y\dx 2 ) \c) 



dx 2 



The real and imaginary parts of this expression must be zero sepa- 
rately, for both y and <j> are real. The imaginary part indicates that 
(d<f>/dx) must be a constant, which we can call (tco/c) for reasons that 
will shortly be apparent. 

Finally, in order that the approximations we have made in the 
previous paragraph be all valid and self-consistent, the real part of 
the equation must also be zero, which means that the effective radius 
of cross section of the horn must (at least approximately) satisfy the 
equation 

S-(f)'o-^-« 

A solution of this equation, properly adorned with constants, is 



y = yo 

S = So 

T 2 = 1 ■ 



co 8 h(|) + rs inh(|) 
co S h(|) + rsi nh(|) 

\uhj \2rhJ 



(24.3) 



and the corresponding approximate solution for the plane wave travel- 
ing out of the horn is 



V 



Q> 



1 pHa/c){rx—ct) 



(24.4) 



The shape of the horn represented by Eq. (24.3) is determined by 
the relative values of the constants T and h. The constants y and 
S = iryl fix the size of the throat of the horn at x = 0. The "scale 
factor" h determines the rapidity of the "flare"; the smaller h is, the 
greater is the curvature of the plot of y against x. The constant T 



VI.24] PROPAGATION OF SOUND IN HORNS 271 

is the "shape factor," determining the general properties of the horn 
near the throat. When T = (h/x ) and h is allowed to go to infinity, 
the horn is a conical horn, with angle tan" 1 (y /x ) between the axis 
and one of the elements of the cone [the "angle of opening of the cone" 
is 2tan- 1 (Wzo)]. When T = 1, y = y e x/h , and the horn is an 
exponential horn. When T = the shape of the generator of the 
surface is a catenary, so the horn will be called a catenoidal horn. 1 
This horn can join smoothly onto a uniform tube at x = 0, whereas 
the conical and exponential horns have a discontinuity in slope of 
y{x) at x = 0, making a worse impedance match with the uniform 
tube. For large values of x, the catenoidal horn is indistinguishable 
from the exponential horn. The shapes of all three types are shown 
in Fig. 58. When the exact wave equation for the coordinates appro- 
priate to the horns of Eq. (24.3) is worked out, it can be shown that 
Eq. (24.4) is a reasonably close approximation to an exact solution 
as long as the radius y of the small end of the horn (the throat) is 
much smaller than the scale factor h. 

We shall next discuss in detail the acoustical properties of each 
of the three main types of horns expressible in terms of Eq. (24.3), 
for cases where the big end of the horn (the mouth) is large enough 
so that no part of the wave coming out of the horn is reflected back 
toward the throat. Finally we shall take up the case where the open 
end is small compared with the wavelength (which is the case of some 
wind instruments), where reflection from the mouth, and consequent 
resonance, occurs. 

The Conical Horn. — In a conical horn of angle # s (angle of opening 
2#„ area of throat irxl sin 2 & a ) the areas of the phase surfaces are 

S = So \l + yyl ; S = iryl y = 2*o sm(^J 

where x is the distance back from the throat to where the apex of the 
cone would be, if the cone were extended back to its apex. 
Equation (24.2) turns out to be 



1 d \, , , 2 dp] 1 d 2 p 



(x + x ) 2 dxl K ' u/ dx] c 2 dt 2 
which is the exact wave equation for a one-dimensional spherical 

1 Although it would also be appropriate to call it a "Salmon" horn, after the 
person who first discussed the acoustical properties of the family of horns of Eq. 
(24.3). 



272 PLANE WAVES OF SOUND [VI.24 

wave. The outgoing-wave solution of this is 

p 

fp = gt(w/c) (x—ct) 

X + X 

The particle velocity at x is 

pz L x + ZoJ ^ 

In this case the phase velocity of the wave inside the horn is the 
same as it is in free space, c. Since the wave front spreads out as it 
travels down the horn, the impedance looking out of the horn from 
the small end is not the same as for a uniform tube, even if there is no 
reflected wave: 

(p\ _ pc _ pc 

uj ~~ 1 + i(c/oox ) ~ 1 + i(\/2rx ) (24 - 5) 

The analogous impedance of the throat of the conical horn, (z /S ), 
is equivalent to a resistance (pc/S ) in parallel with an inductance 
(pXo/So). Therefore any acoustical filter, or other circuit, that is 
terminated by a conical horn has an equivalent electric circuit ter- 
minated by this shunt combination. The power lost in the resistive 
arm represents the power radiated out of the open end. The shunt 
inductance has very little effect on the behavior of the horn at high 
frequencies (X < < 2inro), but at lower frequencies it shunts out the 
resistor (pc/S ). 

The specific acoustic impedance at the throat of the horn is there- 
fore 

z = R — iX = \z\e-^ 

R = pC Y = (pcX/27T3 ) 
1 + (\/2ttXo) 2 ' 1 +. (\/2irXo) 2 
PS: _ pcxpw ^ (24.6) 

\A + (X/27TX ) 2 VC 2 + (XOCO) 2 



\z = 



<p = tan 



— i 



\2irx J \XouJ 



If the particle velocity at the throat of the horn, causing the out- 
going spherical wave, is Woe -2 ™"', then the power radiated out of the 
horn is 

IT - 1 ,,2PO _ i pCMJ}ff _ ! „ , (2t;eop) 9 



VI.24] 



PROPAGATION OF SOUND IN HORNS 



273 



In this case the power radiated is small at low frequencies and rises 
steadily as the frequency is increased, approaching asymptotically 
the limiting value ?Sopcu$. The limiting value is the power that the 
same piston would send into a uniform tube of infinite length. 

Transmission Coefficient. — We are now in a position to define 
what we mean by the term "radiating efficiency of a horn" which has 
been so freely used above. A measure of this efficiency is the ratio 
of the power radiated out of a given horn to the power radiated by the 
same diaphragm, moving at the same velocity, into a cylindrical tube 
of infinite length, having the same cross-sectional area as the small end 
of the horn. This ratio, for any type of horn, is defined as 



2n 



Sopcul 



© 



(24.7) 




and will be called the transmission coefficient of the horn for sound. 
This ratio is not, strictly speak- 
ing, an efficiency, for it is the 
ratio of the actual power radiated 
to the power radiated for a stand- 
ard case, not an ideal case. Con- 
sequently, t sometimes becomes 
greater than unity. In general, 
however, r varies between zero 
and unity. When it is small the 
diaphragm will have to vibrate 
with large amplitude to radiate 
much power. When the veloc- 
ity amplitude of the piston is 
independent of frequency, the power radiated is proportional to the 
transmission coefficient. In any case the power radiated is 

n = iuiSopcr 

The transmission coefficient for a long conical horn with wide 
mouth is 

1 (2tvx ) 2 



0.2 0.4 0.6 

Fig. 59. — Transmission coefficient for a 
long conical horn with open end large 
enough to eliminate resonance. 



1 + (\/2tx q ) 2 c 2 + (2irvx y 



(24.8) 



It is plotted in Fig. 59 as a function of (x v/c) — (x /\) = lb/\(a — b), 
where I is the length of the horn and a and 6 are the radii of its large 
and small ends, respectively. This function has* the value -| when 
(xq/\) = \/2ir or when the frequency equals c(a — b)/2rlb. For fre- 



274 PLANE WAVES OF SOUND [VI.24 

quencies above this the transmission coefficient approaches unity, 
and the horn is relatively efficient as a radiator. For frequencies 
below this the horn is quite inefficient. The smaller the flare of the 
horn [i.e., the smaller the value of (a — b)/l\, the lower the frequency 
at which the horn will be efficient. 

A Horn Loud-speaker. — We can now put some of these formulas 
to work by showing how they can be used in designing horn loud- 
speakers and in analyzing their behavior. The sketch at the top of 
Fig. 60 shows the longitudinal cross section of a simplified version of a 
conical horn loud-speaker. The piston P is considered to move as a 
single mass of mechanical impedance (without the air load) Z p = 
R P — i[wm p — (K p /a))] dyne-sec per cm. Its radiating surface has 
area S p . The area of cross section of the throat of the horn, S , is 
less than this, in order to improve the impedance match between the 
piston and the air. The portion of tube between the piston and the 
horn consists of a "tank" of volume V and a constriction of cross- 
sectional area So, of length I and of effective length l e ~ I -f- 0.5 \/$o 
[this is a modification of Eq. (23.1) suitable for the present case]. 

Since the tube cross section changes, we shall find it easiest to 
make our first analysis in terms of analogous impedances, Z a = (z/S), 
for pressure and volume flow do not change discontinuously at changes 
in tube cross section. The analogous impedance of the horn, looking 
out of its throat, is equivalent to an inductance (px /So) and a resistance 
(pc/So) in parallel. We use Eqs. (23.2) and (23.3) to complete the 
rest of the equivalent circuit, the impedance of which is equal to the 
ratio Z a between pressure and volume flow of air (uS p ) at the piston 
face. This circuit is shown in the middle part of Fig. 60. 

The power radiated out of the horn is represented in this circuit 
by the power dissipated in the resistance (pc/So). This power is 
diminished at the lower frequency end due to the shunting inductance 
(pxo/So), which should therefore be made as large as possible (by 
making xo large) to enable the horn to radiate well at low frequencies. 
The sketch shows that for x to be large the conical angle # 8 must be 
small; therefore the horn must be long in order that the mouth of the 
horn be large enough to radiate away the power. This property of 
conical horns has already been shown in Fig. 59. The power output 
for the horn of Fig. 60 is also limited on the high-frequency side by the 
shunting condenser (V/pc 2 ) and by the series inductance (pl e /S ). 
Therefore the volume V in front of the piston must be as small as 
possible and the constriction as short as possible in order that the 
horn be efficient at high frequencies. 



VI.24] 



PROPAGATION OF SOUND IN HORNS 



275 



To include the effect of the impedance of the piston itself in the 
analysis, we shall find it easier to turn from analogous impedance Z a = 
(pressure) /(air flow) to mechanical impedance Z m = (force) /(velocity) 
= SlZ a . We set up the equivalent circuit at the bottom of Fig. 60 to 
aid in calculating this quantity. The equivalent voltage E equals 




Longitudinal 
Cross Section 
of Conical Horn 
— Loud Speaker 



S =47TX 2 sin z (6 s /2) 
Sp=Areoi Piston 



J 

P 

°1 



(ple/So) 



,_V_ 
■pc« 



Circuit for 
Analogous 
Impedance 
Z a atthe 
Piston P 



(pleSp/S fl 
m p R p 



Sjpc* " 
f/Kp 

— 1|— wvw- 



pXoSpi 



pcSp; 



Equivalent Circuit 
giving Mechanical 
Impedance of Piston 
plus Radiation Load 



"-Impedance due to Air back of Piston 

Fig. 60. — Idealized sketch of conical horn loud-speaker, with corresponding circuits for 
analogous impedance of air load Z a , and for total mechanical impedance Z p + S p 2 Z a . 

the force on the piston, and the current through m p equals the piston 
velocity. 

This circuit is not so convenient as the one above, for the currents 
in the various arms do not everywhere correspond to air flow nor do 
the voltages always correspond to pressures. The pressure at the 
horn throat, for instance, turns out to be equal to the equivalent volt- 
age across the resistance (pcS*/So) divided by the area S (for in this 



276 



PLANE WAVES OF SOUND 



[VI.24 



circuit, voltage corresponds to force = pS). The air velocity in the 
throat equals the equivalent current in the inductance (pl e Sl/S ) 
multiplied by (S p /S ). The power radiated is, of course, still equal 
to the power dissipated in the resistance (pcSl/So)- 

We now can see why it is wcrth while to have the piston area S p 
larger than the throat area So, even though this introduces the unde- 
sirable capacitance (F/£|pc 2 ). The magnitude of the mechanical 
impedance of the piston is usually much greater than pcS , so that there 



r^JM^ 



v 

pe>S 2 p 



= px S 2 p 



Rn"l 



:pcS p 



Equivalent circuit 
giving Mechanical 
Impedance of Mass 
Controlled Piston 
plus Radiation Load 



Lc 



SMAAM 




Circuit giving 
Electrical Impedance 

z E =z c +(r/z m ) 

of Coil of Electro- 
dynamically driven 
Piston 



Motional Impedance 
Fig. 61. — Equivalent circuit at top gives mechanical impedance of piston of Fig. 60 
when piston is mass-controlled. If piston is driven by an electrodynamic coil, as dis- 
cussed on page 34, the electrical impedance of the coil whe n moving is equal to that of 
lower cir cuit. Velocity of piston is equal to \Ao 7 /VE M . Velocity of air in throat 
equals ■\/10 7 /T(S p /So)Er. We have here neglected the air load Z b on the back of the 
piston. 



is a poor impedance match between the piston and its air load unless 
the effective air impedance is increased by the factor (S P /S Q ) 2 . 

If the horn is well designed the stiffness and mechanical resistance 
of the piston are small compared with the mass reactance over the 
useful range of frequencies; in other words, the piston is mass con- 
trolled. In this case the equivalent circuit, representing the mechanical 
impedance, is that shown at the top of Fig. 61. The inductance, 
representing the piston mass, is in series with a parallel combination 
representing the air load. We have omitted the inductance due to 



VI.24] PROPAGATION OF SOUND IN HORNS 277 

the constriction, because this can be made quite small compared with 
m p . 

The large mass of the piston plus driving mechanism is the chief 
difficulty in providing for uniform radiating properties of the loud- 
speaker. By using a horn, we are able to make the piston smaller and 
therefore considerably lighter than the diaphragm-driver unit of the 
"dynamic" speaker. But it is extremely difficult, to reduce this 
mass below 10 or 20 g. Consequently, the reactance due to m p at 
1,000 cps will still be approximately 100,000 dyne-sec per cm. Since 
pc is only 42 dyne-sec per cm 3 , we must make the factor (S%/S ) of the 
order of 1,000 to have a reasonable impedance match. If, for instance, 
So were 3 cm 2 , S p should be about 50 cm 2 (piston diameter ~ 3 in.). 
But this large a piston is difficult to make stiff if it weighs as little as 
10 g. All these practical limitations mean that it is quite difficult to 
design a conical loud-speaker that is efficient over a wide range of 
frequencies. 

To illustrate these difficulties we shall calculate the horn behavior 
for values of the constants which are difficult to attain in practice, 
but which show clearly the general behavior 

m p = 8 g; So = 3 cm 2 ; S p = 27 cm 2 

Xo = 11 cm; # s = 5 deg; V = 8 cm 3 

K v = 12.6 X 10 7 dynes/cm; m r = 3.2 g; R r = 10,000 dyne-sec/cm 

The mechanical impedance of piston plus horn is plotted in the top 
curve of Fig. 62. The mechanical resistance is maximum at 1,000 cps, 
corresponding to the parallel resonance between K v and m r . The 
horn would respond better to high frequencies if this resonance could 
be moved to higher frequencies; but this requires either that #« or V 
be made smaller, both of which are difficult. The reactance is large, 
due to the large value of m p , which is almost impossible to reduce 
much further. 

Now suppose that the piston is driven by an electromagnetic 
coupling system, such as that described on page 34 having coupling 
constant T. The motional impedance of the coil, due to the motion of 
piston and air in the horn, is equal to T times the mechanical admittance 
of the piston-horn system. This motional impedance is in ohms and 
is to be added to the clamped impedance of the coil to give the total 
electrical impedance of the driving coil. The reciprocal of the series- 
parallel circuit at the top in Fig. 61 is the parallel-series circuit inside 
the dashed line in the lower sketch. 

Consequently, this lower circuit has the same electrical behavior 



278 



PLANE WAVES OF SOUND 



[VI.24 



as the driving coil. We can show that the velocity of the piston in 
centimeters per second is equal to \/10 7 /r times the voltage across 
the conde nser ( m p /T) and that the air velocity at the throat of the 
horn is Vl0 7 /r (S P /S ) times the voltage across the condenser (m r /T) 




o ^ 

T5 — 

|^ 

I_-Q 
O 

a. - 




10 Volts Impressed 
Across Driving Coil 



10 
200 



400 600 1000 



2000 



4000 6000 



32 

V. 

-10 15 
3.2 J 
1.0 | 

cc 

0.32 I 
o 
0.! Q - 
10,000 



Fig. 62. — Mechanical impedance of piston-plus-horn of Fig. 60. Electrical imped- 
ance of driving coil if piston is driven by electromagnetic means (see page 34) . Power 
radiated when coil is driven by 10- volt generator. 



and the resistance (T/R r ). The power radiated out of the horn is, 
of course, equal to the power dissipated by the resistance (T/R r ). 

If we assume that r = 10 5 ohms-cm per dyne-sec R c = 0.5 ohm, 
and L c = 0.0008 henry (an improbably small value, chosen to show 
the coil resonance separated from the piston resonance), then we can 
use the equivalent circuit to calculate the total electrical impedance 



VI.24] PROPAGATION OF SOUND IN HORNS 279 

of the driving coil. This is shown in the center graph of Fig. 62. 
The resistance is largest between 200 and 600 cycles; above 3,000 cycles 
the mass load of the piston (the shunt capacitance m p /T) prevents 
much radiation, and the resistance is that of the clamped coil. We 
have adjusted the coil inductance L c so that it resonates with the con- 
densers at 2,000 cps, so as to strengthen the response above 1,000 cps. 
Assuming that we drive the coil of the speaker with a constant 
voltage generator of 10 volts emf but variable frequency, we should 
obtain a plot of output power against frequency of the form given at 
the bottom of Fig. 62. This output is fairly uniform over the fre- 
quency range from 200 to 3,000 cycles, but it drops very rapidly above 
this range, due to the coil inductance L e and to the shunt capacitance 
(Wp/r). In fact, the equivalent circuit corresponds to a rather 
crudely devised low-pass filter, so it is not surprising to have the high 
frequencies cut off. If the coil inductance L e were larger, the cutoff 
would be at a still lower frequency. 

It should be apparent by now that conical horn loud-speakers can 
be made reasonably uniform in response over a range of frequency cor- 
responding to a factor of 10 (if they are carefully designed), but not 
over a very much larger range. If the horn is to be reasonably effec- 
tive at 100 cps, it will probably not be very effective above 1,000 or 
2,000 cps; if it is to be effective clear out to 10,000 cps (such high- 
frequency horns are called "tweeters"), it will not radiate well below 
about 1,000 cycles. This general limitation on frequency range is 
true of horn loud-speakers in general (as, in fact, it seems to be for 
all loud-speakers). New developments in light stiff material (to 
reduce m p ), or in "acoustic transformers'' to improve the impedance 
match between piston and air load, are needed to produce a marked 
improvement in behavior. 

The Exponential Horn. — When the cross section of the horn has 
the form S = S e 2x/h , the horn is called an exponential horn. The 
length h is the distance in which the diameter of the horn cross-section 
increases by a factor e = 2.718. We have already seen in Eq. (24.4) 
that the pressure wave in such a horn can be expressed approximately 
by the formula 

^ _ Pq~ (x/k)+i(oi/c)(TX— ct) I 

r = Vl - (XAo) 2 = Vl ~ {n/vY > (24.9) 

Xo = 2rh; v = (c/2ttK) ) 

This wave travels out of the horn with ever-diminishing amplitude, 
with a velocity (c/t) larger than the speed of sound in the open. 



280 PLANE WAVES OF SOUND [VI.24 

This velocity is different for different frequencies, which makes the 
exponential horn a dispersing medium for sound waves (see page 154). 
When the frequency is equal to v (called the cutoff frequency) , the 
velocity has become infinite; in other words, the air moves in phase 
along the whole extent of the horn [this is, of course, only approxi- 
mately true, since Eq. (24.9) is an approximate solution]. Below this 
cutoff frequency there is no true wave motion in the horn, and the 
equation for the pressure is 

V = P expj ^1 - \^-J - lj | - iaA ( w < Wo ) (24.10) 

where co = (c/h). Thus it turns out that the exponential horn is 
very ineffective below a certain minimum frequency (the cutoff fre- 
quency) which is smaller the larger h, the scale factor, turns out to be. 
Again a very long horn is needed to produce low frequencies effectively. 
The air velocity in the horn for frequencies above the cutoff is 
obtained by differentiating Eq. (24.9) : 

Consequently, the specific acoustic impedance at the throat of the 
exponential horn, for outgoing waves, is 

z Q = R — iX = \z\er**; (« > « ) 






(24.11) 

- / 0)ft\ I c\ I 

\z\ = pc; <p = sin" 



Below the cutoff frequency, no exponential horn can be considered long 
compared with the wave length (inside the horn), so that the impedance 
match at the mouth of the horn must be taken into account. This 
will be considered later. Above the cutoff frequency the impedance 
has a constant magnitude pc, its phase angle being 90 deg at the cutoff 
frequency and approaching zero as the frequency is increased. 

If the velocity amplitude of the driving piston at the throat of the 
horn is Uo, the power radiated is 

n = iulSopcyjl - (^\ [v >v = (c/2Th)] 

Very little power is radiated for frequencies less than v Q . 



VI.24] 



PROPAGATION OF SOUND IN HORNS 



281 



The transmission coefficient for the long exponential horn, the 
ratio of power actually radiated to the limiting value (ulSopc/2), is 



'• - V 1 - (t)' = V 1 - (as)' <'>*>■ < 24 - 12 ) 

The curve for r e is shown in Fig. 63. Comparison with Fig. 59 for 
the conical horn shows that for some purposes the exponential horn 
is better than the conical horn. Although it radiates practically 
nothing below the cutoff frequency, the transmission coefficient rises 




0.6 



0.8 



0.4 
(h/A)=(vh/c) 

Fig. 63. — Transmission coefficient for a long exponential horn with open end wide 
enough to eliminate resonance. 

much more rapidly to unity above the cutoff. If h is made large 
enough (i.e., if the horn does not flare too rapidly), r e will be constant 
over practically the entire range of frequencies. Neither r c nor r e ever 
become larger than unity, however. For this behavior we must turn 
to the catenoidal horn. 

The Catenoidal Horn. — When the cross section of the horn has the 
form S = So cosh 2 (x/h), it is a catenoidal horn. The scale factor 
h is again a measure of how slowly the horn flares out. At large dis- 
tances from the throat the catenoidal horn is indistinguishable from 
an exponential horn, but at the throat the rate of change of ~S with x 
is zero, in distinction to the exponential case.' The catenoidal horn 
can therefore fit smoothly onto a uniform tube of cross section So, 
whereas the conical and exponential types have a discontinuity in 
rate of change of S, which tends to cause reflection back into the uni- 
form tube. 



282 PLANE WAVES OF SOUND [VI.24 

Referring to Eqs. (24.3) and (24.4) we see that an outgoing wave 
is approximately represented by the one-dimensional form 

v = — p e iwo(A-ct) (24.13) 

r cosh (x/h) 

The wave velocity in this horn, (c/t), is larger than the velocity in 
open space, as in the exponential horn, and there is again a cutoff 
frequency v = (c/2irh), below which the horn is a poor radiator of 
sound. 

The corresponding particle velocity is 



1 

u = — 

pc 



+ 



£*■*©' 



p [a > coo = (c/h)] 



so that above the cutoff frequency the specific acoustic impedance at 
the throat (x = 0) of the catenoidal horn is a pure resistance 

' * " " VI -W-)' = "" VI - (X/frrfc)' (24 - W) 

In this case the specific acoustic resistance is greater than pc above 
the cutoff frequency vo, rising to very large values as v — v diminishes 
toward zero. For a constant velocity amplitude u at the throat, the 
power radiated is 

E = WoSopc l f ^ [v > vo = (c/2rfc)] 

The transmission coefficient of the catenoidal horn is therefore 

T - = ( l ) = /i \ /v {v > Vo) (24 ' 15) 

A comparison between the transmission coefficients of conical, expo- 
nential, and catenoidal horns of the same over-all dimensions is shown 
in Fig. 64. The catenoidal horn has a cutoff frequency a little higher 
than the exponential horn, but the transmission coefficient just above 
the cutoff is considerably larger. Evidently, starting the horn with a 
uniform cross section at the throat (dS/dx = at x = 0) can improve 
the output of the horn just above the cutoff frequency. For instance, 
for a catenoidal horn with h = 50 cm = 20 in. (cutoff frequency = 
110 cps) the output at 150-cps is nearly twice that of an exponential 
horn of the same value of h and is very much larger than that of a 
conical horn of the same size. As we have seen earlier, it is difficult to 



VI.24] 



PROPAGATION OF SOUND IN HORNS 



283 



obtain sizable power outputs at the low frequencies, so this property 
of the catenoidal horn is often useful. 

It is also apparent from Fig. 64 that the conical horn is the least 
efficient of the series of shapes we have analyzed. Even though it 
has no cutoff frequency its effectiveness at very low frequencies is 
quite small and does not usually make up for the very slow rise of 
transmission coefficient with frequency. 




Frequency 

Fig. 64. — Comparison between transmission coefficients of long, conical, exponen- 
tial, and catenoidal horns of the same over-all dimensions. Length I is chosen to be 
2.4 times the h for the exponential horn; I turns out to be 11 times the xo for the conical 
horn; and 3.1 times the h for the catenoidal horn. 

Reflection from the Open End, Resonance. — In many cases, horns 
are used with open ends that are not large enough to radiate all the 
sound coming to them from the throat. Many wind instruments, for 
instance, are approximately exponential or catenoidal horns, with 
open mouths designed to be small enough to , ensure resonance inside 
the horn. Even horns designed as loud-speakers, to avoid resonance 
at most frequencies, cannot avoid some resonance at the lowest fre- 
quencies. In this section we shall discuss the effect of such an imped- 
ance mismatch at the mouth of the horn. 

We might as well carry out the calculations for the general form 
given in Eq. (24.3), for any value of T. The formula for S can be 
given in another form: 



S = 



So 

cosh 2 (e) 



cosh 2 



G~> 



T = tanh (e) 



(24.16) 



284 



PLANE WAVES OF SOUND 



[VI.24 



When e = 0, the horn is catenoidal; when e = », the horn is expo- 
nential; and when e = (x /h) + i(ir/2), with h approaching infinity, 
then the horn is conical. Intermediate values of e correspond to 
horns of intermediate shape. 

This can be substituted in Eq. (24.2) to give the approximate 
wave equation and eventually to obtain a solution for the wave. In 
this case we cannot neglect the wave reflected from the horn mouth, 
so our solution has the form 



V 



Pe~ 



cosh 



sinh^ + UaTX 



.) 



V 

u = — 

pc 



z = pc 



.,«h(, + =) + £—(! + .) 



■(■ 



t coth( \j/ + l —\ + ^ tanhl t 



•i 



(HI 



(24.17) 



where t 2 = 1 - (c/co/i) 2 . This is similar to Eqs. (23.8) for the uniform 
tube. The differences in the expressions for the specific acoustic 
impedance z lie in the factor t, corresponding to the fact that waves in 
these horns travel with a speed different from waves in the open, and 
in the term including tanh[(rr//i) + e], corresponding to the fact 
that the surfaces of constant phase are curved surfaces, not planes as 
in the uniform tube. 

Our first task is to fit the wave at the mouth of the horn, at x = I. 
Since the surfaces of constant phase are not planes, this fit is not so 
simple as that for the uniform tube, which resulted in Eq. (23.13). 
To a reasonable approximation, however, the joining of the wave 
from the horn with the wave in the open effectively cancels the term 
in tanh [(x/h) + e] and results in the equation for \J/ 

£ tanh(* + ^) = pc [l - 2 ^] - ipcMiw) (24.18) 

which is analogous to Eqs. (23.13). The quantity w is (2coa/c), where 
a is the radius of the mouth of the horn. This joining equation holds 
as well for frequencies below the cutoff frequency as for those above. 
When (wh/c) < 1, we set 



-'^-(t)' 



and carry on with the calculations. 



VI.24] PROPAGATION OF SOUND IN HORNS 285 

We solve Eq. (24.18) for \f/ + (ioirl/c) = *-(«& — ifa) and from this 
compute the impedance at the throat of the horn: 

z = pc <t coth ir(a h — tj8*) — ( -^— J H r tanh (e) > 

For large values of w = (2«a/c), a h becomes large, and 

*« » -l / • / P ^ + u ^ (24.19) 

w-* oo t + {ic/a)h) tanh (e) 

which corresponds to Eqs. (24.6), (24.11), and (24.14) for the special 
cases discussed before. For these higher frequencies no reflected 
wave is present in the horn. 

For small values of w, the approximate expressions for ah and fih are 
similar to those of Eqs. (23.15) : 



ah 



teX?)"' *-©(-) 



with a the radius of the mouth of the horn. The effective length of 
the horn is l p = I + (8a /Sir), as was the case for the uniform tube. 
In terms of this we can write the approximate expression for the 
acoustic admittance ratio at the throat : 

* =* \-*™™' c) *^}f} + 4 tanh («) (24.20) 

z (T/2)(aco/c) 2 — t tan (corZp/c) <ah ' 

(OW/C) « 1, T 2 = 1 - (C/C0/1) 2 

Wood-wind Instruments. — If the tube of the flute or open organ 
pipe has a shape corresponding to Eq. (24.16) for some value of c, 
then the frequencies picked out are those for which z is real and as 
small as possible. This corresponds to the requirement that (oyrl p /c) 
be approximately equal to (nx), (n = 1,2,3 • • •). By using the 
equation for t, we obtain the following equation for the resonance 
frequencies : 

In comparison with the uniform tube [see Eq. (23.17)] the overtones 
here are not strictly harmonic, though if the tube flare is small (h large), 
the discrepancy is small. The specific acoustic resistance at the throat 
for the nth. "harmonic" is 



l -~ i "l\'V) + WJ (24 ' 22) 

which is to be compared with Eq. (23.18) for the uniform tube. 



286 PLANE WAVES OF SOUND [VI.24 

For a reed instrument with tube shaped according to Eq. (24.16), 
the frequencies picked out will be those for which z is as large as 
possible. We must choose a value of (cotZ p /c) so that (pc/z ) in Eq. 
(24.20) is real and as small as possible. This is not obtained by setting 
the tangent equal to infinity, for the term (c/oh) tanh e is usually 
larger than (aco/c) 2 , and (pc/z ) would not be real or as small as pos- 
sible. Instead, we set (o)tI p /c) = ir(m + i) + 5, (m = 0,1,2 • • • ) 
where 5 is quite small. This corresponds to tan (o>tI p /c) ~ —(1/5), 
which is quite large, but not infinite. We then adjust 5 so that, to 
the first order in the small quant ties 5 and (aco/c) 2 , (pc/zo) is small 
and real. This corresponds to making 5 = (c/toJi) tanhe, which is 
small enough, since we are assuming that we are above the cutoff 
frequency (co > coo = c/h). Setting this value for r in the equations 
for co and for z , we have 



Zanti 



(iW m ; + © !+ P taiih ' 

C I p 1 I 1 ( p I p + Vi 

Xiram) |_ \irhmj ir 2 hm 2 



(24.23) 



where ra = 1, 3, 5 • • • . This is to be compared with Eqs. (23.19) 
and (23.20) for the uniform tube. We note that for the catenoidal 
case (e = 0) the term (8l p /Tr 2 h) tanh e is zero ; for the exponential horn 
the term is (8l p /T 2 h) ; and for the conical horn this term is (SI p /t 2 x ) 
and the term (2l p /irh) 2 is zero. 

The introduction of a nonuniform cross section thus makes the 
overtones nonharmonic by a small percentage, makes the resistance 
at resonance (z minimum) somewhat larger than for the uniform tube, 
and makes the resistance at antiresonance (z maximum) somewhat 
smaller than for the uniform tube. A certain small amount of flare 
(h large but not infinite) does not alter v m and z very much, however, 
and the larger mouth radiates more efficiently. 

The case not included in this approximate solution is the one for a 
conical horn when x is smaller than the wavelength X. This is the 
case where the horn throat is extremely small compared with the open 
end and is approximately the case for the oboe and the English horn. 
For the conical horn, Eq. (24.20) is 

pc ^ 1 — (t'/2)(aco/c) 2 tan(coZ p /c) , ic_ 
Zq (l/2)(aco/c) 2 — i tan(co? p /c) o)X 

and when x is small enough, (c/W ) is large, instead of small, though 
it is not usually so large as (c/aco) 2 . In this case, tan(coZ P /c) turns 



VI.24] PROPAGATION OF SOUND IN HORNS 287 

out to be approximately equal to — (wx /c), and the allowed frequencies 
and the throat impedance at these frequencies turn out to be 






) 



Here the allowed frequencies include both the even and odd harmonics, 
as is the case with the flute, rather than the clarinet. The specific 
acoustic impedance at the throat is not particularly large, but it is 
larger than any value of the resistance for any other frequency; so 
the frequencies v n are the ones that are heard in the emitted note. 
This lack of a pronounced resonance peak may partly explain the 
difficulty of playing the oboe. 

As a final item in this section, we note that the impedance at the 
throat of a long horn, below the cutoff frequency, can be easily obtained 
from Eq. (24.19). It is a pure mass reactance 

z ~ -io)(ph) Jl — I — J + tanhe (w < c/h) 

It is approximately equal to the mass reactance of all the air in 
the horn within a distance h of the throat. 

Transient Effects. — Calculation of transients in horns of general 
shape is difficult because of the complicated behavior of the impedance 
function. We shall carry through one example here, however, to 
show the general nature of the results. We consider the conical horn, 
to keep the calculations as simple as possible. Suppose that we have 
a piston in the throat of the horn, moving with specified velocity 
C/(w)e _iw '. The pressure wave out of the horn, if it is long enough so 
that reflections from the mouth are negligible, is 

pCX e i(»/c)(x-ct) 

x + x 1 + (ic/coxo) 

We refer now to Eqs. (6.15) and (6.16) and the related discussion. 
The expression for the pressure wave due to any velocity motion 
u (t) of the piston is 

OCX* f°° e -^>[t-(x/e)] C °° 

2ir(x + Xo) J- «, 1 + (tc/oiXo) J-« 



288 PLANE WAVES OF SOUND [VI.24 

To improve convergence, we shall utilize the trick mentioned in the 
last part of Sec. 6: 

1 = 1 — (* c /zo) 



1 + {ic/wxo) (a -f- (ic/xo) 

Therefore the integral becomes 

= _£opc_ ( 1 | e - iu[t - {x/c)] do} J u^ivr dT 
x + x Q {2t J_„ J-» 

r °° r °° e -Mt-T-(x/c)] 

- I w (t) <2t I i /• / \ dco 

oJ-» J-» o) + (*c/a;o) 



27TiCn .]- 



We now use Eq. (2.20) to calculate the first integral and use contour 
integration to calculate part of the second: 



Xopc 
V = — r — 

X + Xo 



t--\- y -^ T Mo ( T ) e -(o/.o)(«-r)+(^o) dr 

c) (x + XojJ-* 

Therefore part of the wave has the same "shape" as the "velocity 
shape" of the motion of the piston at the throat. Behind this wave, 
however, is a "wake" represented by the second integral, which dies 
out exponentially. This behavior is best seen if we cause the piston 
to move suddenly forward at t = 0. Then u (t) = 8(t), and 

( [t< (x/c)] 

p,(0 = J _p^o_ 8 ( t _A_ • /*!_ e-cc/^^/o] [t > {x/c)] 

\X + Xz \ c) (x + Xo) 

The original pulse moves out of the horn with velocity c and with 
amplitude diminishing as (x + x ) _1 . After the sharp pulse comes a 
negative pressure wave which dies out exponentially; the smaller 
the constant x is, the more rapidly does this "wake" damp out but 
the larger is its initial amplitude. 

Problems 

1. What is the speed of sound inside a gasoline-engine cylinder just after com- 
bustion, when the pressure is two hundred times atmospheric and the temperature 
is 1000°C; if the gas mixture has a value of y = 1.35 and would have a density of 
0.0014 g per cc at 0°C and atmospheric pressure? 

2. A plane wave in air has an intensity of 100 ergs per sec per sq cm. What 
is the amplitude of the temperature fluctuation in the air due to the wave? 

3. The specific acoustic impedance at a point in a tube is R = 42, X = —42 
when v — 344. What is the specific acoustic impedance at points 25, 50, 75, and 
100 cm farther along the tube? 

4. A vibrating piston is placed at one end (x = 0) of a tube whose cross 
section is 10 sq cm, and a second piston whose mechanical impedance is to be 



VI.24] 



PROPAGATION OF SOUND IN HORNS 



289 



measured is placed at the other end (x = 30). When the pressure due to the 
sound wave is measured at different points along the tube, it is found that the 
pressure amplitude is a maximum at the points x = 3, x = 15, x = 27> having an 
amplitude of 10 at these points. The pressure amplitude is a minimum at x = 9, 
x = 21, with a value of 6.57. From these data find the mechanical impedance 
of the driven piston, the frequency of the sound used, and the amplitude of vibra- 
tion of the driving piston. 

5. A vibrating piston is set in one end of a tube whose length is 86 cm and 
whose cross section is 10 sq cm. Closing the other end of the tube is a diaphragm 
whose over-all mechanical impedance is to be measured. The measured radiation 
impedance of the driving piston (not including the mechanical impedance of the 
piston itself) has the following values: 



V 


100 


200 


300 


400 


500 


600 


700 


800 


R p 





42 


147 


420 


189 


840 


67 


1,260 


-X-p 


- 71 


1,260 


-365 





126 


-840 


76 


-1,680 



What are the real and imaginary parts of the mechanical impedance of the 
diaphragm at these frequencies? Plot a curve of the magnitude of the impedance. 
6. A tube of length 86 cm and cross section 10 sq cm is closed at one end and 
has a piston fitted in to the other end. The piston is driven electromagnetically 
by a coil whose coupling constant is r = 1,000. The impedance of the coil is 
measured at different driving frequencies, and the part of the impedance due to 
the motion of the piston is found to be 



V 


100 


150 


200 


250 


300 


350 


400 


450 


500 


Rm 


1.92 


1.17 





1.68 


1.47 


0.54 





1.75 


0.67 


Xm 


-0.37 


0.98 





-0.75 


0.88 


0.89 





0.64 


0.95 



Compute and plot the real and imaginary parts of the mechanical impedance of 
the piston and of its radiation impedance. 

7. Two similar pistons, each of mass 10 g, slide freely in opposite ends of a 
uniform tube of length 34.4 cm and cross section 10 sq cm. One piston is driven 
by a force 100e _2 ' r * > ' dynes. Plot the amplitude of motion of the other piston as 
a function of v from v = to v = 500. 

8. A piston of negligible mechanical impedance is fitted into the end of a 
uniform tube of cross section 10 sq cm. The other end of the tube, 34.4 cm from 
the piston, is closed by a rigid plate. The piston is driven by a force 

: F v - 42 sin (Stt*) - 28 sin (24irt) + 8.4 sin (40ttO 

Show that the velocity of the piston is 

u P = ^ tan ( 2^5 J cos (8nrt) — ^ tan ( ^ J cos (24x0 + ^ tan (^q) cos ( 407rf ) 

Plot the values of F and of the piston displacement as a function of t from t => 
to I ■'* (I), What is the curve for piston displacement when the tube is 68.8 cm 
long? 



290 PLANE WAVES OF SOUND [VI.24 

9. How long must an open organ pipe be to have its fundamental frequency 
equal to 250 cps, if the pipe is a cylinder, with radius 2 cm? 

10. When a gasoline exhaust valve first opens after the explosion cycle, the 
volume occupied by the gas in the cylinder is 200 cc. The effective area of the 
valve opening is 1 sq cm, and its effective length is 1 cm. Assuming the cylinder- 
valve system to act like a Helmholtz resonator, what will be its natural frequency 
of oscillation if the air in the cylinder is at normal pressure and temperature? 
What will be the frequency if the gas is at a temperature of 546°C and is at normal 
pressure? 

11. The exhaust valve in Prob. 10 opens into an exhaust pipe of 5 cm inside 
diameter, 100 cm long. What will be the lowest natural frequency of the cylinder- 
valve-pipe system for air at normal pressure and temperature? 

12. A Helmholtz resonator has a cylindrical open neck 1 cm long and 1 cm 
in diameter. If the resonating vessel is spherical, what must its radius be to 
have the resonance frequency be equal to 400 cps? 

13. A piston of mass 10 g, frictional resistance 1,000 dynes cm per sec, is at 
one end of a tube 13.69 cm in diameter and 43 cm long, open at the other end. 
Plot the real and imaginary parts of the total mechanical impedance of the piston 
as functions of the frequency from v = to v = 2,000. 

14. A simple "muffler" consists of a tank of volume V with input pipe at one 
end and a cylindrical output pipe, of length I and radius a, at the other end. 
Draw the equivalent circuit for the analogous impedance at the input end; include 
the analogous radiation resistance R. If the volume flow into the tank at the 
input end is simple harmonic, v e~ iat cm 3 per sec, what is the volume flow out 
of the output pipe? What is the power radiated into the open? How does it 
depend on frequency? What is the frequency above which the output radiation 
falls off rapidly? Above what frequency will the equivalent circuit be invalid? 

15. The back of a loud-speaker diaphragm looks into a "tank" of volume V, 
which connects with free space by a constriction of negligible length, and area S 
of opening. Set up the equivalent circuit for the analogous impedance at the back 
of the diaphragm, and give the formula for the additional mechanical impedance 
load on the diaphragm. Over what range of frequencies will the motion of air 
in the constriction be in phase with the motion of the diaphragm with respect to the 
outside (i.e., move out as the diaphragm moves out; be sure to express the phase 
relations between input current. in the equivalent circuit and diaphragm motion 
correctly) and be larger than the diaphragm motion? If the area of the constric- 
tion is 100 cm 2 , what volume must the "tank" have in order to have this reinforce- 
ment of the diaphragm motion come at and below about 100 cps? Above what 
frequency will the equivalent circuit be invalid? 

16. A tube 1 cm in radius and 68.8 cm long has a hole in its side, placed mid- 
way along its length, which is 1 cm in diameter and is provided with a short open 
tube of the same diameter, 1 cm long. Plot the specific acoustic impedance at 
one end of the long tube when its other end is closed. What are the lowest three 
resonance frequencies of the system? Plot the specific acoustic impedance at one 
end of the tube when its other end is open. What are the lowest three resonance 
frequencies for this case? 

17. An air-conditioning system has a circulation fan that produces noise of 
frequency chiefly above 200 cps. Design a low-pass acoustic filter, consisting of 
two vessels and three narrow tubes in series, which will filter out the noise. The 



VI.24] 



PROPAGATION OF SOUND IN HORNS 



291 



narrow tubes cannot be less than 5 cm in diameter, and the vessels cannot have a 
volume larger than 30,000 cc. 

18. A condenser-microphone diaphragm is stretched across one end of a tube 
2.5 cm in radius and 1.2 cm long, open at the other end. Compute and plot the 
ratio between the pressure at the diaphragm and the pressure at the open end of 
the tube as a function of v from v — to v = 10,000. 

19. When the damping of wave motion along a tube must be taken into 
account, show that the pressure amplitude at the point x is given by the formula 

[ V \ = A -\/cosh 2 Lvcci + (-} (I - z)l - COS 2 \irf3i + (y) (I - X) J 

where k is the damping constant. Plot the pressure amplitude as a function of 
x, for a tube having 

I = 100 cm, X = 50 cm, (£) = (j^A «* = 0.1, ft = 0.4 

What is the acoustic impedance atz = Z? At a; = 0? 

20. The pressure amplitude is measured in a tube of length 100 cm, as function 
of the distance re along the tube from the driving end. The values of the maxima 
and minima, and their positions along the tube are 



Distance from driver x 


12.5 


25 


37.5 


50 


62.5 


75 


87.5 


Pressure |p|, max 




1.899 




1.478 




1.204 




Pressure |p|, min 


1.934 




1.335 




0.869 




0.489 



From these data compute X, a*, ft, the damping constant k, and the specific acoustic 
impedance at x = and x = 100. What is the pressure amplitude at x = 0? 
At x i- 100? 

21. A tube of length I has a termination impedance such that ai and ft are 
both independent of frequency. The damping constant k for waves in the tube 
is likewise independent of frequency. The end at x = is fitted with a piston, 
which is given a sudden push at t = 0; uo = 8(t). Show that the "flutter echo" 
in the tube is represented by the expressions: 



e -KX/C g 



e 



-2x(aj-t/3i) 
? 

— 4x(ai— i/3i) 



: i-ifld-(f)(2J-*) s f f _ 21 - x \ 



[0 < t < (l/c)] 
{{1/6) < t < (2l/c)) 
[{21/ c < t < {3l/c)} 
[(3Z/c) < t < {4l/c}} 



22. A conical horn has an angle of opening fi = 20 deg. The area of the neck 
of the horn is 5 sq cm, and that of the mouth is large enough so that no sound is 



292 PLANE WAVES OF SOUND [VI.24 

reflected back into the horn. Plot the transmission coefficient of the horn as a 
function of v from v = to v = 1,000. When the air in the throat is set into 
vibration at a frequency of 500 cps, what will be the specific acoustic impedance 
in the throat? If the velocity amplitude of the air in the throat is 1 cm per sec, 
what is the intensity of sound in the horn 200 cm from the throat? 

23. The conical horn of Prob. 22 is 200 cm long, and its open end has an area 
of 4,000 sq cm. What value of x must be used in the formula for an exponential 
horn that has the same area of throat and open end as the conical horn and is as 
long? Plot the transmission coefficient of such an exponential horn as a function 
of frequency from v = to v = 1,000. What is the specific acoustic impedance 
in the throat of the horn at x = 500? If the air in the throat vibrates with a 
velocity amplitude of 1 cm per sec, what is the intensity of the sound in the horn 
200 cm from the throat? 

24. The conical horn of Prob. 22 has its small end fitted to a driving piston of 
area 25 sq cm and mechanical constants to = 1, R = 1,000, K = 1,000. Assum- 
ing that the horn is large enough so that Eqs. (24.6) are valid, compute and plot 
the total mechanical impedance of the piston as a function of v from v = 100 to 
v = 5,000. Is the radiation impedance an important part of this? A similar 
piston is attached to the exponential horn of Prob. 23.. Plot its total mechanical 
impedance as a function of v from v = 100 to v = 5,000. 

25. The pistons of Prob. 24 are each driven by a force of 10,000e- 2Xi,,< dynes. 
Plot the power radiated by each of the horns as a function of v from v = 100 to 
v = 5,000. In the light of these curves discuss the relative advantages of the 
two types of horns. 

26. What are the constants for a catenoidal horn having the same over-all 
dimensions as the horn of Prob. 23? Plot the transmission coefficient of such a 
horn over the frequency range from to 1,000 cps. What is the specific acoustic 
impedance in the throat at v = 500? What is the intensity at x = 200, if the 
air in the throat vibrates at 500 cps with a velocity amplitude of 1 cm per sec? 

27. The loud-speaker of Fig. 62 has its conical horn replaced by an exponential 
horn of cutoff frequency v = 100 cps. What is the value of h? Recompute the 
curves of Fig. '62 for frequencies from 100 to 4,000 cps for this horn. Is this a more 
satisfactory loud-speaker? Can you suggest changes in values of the constants to 
improve results? 

28. Repeat the calculations of Prob. 27 for a catenoidal horn having a cutoff 
frequency of 100 cps. Suggest further improvements. 

29. One unit of a "tweeter" loud-speaker consists of a catenoidal horn of cutoff 
frequency 1,000 cps. The driving piston, of effective mass 1 g, is set in the throat 
of the horn, which has area So = 3 cm 2 . The piston is driven by electromagnetic 
means, having a coil of inductance L c = 0.001 henry, resistance R c = 1 ohm, and 
coupling constant r = 10 5 . Set up the equivalent circuit for the total electrical 
impedance of the driving coil. Compute the power radiated for a 10-volt driving 
emf for the frequency range 1,000 to 10,000 cps. What changes in constants will 
improve this behavior? 

30. The "tweeter" unit of Prob. 29 has a conical horn with x& = 5 cm. Calcu- 
late the power radiated over the frequency range 1,000 to 10,000 cps. 

31. The piston of Prob. 29 is driven by an x-cut crystal of Rochelle salts, of 
dimensions a = 2 cm, b = 2 cm, d = 1 cm (temp. = 15°C). Set up the circuit 
for the total electrical impedance of the crystal. Suppose that the crystal is 



VI.24] PROPAGATION OF SOUND IN HORNS 293 

driven by an emf of 100 volts, compute the power radiated from v = 1,000 to 
v = 10,000. 

32. A catenoidal horn of shape formula S = So cosh 2 (x/h), of length I, is used 
as a "speaking trumpet" for a dictating machine. Assume that the impedance 
of the diaphragm at the throat is infinite and that a plane wave strikes the mouth 
parallel to the axis of the horn. Modify Eqs. (24.18) and (23.28) to show that the 
specific acoustic impedance presented to the plane wave at the mouth (x = I) is 

*o = pc j [l - 2 ^pH - i \m(w) - i cot (corZ/c)] | 

where « = (4xa/X) = (2cm /c), t 2 = 1 — (c/coh) 2 , and a is the radius of the mouth 
of the horn. Show that the pressure in the horn a distance x from the throat is 

ipc po cosh (l/h) cos (utx/c) 
t Zo cosh(x//i) sin (oxrl/c) 

where po is the driving pressure at the mouth of the horn due to the incident plane 
wave. Plot the ratio of the pressure at the throat (x = 0) to this driving pressure 

-gdh 1 - 2 ^]Mt) + [*<•>-(?) — mv 

for a horn of I = 34.4 cm, h = 17.2 cm, a = 3.44 cm as a function of « = 2wv from 
v = to 500. 

33. Use the relation 



X' 



/ : — pr 6 V P 1 + a 2 

e-* Jo(« Vt 2 - b 2 ) dt = / 

V p 2 + a 2 



to help calculate transient sound in exponential or catenoidal horns. For a long 
horn of shape given by Eq. (24.16), show that if a unit pressure impulse starts at 
the throat at t = it will travel out of the horn according to the equation 



1 cosh 

V = 



2tt cosh [(x/h) + e] 
coshe 



/- e <(?)^RIy-^ 

M< : V4-!)^^4V(iRly]} 



cosh [(x/h) + e] } \ cj \ c) y/ t % - ( x / c ) 

where S and u are defined in Eqs. (6.9) and (6.11). 

34. Use the equation quoted in Prob. 33 to show that the velocity wave out of a 
long catenoidal horn due to an impulsive motion of a piston at its throat is 

"* - o \tt-7i\ f" T 1 +—■ } C/k) tanh(s/ft)~|. 

2x cosh (x/h) J- « L -*« VI - (c/coh) 2 J 



. e< („*/c)Vi-(^) 2 -^ do 



, c sinh (x/h) 



O-CKV^F© 1 ) 



h cosh 2 (x/h) 
Obtain an expression for the pressure pulse accompanying this velocity pulse. 



CHAPTER VII 
THE RADIATION AND SCATTERING OF SOUND 

25. THE WAVE EQUATION 

We must now consider the more complicated case of sound waves 
that are not plane waves. When we studied the motions of a mem- 
brane we found that waves other than parallel waves are very difficult 
to analyze; in the case of sound wave's we shall find an even greater 
increase in complexity. Nonplane waves do not all travel with 
constant speed or shape. Their motions must be expressed in terms 
of coordinate systems which often are too complicated to handle. We 
shall content ourselves, in this book, with a discussion of cylindrical 
and spherical waves. 

The Equation for the Pressure Wave. — The first task is to obtain a 
wave equation for three dimensions. This can be obtained by a simple 
generalization of the argument of Sec. 22. The air particle at the 
point (x,y,z) can move in any direction, so we must find the three com- 
ponents of its displacement from equilibrium. This displacement is 
a vector d with components £, 17, and f in the x-, y-, and 2-directions, 
respectively. The corresponding particle velocity u = (dd/dt) has 
components u, v, and w. All these quantities, together with the 
pressure p, are functions of x, y, z, and t. When displaced by the 
sound wave, the elementary volume dx dy dz becomes a parallelepiped 
of volume J dx dy dz, where 



J = 



dx 
dx 
dx 



dj 

dy 

1+p- 
dy 

it 

dy 



dj 
dz 
dr] 

dz 

dz 



~ 1 + div(d) 



where div(d) = — + — + ^- is called the divergence of the particle 

displacement d. 

Therefore the counterpart of Eq. (22.2) for the equation of con- 
tinuity is 

294 



VII.26] THE WAVE EQUATION 295 

14-5 = 1; 5 ~ -^div(d) 

Corresponding to Eq. (22.5) for the adiabatic compression of the gas 
is the relation 

1 + ■?- = (1 + 8)t = ^- e ; V ^ -VcPo div(d) 
Finally, Newton's equation of motion is 

= -grad(p) 



■©- 



where grad (p) is a vector, called the gradient of the pressure, with com- 
ponents (dp/dx), (dp/ By), and (dp/dz). 

These equations can be combined as follows: 

^ = - 7c Po div(^ = ^ d iv[grad(p)] = Wp (25.1) 
where c 2 = (7<JVp) and where 

divferad( P )] = §* + $ + g = V*p 

is called the Laplacian of the pressure. This is the wave equation 
in three dimensions for the pressure. The particle velocity for simple 
harmonic waves can be obtained from the pressure by using the 
relation 



\io>p/ 



grad(p) (25.2) 



The potential energy of a unit volume of the gas at any instant 
can be given in terms of the absolute temperature 

pCvT ~ R (1 + 8) " ( 7 . - 1) K + } (7c ~ IV- 1 

The equilibrium value of this, Po/(y c — -1), is the internal energy of 
the gas at equilibrium and is not included in the expression for the 
energy of the sound wave. When we expand the quantity (1/J 1 " -1 ) 
in a series of the small quantities (d£/dx), etc., we can leave out the 
terms in the first powers of these quantities and terms involving cross 
products, such as (d£/dy)(dn/dx), etc., because these terms have zero 
average value. The only terms that are not zero on the average 



296 THE RADIATION AND SCATTERING OF SOUND [VII.25 

(unless we go to terms smaller than second-order ones) will be those 
involving the square of the quantities (drj/dy), etc., 

(7. - i) v**- 1 / 

~ r~TT) | ~ 2 l)y ° div 2 (d) + neglectable terms! 
The total average energy of the sound wave is therefore the integral 

W ~ ip f f f [Y^ + c 2 div 2 (d)l dx dy <fc 

= JJJ Up" 2 + 2p7 2 P 2 J rf z dy dz 



(25.3) 



Curvilinear Coordinates. — The equations for pressure, particle 
velocity, and energy have been given in terms of the rectangular 
coordinates (x,y,z); they can also be given in terms of other coordi- 
nates. The differential operator V 2 in Eq. (25.1) can be written as 

d 2 d 2 d 2 

-z—z +.-^-5 + t~i m rectangular coordinates 
dx 2 By 2 dz 2 

d 2 d 2 . 

t— 5 + r-:in cylindrical coords 
V 2 = ( ■'■ °> \ or/ r- d<i> 2 dz' ^ (25 4) 

1 d 



r dr\ drj r 2 



1 d 2 

+■ , • o « t - 5 in spherical coords. 
r 2 sm 2 #d99 2 

The operator V 2 is called the Laplace operator, or the Laplacian: it 
measures the concentration of a quantity (or, rather, the negative of 
the concentration). The value of V 2 p at a point is proportional to 
the difference between the average pressure near a point and the 
pressure right at the point. When this is negative, there is a con- 
centration of pressure at the point, and when it is positive there is a 
lack of concentration there. The wave equation simply states that, if 
there is a concentration of pressure at some point, the pressure there 
will tend to diminish. Compare this with the discussion on page 175 
for the membrane. 

In order to compute the particle velocity in curvilinear coordinates 
we must have available expressions for the components of the vector 
grad(p) along the coordinate axes. If, for instance, we label the 



VTI.26] RADIATION FROM CYLINDERS 297 

component of the gradient parallel to the cylindrical radius, grad r ( ), 
that perpendicular to r and to z, grad^,( ), and so on, we have 

grad,( ) = A; gradj/ ( ) = |. ; g r ad 2 ( ) = A 

in rectangular coordinates 

grad r ( ) = A; grad,( ) = A; grad,( ) = i A 

in cylindrical coordinates / v^^.o) 

grad r ( ) =-; grad^ . ) = - -; 

1 8 
grad»( ) = — : — r — - in spherical coordinates 
rsm&d<p 

These expressions can be substituted in Eq. (25.2) to compute the 
components of particle velocity in these coordinates. 

Finally, to calculate the sound energy in terms of particle displace- 
ment, we must give expressions for the divergence of a vector A : 

dA x 6A y dA z . . . .. x 

-^ r -r h -t— m rectangular coordinates 

in cylindrical coordinates 

in spherical coordinates 

With these formulas giving the different mathematical forms of the 
physical quantities V 2 (p), grad(p), and div(d) when expressed in 
these three coordinates, we are ready to study plane, cylindrical, and 
spherical waves in free space. 

26. RADIATION FROM CYLINDERS 

We have already discussed circular waves on a membrane in 
Sec. 19 and have developed there the set of functions needed to discuss 
cylindrical waves. We showed there that circular waves have a more 
complicated behavior than plane waves — for instance, that circular 
waves change shape as they spread out, leaving a "wake" behind 
them. Cylindrical waves, the three-dimensional generalization of 
circular waves, show the same behavior. We shall spend most of our 
time here in discussing simple harmonic cylindrical waves. 



298 THE RADIATION AND SCATTERING OF SOUND [VII.26 

The General Solution. — The general solution of the wave equation 
in cylindrical coordinates is a combination of functions of the type 

p = cos (m<f>)[AJ m (kr) + BN^krW*- 2 ™' (26.1) 

Sill 

where v = (c/2ir) s/k 2 + k 2 . The variable z measures distance along 
the cylinder axis, r the perpendicular distance from the axis, and 
is the angle that r makes with the reference plane. The functions 
J m and N m have already been discussed; their properties are given in 
Eqs. (19.4), (19.5), and (20.1), and the values of some of them are 
given in Table III at the back of the book. 

Uniform Radiation. — For waves spreading uniformly out from a 
cylinder, we use the function for k z = and m = 0, which represents 
outgoing waves [see Eq. (20.2)] : 

>aJ-%- ^r-cD-iWA)^ k = (2tv/c) = (2tt/\) > (26.2) 

— l (v) ln ^ e ~ 2 * irt 



r— >0 



Suppose that we have a long cylinder of radius a which is expanding 
and contracting uniformly in such a manner that the velocity of the 
surface of the cylinder is u = U er 2irivt . The constant A, to corre- 
spond to the radiated wave, must be chosen so that the velocity of the 
air perpendicular to the cylinder surface u r = (l/2Trivp)(dp/dr) is 
equal to u at r = a. If a is small compared with the wavelength, 
this velocity is (A/ic*vpa)(r 2rivt } so that A must equal (T 2 v P aU ). The 
pressure and velocity at large distances from the cylinder are then 

\(*V 

V -> TrpaUo J— e*<~»-«*/*> 



Mr 



yjcr 



TaU * - e*<~»-*<* /4 » 



The product of the real part of each of these expressions gives the 
flow of energy outward per second per square centimeter, and the 
average value of this 

T~U*pa 2 U 2 0?\ (26.3) 



VH.2G] RADIATION FROM CYLINDERS 299 

is the intensity of the sound at a distance r from the cylinder's axis. 
The total energy radiated in ergs per second per centimeter length of 
the cylinder is 

n = T* P va*Ul 

Radiation from a Vibrating Wire. — A somewhat more complicated 
wave is generated by a cylinder of radius a vibrating back and forth 
in a direction perpendicular to its axis, with a velocity U e~ 2Tipt . If 
the plane of vibration is taken as the reference plane for <f>, the velocity 
of the part of the cylinder's surface at an angle <£ from the plane of 
vibration has a component U cos <j> e- Mvt perpendicular to the sur- 
face. In this case we take the radiated wave to be 

p = A cos0[J"i(At) + iNi(kry\er M '*, k = (2rv/c) 

.Ac „ . , 

» — i—j— cosd>e~ 2rtrt 

r _»o i r'rr y (26.4) 

> A J~ ^(r~ct)-i^/i) cog( x 

r-*co \TT l vr ^ 

If a is small, the radial velocity at r = a is 

cos<f>e~ 2Tipt 



\2ir 3 v 2 pa 2 J 



which must equal the radial velocity of the surface, so that A = 
(2ir z v 2 P a 2 U /c). The radial component of the particle velocity at 
large distances from the cylinder is 



Ur > I — J J-^ e^(r-c*)-*(3x/4) cog ^ 



j— >«> \pc/ \7r J vr 



There is a component of particle velocity perpendicular to the radius 
r, but it diminishes as r-* at large distances and so is negligible there 
compared with u r . The intensity at large distances and the total 
power radiated per centimeter length of vibrating wire are 

t-?^3co..* n = ^W3 (26 .5) 

The amount of sound energy radiated by a vibrating string therefore 
diminishes rapidly as the frequency of vibration of the string decreases 
and diminishes very rapidly if the string's thickness is decreased. As 
we have mentioned in Chap. Ill, a vibrating string is a very inefficient 
radiator of sound. 



300 THE RADIATION AND SCATTERING OF SOUND [VII.26 

The reaction of the air back on the vibrating wire is obtained from 
the expression for the pressure at r = a: 

p ~ — iupaUo cos0e -iw< 

The net reaction force on the wire per unit length in the direction of 
its motion is 



•J". 



2*- 

p cos <pd<t> = -Foe-™ 1 = —i(ap(ira 2 )UQer i » t 



The ratio of this to the velocity of the wire is the mechanical impedance 
per unit length of wire, due to sound radiation, 



(jff) ~ - to(*-a 2 P ) (a « c/«) 

which is equivalent to the reactance of a mass of air of volume equal 
to that of the wire. The resistive part R of this impedance is too 
small to be included in this approximation when a is small. We can 
find the resistive part from Eq. (26.5) for the power radiated, for of 
course II = %RUl, 

Z tad c* -i^a^p) + ( 7 ^) (a « oh) 

This can be used to compute the effect of the presence of air on the 
motion of a vibrating string, as discussed on page 105. 

Radiation from an Element of a Cylinder.— To solve more com- 
plicated problems, where the velocity of the surface of the cylinder is a 
less simple function of <p than the preceding examples, it is convenient 
first to solve the problem where only a single line element on the 
surface of the cylinder does the vibrating. Suppose that the radial 
velocity of the surface r = a is 

(da , da\ 

- — < 4> < + -g J 

( , da ^ , . da\ 

(^ + -2 < <t> < 2tt - T J 

The Fourier-series expansion for this function of 4> is 

\ / L OT= 1 



U a = 



2rivt 



V1I.26] RADIATION FROM CYLINDERS 30 1 

To fit this distribution of velocity at the surface we choose a general 
sort of outgoing wave 

00 

V = 2 Am cos (- m ^ J ^ kr ) + iN m {kr)-\e-^\ k = (2rv/c) 

m = 

The corresponding radial particle velocity at r = a is 

a = ( X 02) = \A^1 e'y + ^ AmCm e 1 ^ cos(m<f>)\ e~ 2 * irt 
\2irivpo drj |_ 2p c -£J pc J 



u a = 
where 



2 ^ [Jo(n) + iATo(M)] = iCoe**; 

^ [/*(/*) + ii\T„Gz)] = iC«e^- (m > 0) 

and where p. = ka. Therefore, using Eqs. (19.5), 

Ji(ka) = %Co sin (70) ; Ni(ka) = — iC cos (70) 

J m+ i(ka) — J m -i(ka) = 2C m sin (y m ) (m > 0) 

N m -i(ka) — N m +i(ka) = 2C m cos (y m ) (m > 0) 

Note the additional factor of two in C , to anticipate the factor ^ in the 
sum for ra = 0. Limiting values of the amplitudes C m and phase 
angles y m are given by the following approximate formulas: 

When (ka) = (2ira /\) » m + £ 

Co c^ \/8/irka; y ~ka — (t/4) 
C m ~ y/2/wka; y m ~ ka — hf(m + i) (m > 0) 

When (ka) = (2ira/\) « m + i ) (26 6) 

C ~(4Ma); 7o^7r(fca/2) 2 
„ m!/2V +1 **» (ka\ m 

Values of some of the C's and 7's are given in Table X at the back of 
the book. 

To fit the expression for u r at r = a to the expression for the velocity 
of the cylinder u a , we must make 

. pcU da . 

A m = - — ~ — e tym 

Since, at very large distances from the cylinder 

J m (kr) + iN m (kr) ~ ^-| e f [^-i^+i)] 



302 THE RADIATION AND SCATTERING OF SOUND [VII.26 

we have the following expressions for the pressure, particle velocity, 
and intensity of sound at the point r, <f> (when r is a large number of 
wavelengths) and for the total power radiated by the element per unit 
length of cylinder: 



a U da */- e ik{r - ct) \{/(<p) ; p ~ pcu 
\w 3 ka ^u (J m 



r 

00 

_ / .N r Tm + £ (2m+1) 



T 



pc\Uda) 2 *S? cos (m<f>) cos (n0). } (26.7) 



2r*pr ^J n C m C n 



•cos [y m — y n + ^7r(m — n)] 

n = pc 2 (£/^) 2 [_2_ , ^ J. 1 
Wv LCI £{ ell 

To find the intensity we have, of course, multiplied the real parts of 
u r and p together and averaged over time. The total power II per 

Long- and Short-wave Limits. — When the wavelength is quite 
long compared with 2ira, we can use the second part of Eqs. (26.6) to 
compute the radiation. The largest values of (l/C m ) are for m = 0, 
so that to the first approximation 

^(0)c- A /pe-** /4 (26.8) 

T ~ (*£-') (IT *»)•; TL^(^)(Uday 

At these low frequencies the sound radiates out with equal intensity 
in all directions, and the amount radiated is small. The expression 
for intensity is the same as that given in Eq. (26.3) for a uniformly 
expanding cylinder, if we substitute for U in the earlier expression 
the average velocity (U da/2-n) of the surface. 

Values of T and II are plotted in Fig. 65. Polar curves of the 
intensity are shown for different values of n = ka = (2-ira/X) and a 
curve is given for II as a function of /z. We notice that at long wave- 
lengths very little power is radiated, and the intensity has very little 
directionality. As the wavelength is decreased more power is radi- 
ated, and the intensity has more directionality; the cylinder begins 



VII.26] 



RADIATION FROM CYLINDERS 



303 



to cast a "shadow," and a smaller proportion of the energy is sent 
out on the side of the cylinder opposite the radiating element. For 
very short waves the intensity is large and uniform from <£ = — {ir/2) 
to d> = +(x/2) and is zero from <t> = (ir/2) to <t> = (3tt/2), in the 
shadow. In the intermediate range of n, where the wavelength is 
about the same size as a, interference effects are noticeable. The polar 
curve for ju — 3 shows that a fairly intense beam is sent out from the 
cylinder in a direction diametrically opposite to the position of the 
line source (<£ = 180 deg). 

The general properties illustrated by this set of curves are a charac- 



Cylinder 



2k<* 



teristic of all wave motion when 
it strikes an obstacle. When the 
wavelength is large compared with 
the size of the obstacle, the wave 
pays hardly any attention to its 
presence. The first polar curve 
in Fig. 65 shows that for long 
waves the intensity is distributed 
in approximately the same manner 
as it would be if the line source 
were not in the side of a cylinder 
but were all by itself, radiating 
into free space. On the other 
hand, when the wavelength is 
very small compared with the size 
of the obstacle, the motion resem- 
bles the motion of particles, the 
waves traveling in straight lines, 
and the obstacles casting sharp- 
edged shadows. Light waves 
have this raylike property in most 
cases; geometrical optics is a valid 
approximation because the light waves are very much shorter than the 
size of most of the obstacles that they encounter. 

When the wavelength is about the same size as the obstacle, com- 
plicated interference effects can sometimes occur, and the analysis of 
the behavior of the waves becomes quite complicated. 

Radiation from a Cylindrical Source of General Type. — If the line 
source is not at <f> = on the surface of the cylinder but is at <f> = a, 
the pressure and velocity at large distances are, by Eqs. (26.7), 




Fig. 65. — Power radiated and distri- 
bution-in-angle of intensity from a vibrat- 
ing-line source set in a rigid cylinder, for 
different values of n = 2ira/X. 



304 THE RADIATION AND SCATTERING OF SOUND [VH.26 

p — > pc(U da) .J- \}/(<j> — a)e ik< - r - ct \ 

U r -+(Uo da) J- 4>(4> - a )e mr - c » 

the axis of the polar diagram being turned through an angle a. If 
several sources are distributed over the surface of the cylinder, each 
for a different value of a, the resulting radiated wave will be the sum 
of all the waves for the individual sources taken separately. This fact 
can be used to express the radiation from a cylinder whose surface 
vibrates with any arbitrary distribution of velocity amplitude. If the 
distribution is such that the surface at <f> = a has the radial velocity 
TJ{a)e~' iTivt , then the wave may be considered to be the result of an 
infinite sequence of line sources, the one at <f> = a having the velocity 
amplitude U(a), etc. The pressure and radial velocity at large 
distances are then obtained by integrating the expressions given 
above for a single line source: 

p~ p cj® e ik ( r - c » J iK0 - a) U(a) da (26.9) 

-fe) 

For instance, if a section of the cylinder between a = —a and 
a = -\-a is vibrating, so that U(a) = U for — a Q < a < a , and 
is zero for the rest of the values of a, then the pressure wave at large 
distances from the cylinder is 



V = pcUo J^ e ik (*-°» J ° iK0 - «) da 

= 2pcU /7 tt(fWrf) >h sin (map) cos (m<ft) g -i[\m+f(2m + i)] 
t 2 \j/r ^J mC m 



(26.10) 



where we use the convention that - — - — — = a when m = 0. 

m 

The intensity and total power radiated are 

_ 2pc 2 Ul >ri sin (map) sin (na ) cos {m<j>) cos (n<f>) 
T*vr ^Li mnC m Cn 

• cos 7m — 7n + (m — n) I 



Vn.26] RADIATION FROM CYLINDERS 305 

_ 2pc 2 U 2 [2a| ^h sm 2 (m<x ) ] 
11 ~ tt 3 ? C 2 "^ 2j m*Cl J 

When the frequency is very small, so that X is much larger than a, 
the intensity and power radiated are 

T -> P^H n->7rpm 2 t/^ 

to the first approximation. The expression for T is quite similar to 
Eq. (26.3) for a uniformly expanding and contracting cylinder. Even 
though the velocity of the surface of the cylinder, in the present case, 
is not symmetrical about the axis, nevertheless the radiation at very 
long wavelengths is symmetrical, behaving as though it came from a 
uniform cylindrical source whose velocity amplitude is (ZJo<xo/t) (the 
average velocity amplitude of the actual surface). This is another 
example of the fact, mentioned above, that wave motion is insensitive 
to details smaller in size than the wavelength. 

Transmission inside Cylinders. — Now that we have discussed the 
behavior of cylindrical waves of general type, it is appropriate to 
return to the subject discussed in Chap. VI, the transmission of sound 
inside tubes, and indicate what happens when the sound is not a one- 
dimensional wave. As an example, we take a tube of circular cross 
section of radius a, driven at the end x = 0. The three coordinates 
are r and 4>, giving position in the cross-sectional plane, and x, the 
distance along the axis (we use x instead of z to correspond to the 
usage of the last chapter). The solutions for the pressure waves 
inside the tube will be some combination of the characteristic waves 

™(»*V«(Av)« l( *^" l> ; *? + *2= , (fY 

The functions N m cannot be used here, for they become infinite at 
r = 0. The values of k r are determined by the boundary conditions 
at the inner surface of the tube, r = a. 

Ordinarily the tube is fairly solid, but it does yield slightly to the 
pressure of the waves on the inside, and in many cases this yielding 
has important effects. At any rate we cannot always neglect it 
entirely. We shall assume that each portion of the tube moves out- 
ward with a velocity proportional to the pressure of the wave at that 
point: 

u r = —) at r = a 

Zt 



306 THE RADIATION AND SCATTERING OF SOUND [VII.26 

The factor of proportionality z t is called the specific impedance of the 
inner tube surface and the ratio of this to pc is the impedance ratio 
of the wall, ft. It may be due to the elasticity of the tube walls (in 
which case z t is mostly reactive), or it may be due to the fact that the 
inner surface of the tube is covered with porous material which allows 
a small amount of air to penetrate the pores (in which case z t is often 
mostly resistive). Of course z t will depend on frequency. The 
reciprocal of the impedance ratio, the admittance ratio, will be written 
(pc/z t ) = (1/ft) = (i<t — i<r t ), where a t is the conductance ratio of the 
tube and c t is its susceptance ratio. 

We can compute the radial velocity from the pressure wave by 
using Eqs. (25.2) and (25.5). Confining ourselves here to the radially 
symmetric waves, the boundary condition is 



p = AJoiKrfeW**-^; u r = i l — J AJi(k r r)e i(kxX - iut ^ 
Jo(k r a) = i(^f)ji(k r a); k = (£), ft = (ji\ 



(26.11) 



If the specific impedance z t is large compared with the characteristic 
impedance of air (as it usually is), then ft is large compared with unity, 
and the first approximation to the solution for k r is that —Ji(kra) = 

jfj^x [Jo(kra)] is equal to zero. Solutions for this can be obtained 
from tables of Bessel functions: 

fc r a~xa „; aoo = 0; a i = 1.220; a 02 = 2.233 • • • (26.12) 

(For further values of the a's, see page 399.) The case a o = is the 
plane wave case we discussed in the previous chapter. We note 
that a 0n equals the quantity /3i„ given in Eq. (19.6) for the circular 
membrane. 

The next approximation is obtained by expanding the functions J 
and J i about k r a = wa 0n and taking the first terms: 



+ 



{k r a)Ji(k r a) = (xaon)«/i(7ra n) + (k r a — ira 0n ) -j- zJ x (z) 

Therefore Eq. (26.11) becomes, to the next approximation (assuming 
that ft is large), 



7 ■ I ka . \ . / oioa ,\ 

kra ~ ira n — 1 1 U ] ~ ira 0n — * [ U I 

\ira 0n / yraion / 



(26.13) 



k = ( w /c); ft = (z t /pc) 



VII.26] RADIATION FROM CYLINDERS 307 

except in the case n = 0. In this special case, Eq. (26.11) reduces to 
1 = li(k*a$i/k), or 

*'~V!s ; M+ fe) (n=0) ■ 

The general expression for axially symmetric waves in the tube is 

(26.14) 



+ 2^.[(^)-<^) 



. g— (iKi/o)+i[ (&)T 0n /c) + (<rj/a) Jx— iu « 

where (1/ft) = x t — *Vt = (pc/z t ). 

Wave Velocities and Characteristic Impedances. — These waves all 
travel with different velocity, even in the case of a rigid tube (f t = oo ). 
This is due to the factor ro n entering into the term (r „oja;/c) and is 
fundamentally due to the fact that all the waves except that for n = 
(a o = 0) have components of motion perpendicular to the tube axis. 

The lowest mode {n = 0) represents the plane wave solution we 
discussed in the previous chapter, for the case of ft = oo . When the 
tube wall impedance is riot infinite, even this wave has a small com- 
ponent of motion in the radial direction, for p depends to some extent 
on r. To the second approximation the pressure wave for this lowest 
mode is (from the first two terms in the expansion for J ) 



V 



( r 2 \1 

\ _1_ £ w I 1 g— (xKt/a)+i[(a/c)+(<rt/a)]x— iwt 

\2acf t /J 



Therefore, when the tube wall impedance is taken into account, the 
pressure wave is not quite uniform over a plane perpendicular to the 
z-axis; there is a part of the wave, out of phase with the usual plane 
part, which is zero at the axis r = and reaches a maximum at r = a. 
If the tube admittance (l/z t ) = (1/pcft) = (l/pc)(/c, — ia t ) has a 
real part {n t /pc = conductance) not equal to zero, the wave dies out 
exponentially as it travels down the tube. If the tube susceptance 
(<r t /pc) is not zero, the wave velocity differs from that in free air, even 
for this "plane-wave" mode. To the first approximation, the wave 
velocity is c[l - <r t (\/2ira)], as long as f< is very large. A stiffness 
reactance corresponds to a positive susceptance (f ~ ipc{K/u) ; 
(1/f) — —if = ~i((a/Kpc), so that a tube with "stiff" walls has a 



308 THE RADIATION AND SCATTERING OF SOUND [VII.26 

wave velocity somewhat less than c, whereas a tube with walls pre- 
senting masslike reactance has a wave velocity somewhat larger than 
c. The departure from velocity c is greatest at long wavelengths 
(unless the variation of <x t with co cancels this effect). 

The higher modes travel with a wave velocity which is larger than 
that in free air. Even when the tube impedance is infinite the veloc- 
ities are (c/ron) = c/s/l — (Xa n/2a) 2 . Below a certain frequency 
v 0n = (ca 0n /2a), called the nth cuioff frequency, the nth mode cannot 
be transmitted down the tube as wave motion. The quantity t » 
becomes imaginary, and the pressure dies out exponentially for any 
lower frequencies. In other words, for a given frequency v only the 
first n modes are propagated down the tube, where n is such that 
a 0n is just smaller than (2av/c). For a tube of 1 ft diameter only the 
lowest, "plane-wave," symmetrical mode is transmitted for fre- 
quencies below about 1,300 cps, and for a smaller tube the higher 
modes are damped out for even higher frequencies. At the cutoff 
frequency the wave motion for the "cutoff" mode is purely transverse. 
Only at higher frequencies does this mode have any motion "to spare" 
for travel along the tube. 

The nonsymmetrical modes, involving cosine and sine of (ra<£), 
also have cutoff frequencies v mn = (ca mn /2a), where 

aio = 0.586; an = 1.697; a 12 = 2.717 

a 20 = 0.972 • 

a 3 o = 1.337 } (26.15) 

TYh 1 

a mn — w + 9 + 7 ( n large) (see also page 399) 

Only the "plane-wave" mode (0,0) is transmitted at all frequencies. 
We note that these a's are not equal to any of the /3's in Eq. (19.6) 
for the circular membrane. 

The ratio of pressure to air velocity in the direction of wave 
motion for a wave traveling in one direction is defined as the char- 
acteristic impedance for the mode in question. For free air we have 
found that it is pc. In a tube with walls of impedance pc{ t the char- 
acteristic impedance for the "plane-wave" mode is approximately 
pc[l — (\/2Ta)((Tt + iKt)]. Therefore if the tube impedance is purely 
reactive, the characteristic impedance is purely resistive; if the tube 
impedance has a real component, the characteristic impedance has a 
reactive term. The difference between the characteristic impedance 
for this mode and that of free air is proportional to the tube specific 
admittance (pc/£ t ) as long as f t is large. 



VH.26] RADIATION FROM CYLINDERS 309 

The nth symmetrical mode has a c haracteristic im pedance (pc/T 0n )- 
•[1 — i(X/2iraT 0n ^t)], where r 0n = \A — (Xaon/2a) 2 . Therefore the 
characteristic impedance for the higher modes differs from pc even 
when the tube impedance is infinite. At frequencies well below the 
cutoff for the mode, the impedance is reactive, being approximately- 
equal to that of a mass of (pa/ira n) g per sq cm of piston surface. 

If a reflected wave is also present, due to mismatch at the far end, 
the impedance for each mode has a hyperbolic tangent factor in it, 
analogous to the cases worked out in the previous chapter. 

Generation of Wave by Piston. — The sort of wave that travels 
along the tube is determined by the nature of the driving piston at 
x = 0. To illustrate this part of the calculation, we consider a rigid 
tube and confine ourselves to symmetric oscillations of the piston; 
hence only the symmetric modes will be present. The extension to 
nonsymmetric modes and to nonrigid tubes will then be evident. 

Suppose that the velocity of the driving piston is given by u x = 
Ua{r)e~ iat at x = 0, and suppose that the tube is long enough so we 
can neglect reflected waves. The functions Jo(ira 0n r/a) are orthogonal 
characteristic functions, analogous to those discussed in connection 
with Eq. (19.10). Their integral properties are 



jJ*J^)j (^) rdr = K 2 (26.16) 

o \ a / \ a / )-x JKiraon) (to = n) 



Therefore the function u (r) can be expanded into the series 

U n Jo(Tra 0n r/a) 



n = 

where 



Uo(r) = > =-) f 

^4 Joiiraon) 

n = 

^» ~ TTl "T I u (r)J ( 7 ^IL_ \ r dr 

la 2 Jo{ira Qn )_\ Jo \ a / 

The coefficient Uo = (2/a 2 ) f u (r)r dr is the average velocity of 

the piston. 

We next fit the series of Eq. (26.14) (for K t = a t = 1/ft = 0) to 
this series as a boundary condition at x = 0, by first computing u x and 
then equating term by term. We find that A n = {pc/tq^} U n /J (Traon), 
so that the pressure wave traveling away from the piston is 



310 THE RADIATION AND SCATTERING OF SOUND [VII.26 

(pc/r 0n ) U n Jo(Tao n r/a)e i( - a/c) fro.*-*) 



-2 

n = 



Jo(Traon) 



•Jo (~~) e^Wo**-^ (26.17) 

TOn = Vl — (TTCQIOn/wa) 2 

If the motion of the piston is uniform over its surface, then only the 
first term (n = 0) remains, and we are back to the plane-wave case 
discussed in the previous chapter. If the piston is not perfectly stiff, 
the motion will not be uniform and some of the higher modes will 
come in. 

As an example, suppose that the driver is a membrane, driven so 
that its motion is u (r) = UJ (Tt3 ir/a), where O1 is adjusted so that 
Jo(irpoi) = [see Eq. (19.6)]. Then, from formulas for the integration 
of two Bessel functions, it turns out that 

TT _ (2C7/3oiA) t ( r \ 0.432E/ 

Un ~ ~Q2 -2- «/ lW30lj = "H /i or . a Ton 

POI — «0n U — (1.306o!0n) 2 ] 

The average velocity of the piston is U Q = 0.432 U. 

The average pressure back on the diaphragm is the coefficient of 
the first term in the series, pc times the average velocity. The pres- 
sure at a point on the diaphragm a point r from its center is 

oo 

Voir) = pcUo > pj n Qnft — '' . r — (26.18) 

^-Jll — (1.306o:oJ 2 l/n(7ra;n„.)Tn» ' 



n = 



which is not equal to the average pressure. In fact this pressure is 
no t exactly in pha se with the velocity because of the factors t „ = 
\/l — (caon/2av) 2 , which become imaginary for the larger values of n. 
At this point we can refer back to our discussion of the air reaction 
on a membrane, given on page 193. There we said that if the velocity 
of waves on the membrane was very much smaller than the velocity 
of sound waves in air, then the pressure was fairly uniform over the 
membrane and was proportional to the average velocity of the mem- 
brane. In the present case, if the membrane is in free vibration at 
its lowest frequency, the driving fr equency will be v = (c TO /3 i/2a), so 
that the factors t » will equal \A — (ca 0n /c m ^oi) 2 (where c is the 
sound velocity and c m the velocity of waves in the membrane). If c is 



VII.27] RADIATION FROM SPHERES 311 

very much larger than c m , then all the t's, except t o = 1 (for apo = 0), 
will be large and imaginary. Consequently, the most important term 
in the series (26.18) for the pressure will be the one for n = 0, which 
is just pc times the average velocity of the membrane, the higher terms 
being out of phase and smaller the larger the ratio (c/c m ) is. 

On the other hand, if (c/c m ) is quite small, a large number of t's 
will be approximately equal to unity; therefore series (26.17) will 
become p c^pcwo(r)e i(o,/c)<:af ~ c<) , except for small terms for higher n. 
In this case, therefore, the reaction at a point on the membrane is 
just pc times the velocity at that point. This is the other limiting case 
mentioned on page 198. The intermediate case is, as we see, quite 
complicated. 

The same sort of calculation can be made for waves radiating into 
the open (we shall discuss this in the next section) and the same 
general conclusions concerning wave-velocity ratios and reaction on 
the membrane can be reached. 

The transmission of sound through ducts will be taken up again 
in this chapter, after we have discussed the nature of wall impedance 
more in detail. 

27. RADIATION FROM SPHERES 

More important for radiation problems than cylindrical coordi- 
nates are the spherical coordinates r, &, and <p; r being the distance 
from some center, # being the angle between r and a polar axis, and 
<p the angle between the plane through r and the polar axis and a 
reference plane. Few radiators of sound are so much longer than 
they are wide that they behave like long cylinders, but many radiators 
behave like spherical sources, especially when their dimensions are 
small compared with the wavelength of the sound emitted. 

Uniform Radiation. — The simplest sort of outgoing spherical wave 
is one that is radiated by a sphere uniformly expanding and contract- 
ing, so that the wave does not depend on # or <p. The wave equation 
in this case is 

d 2 p 
dt 2 

A general solution of this equation which is finite everywhere except 

at r = is 



r 2 dr\ dr ) c 2 



p =±F(r-cQ+±f(r + cQ 



consisting of a wave of arbitrary form going outward from the center 
and another wave focusing in on the center, 



312 THE RADIATION AND SCATTERING OF SOUND [VII.27 

This solution, except for the factor (1/r), is similar to Eq. (8.1) 
for waves on a string and to the equation for plane waves of sound 
given on page 238. This means that spherical waves are more like 
plane waves than they are like cylindrical waves. Plane waves do 
not change shape or size as they travel ; spherical waves do not change 
shape as they spread out, but they do diminish in amplitude owing 
to the factor (1/r) ; whereas cylindrical waves change both shape and 
size as they go outward, leaving a wake behind. Figures 40 and 41 
show that, if a cylinder sends out a pulse of sound, the wave as it 
spreads out has a sharp beginning but no ending; the pressure at a 
point r from the axis is zero until a time t = (r/c) after the start of the 
pulse, but the pressure does not settle back to its equilibrium value 
after the crest has gone by. With both plane and spherical waves 
the wave for a pulse has a sharp beginning and ending, the pressure 
settling back to equilibrium value after the pulse has gone past. This 
behavior is another example of the general law (proved in books on 
the mathematics of wave motion) that waves in an odd number of 
dimensions (one, three, five, etc.) leave no wake behind them, whereas 
waves in an even number of dimensions (two, four, etc.) do leave 
wakes. 

Spherical waves do resemble circular waves on a membrane, how- 
ever, in that they have infinite values at r = 0. As we have seen on 
page 176, this simply means that the size of the source must be taken 
into account; every actual source of sound has a finite size, so that the 
wave motion never extends in to r = where it would be infinite. 

Suppose that a sphere of average radius a is expanding and con- 
tracting so that the radial velocity of its surface is everywhere the 
same function of time U{t). The rate of flow of air away from the 
surface of the sphere, in every direction, is Anra % U{t) = Q(t). To 
obtain an expression for the pressure wave radiated from the sphere, 
we write Newton's equation as p(du r /dt) = — (dp/dr). If p is chosen 
to be an arbitrary outgoing wave p = P(r — ct)/r, the requirement at 

the surface of the sphere, - -^ 5 = — -r 1 — - 9 —> at r = a serves to 

7 r or r z 4xcr at 

determine the shape of the wave P. 

The Simple Source. — If the vibrating sphere is very small (more 

specifically, if a is small compared with the wavelength of the sound 

radiated), the sphere is called a simple source of sound. In this case 

(P/r) is much larger than (dP/dr) at r = a, and P ~ (p/4nr)(dQ/dt) 

at r = a. The pressure wave at a distance r from the center of the 

simple source is therefore 



VH.27] RADIATION FROM SPHERES 313 

(27.1) 



&«■('-§) 



where Q'(z) = -r- Q(z). The function Q gives the instantaneous value 

of the total flow of air away from the center of the source. The pres- 
sure at the distance r is proportional to the rate of change of this flow 
a time (r/c) earlier. 

In some cases the source of sound is small enough or is so placed 
with respect to boundaries that source plus boundaries (if they are 
present) can be replaced by a distribution of simple sources, such that 
the element of volume dx' dy' dz' at (x',y',z') has an equivalent outflow 
of air q(x' ,y' ,z' ;t) dx' dy' dz' cm 3 per sec. The function q is called the 
source function or density of sources. This method of calculation will 
be particularly appropriate for the calculation of the radiation from a 
piston and for the calculation of sound radiation in a room. 

It is not difficult to see that a logical extension of Eq. (27.1) gives 
for the pressure wave at the point (x,y,z) 

< a d C C C dx' dy' dz' ( , , , R\ /nfr „ N 

V^y,z;t) = p - j J J —JL- a [x',y',z';t - -) (27.2) 

where R is the distance between (x,y,z) and (x',y',z'). The differential 
equation for the pressure wave in the presence of a source distribution 
of density q is 

An important special case is when the simple source has a simple 
harmonic flow of air, so that Q(t) = Qoe-™*, where Q , the magnitude 
of the volume flow of air at the surface of the source, is called the 
strength of the simple source. In this case, at large distances from the 
source (i.e., many wavelengths away) the pressure wave is 

p ~ -iw U~\ Qoe^o-"), k = (u/c) = (27I-/A) (27.4) 

The corresponding particle velocity at great distances is (p/pc), and 
the intensity and total power radiated from a simple source are 



8cr 2 2c 

When the wavelength of sound is much longer than the over-all 
dimensions of the radiator, the radiation will be much the same no 



314 THE RADIATION AND SCATTERING OF SOUND [VII.27 

matter what shape the radiator has, as long as the motion of all parts 
Of the radiator is in phase. In such limiting cases the formula (27.4) 
for the simple source can be used. For instance, the open end of an 
organ pipe, or of any wood-wind instrument, is usually small enough 
to be considered as a simple source. If the average velocity of the 
air in the mouth of the tube is U e- 2rivt and the cross section of the 
tube has area S, then the strength of the equivalent simple source is 
UoS, and the power radiated is (wp/S V £7g/2c) . 

Spherical Waves of General Form.— To discuss any more com- 
plicated waves radiating from a sphere, we must consider the solutions 
of the wave equation that depend on # as well as on r and t. In this 
book we shall limit the discussion to waves that depend on # but are 
symmetric about the polar axis and therefore are independent of <p. 
To consider the waves that depend also on <p would add more compli- 
cation than is needed for the problems to be treated. 

The wave equation to be solved is 

r 2 dr V dr J ^ r 2 sin tf d& \ * d&) c 2 dt 2 
If p = R(r)P(&)e- 2wirt , the equation becomes 




The left side of the equation is a function of r only, and the right- 
hand side is a function of # only. Since they are equal for all values 
of r and #, they must both have the same constant value, which we 
can label C. 

Legendre Functions. — We shall first solve the equation for P 



1 * ( shi »§) + CP =° < 27 - 5 ) 



sin # d& 



or, letting cos# = 



x, 



d [d " **) ^] + CP = 



dx\ dx 



We solve this equation by setting P equal to an arbitrary series 
p = a + aix + a 2 x 2 + • • • and substituting this in the second 
equation. We find that the coefficients a must be such that 



VII.27] RADIATION FROM SPHERES 315 

P = a 



1 _ £ X 2 _ C(6 ~ O x < _ C(6 - 0(20 - O r6 
2! 4! 6! 



+ ax he H g|— x s + i ^ i a; 5 + 

The equation and solution have been treated in Prob. 5 (Chap. I) and 
on page 118. We have found that unless C has certain specific values 
the function P will become infinite at x = ±1. The only cases where 
P stays finite are for the following sequence of values of C and a or 
a x : C = 0, a x = 0; C = 2, a = 0; C = 6, a x = 0; C = 12, a = 0; etc. 
The allowed values of C are therefore 

C = m(m +1) (to = 0,1,2,3 • • • ) 

The solution of Eq. (27.5) which is finite over the range of x from —1 
to +1, corresponding to C = m(m + 1), is labeled P m (x). It can 
be obtained by substituting the proper value for C in the series given 
above, making one a equal to zero and giving the other a the value 
that makes P m (l) = 1. The resulting solutions are 

m = o, C = 0, P (x) = 1 

Po(cos#) = 1 



w = 1, C = 2, Px(x) = X 

Pi(cos#) = cos# 
to = 2, C = 6, P 2 (x) = i(3z 2 - 1) 

P 2 (cos#) = -K3 cos 2* + 1) 
to = 3, C = 12, P,(x) = -H5* 3 - Sx) 

P 3 (cos#) = |(5 cos 3^ + 3 cos)* 



(27.6) 



The function P TO is called a Legendre function of order to. It can 
be shown to have the following properties: 

d 2 P dP 

{x2 ~ l) ^ + 2x ^~ w(m + l)Pm = (* = c °s*) 

1 d m 

Pm{x) = ¥^.dx^ {x2 ~ l)m 



(2m + l)a;P m (aO = (m + l)P m+1 (z) + mP m ^{x) 
(2to + l)P- m (s) = *- [P m+1 (x) - P m -i(x)) 

(n J* 5 m) 
(n = to) 



(27.7) 




316 THE RADIATION AND SCATTERING OF SOUND [VII.27 

Values of some of these functions are given in Table IX at the back 
of the book. 

The last equation of (27.7) shows that the functions P m (x) constitute 
a set of orthogonal characteristic functions. Any function of x in the 
range from a; = 1 to a; = — 1 can be expanded in terms of a series of 
these functions: 

CO 

F(x) = 2) B„P m (x), B m = (m +i) f 1 F(x)P m (x) dx (27.8) 

m = 

The expression for the coefficients B m can be obtained by the method 
discussed on page 108. 

Bessel Functions for Spherical Coordinates. — We must now solve 
the equation for the radial function R: 

Only the allowed values of C can be used; otherwise the related func- 
tion of # will not be finite everywhere. Changing scale, to get rid of 
numerical factors, we obtain 

g+si+O-^)*-*-^-* (2 , 9 ) 

This equation looks very much like the Bessel equation (19.3), and 
in fact solutions of the equation are R = (l/\/z)J m+ i(z). The solu- 
tions can be expressed in a simpler way, however, in terms of trigono- 
metric functions. Direct substitution will show that two solutions of 
Eq. (27.9) are the spherical Bessel and Neumann functions j m (z) 
and n m (z), where 



3 


»(*) 


= 


smz 

z 


n (z) = — 


cosz 
z 




3 


iGO 


= 


sin z 


COS2 

z 


n\(z) = 


sinz 

z 


cosz 


3< 


.(*) 




ft- 


- J smz 

2/ 


3 

k cosz 

z 2 

ni{z) = 


3 . 

5 sinz 

z 2 


_(3. 
\z 3 



(27.10) 



- I cosz 



(27.11) 



VH.27] RADIATION FROM SPHERES 317 

These functions have the following properties: 

3m(z) = yJ^Jm+itz), n m {z) = J^N m+i (z) 

jm( - z) T^l'3-5 ••• (2m + 1) 

„.(,) _ - 1-1-8-5 u;( 2m-l) 

... 1 / m-f 1 \ 
3m(z) ~^~ cosl 2 2"— T ) 

. . 1 . / m+ 1 \ 

n m (z) > -sinlz £ — t ) 

Z— » oo Z \ if 

fft(z)z* dz = ~ [jl(z) +' no(*)iiOO] 

2 3 
/n§(2)2 2 dg = - [ng(«) - io(2)n!(2)] 

n m _i(2) < 7* OT (2) - n m (2)j m _i(2) = f-^J 
and have the following properties, for either j m or n m : 

3m-AZ) + Jm+l(2) = J m (z) 

J -J 

di* 7 '" 1 ^ = 2m + 1 W»-i(*) ~ ( m + l)i-+i(2)] 
g^V-OO] = ^ + %-i(2), ^[2-^(2)] - ~z-j m+1 (z)) ( 27 -12) 

Jj'lCa) dz = -jo(z), Jjo(z)2 2 cfe = 2 2 Ji(2) 
2 3 

Jim(2)z 2 dz = g \.Jl(z) - i»-i(2)y*+i(2)] (m > 0) 

In all the equations (27.12) the function n can be substituted for j. 
Values of some of the functions j and n are given in Table VII at the 
back of the book. 

Returning to the problem of wave motion, we can now see that 
the general solution of the wave equation which is symmetrical about 
the polar axis and is finite everywhere except at r = is a combination 
of the functions 

V = P.(cos^) [Aj m (?^) + Bn m (^] e -w (27<13) 
This corresponds to Eq. (26.1) for cylindrical waves. 



318 THE RADIATION AND SCATTERING OF SOUND [VII.27 

The Dipole Source. — The first example of the use of the functions 
discussed above will be that of the radiation from a sphere of radius a 
whose center vibrates along the polar axis with a velocity Uoe-~ 2 * ivt . 
The radial velocity of the surface of the sphere is Uo cost? er 2Tivt } 
where # is the angle from the polar axis; and this must equal the radial 
particle velocity of the air at r = a. Therefore the pressure wave 
must be 

p = AP!(cos#)[ji(/cr) + in x {kr)]e-^ ivt , k = (2*-/X) (27.14) 

> — A ( ^— ) cost? e ik( - T ~ ct) 

r—>«> \lirvr/ 

The radial particle velocity is 

- * dP 
r 2wivp dr 

= Acos * [j (kr) + in (kr) - 2j 2 (kr) - 2in 2 (kr)]e-** i * 
oipc 

cost? e~ 2 * ivl 



»o \ pc ) \kr) 



If the radius of the sphere is small compared with the wavelength of 
the sound radiated, the limiting expression for u r can be equated to 
Uo cos t? e~ 2wivt at r = a to determine A : 



A ~ 



/4fVW \ 



A small vibrating sphere of this sort is called a simple dipole source. 
The intensity and the total energy radiated are 

2irV 4 a 6 ^oCOS 2 <? 

(27.15) 



J»2ir Pit 

d<p 
o Jo 



Tr 2 smt? dt? = — ~x — - 
6c 3 



We note that the dipole source is less efficient than the simple source 
[given in Eq. (27.4)] for radiating at low frequencies, for II diminishes 
as v 4 instead of as v 2 . 

To find the reaction back on the source due to its motion, we first 
obtain the pressure at the sphere's surface. To the first approxima- 



VII.27] RADIATION FROM SPHERES 319 

tion, this is p r==a ^ —rivpaUo cos# e~ 2irivt , for a small. The net' force 
on the sphere is in the direction of vibration and is 

F = ° 2 Jo' d<p Jo ( pr=a cos ^ sin ^ d * — ~ (^'"Xl^pa 3 ) Uoe- 2 ™ 1 

This force is entirely reactive (since it is 90 deg out of phase with the 
velocity) and is equivalent to the reaction of a mass of air having 
volume equal to half the volume of the sphere. The resistive reaction, 
the part in phase with the velocity, is not given by this approximate 
expression, which simply means that the resistive part of the reaction 
for the dipole is so much smaller than the reactive part that it is not 
included in the first approximation. To find it we must compute the 
second approximation for A and p r=0 . However, since we know that 
the total energy radiated must equal (E/o/2) times the real part of F, 
we can find this real part from Eqs. (27.15). The radiation impedance 
of the simple dipole is therefore 

7 F-Ww*pv*a? , -.-w-i ax - '"■' 
r = U 3c 5 (2Tiv)(hrp a*) 

Radiation from a General Spherical Source. — We take up next the 
general case of a sphere, not necessarily small, whose surface vibrates 
with a velocity U{d)e- 2Tivt , where U is any sort of function of #. We 
first express the velocity amplitude £/(#) in terms of a series of Legendre 
functions 



m = 

U m = (m + i) £ U(&)P m (cos&) sin# d& (27.16) 

To correspond to this we also express the radiated pressure wave in a 
series 

00 

p = 2) AmPmicoB^lj^kr) + in m (kr)]e- 2 " rt 

m = 

where the values of the coefficients A m must be determined in terms 
of the known coefficients U m . The radial velocity of the air at the 
surface Of the sphere is 



= - ^ A TO Z> m P m (cos#)e* 8 »'- i 



pc 

n» = 



320 THE RADIATION AND SCATTERING OF SOUND [VII.27 

where 

- j- [jm(ij) + in m {n)] = iD m e^ m , and n = (ka) = I -^ J 

Therefore, using Eqs. (27.11) and (27.12), 

mn m -i(ka) — (m + l)n m+ i(ka) = (2m + l)D m cos 8 m 
(m + l)jn^i(ka) — mj m -i(ka) = (2m + \)D m sin 8 m 
When ka » m + -g- 

D m ~lj-); 8 m ~ ka — £n-(m + 1) 
When k«m + | 

^ 1 - 3 - 5 -';f; + r 1)(m+1) («>o) 

5 ™ — 12 . 32 . 52 . . . (2m - l) 2 (2m + l)(m + 1) 

Values of the amplitudes D m and of the phase angles 8 m are given in 
Table XI at the back of the book. 

The radial velocity of the air at r = a must equal that of the sur- 
face of the sphere, and equating coefficients of the two series, term by 
term, we obtain equations for the coefficients A m in terms of U m : 



(27.17) 



Am — 



( pcU m \ 
\D m ) 



The pressure and radial velocity at large distances {i.e., many 
wave lengths) from the sphere can then be expressed 



-© 



w r ~ C/ [ - J e ik( - r - ct) ^(&) ; p ~ p cu r 



\ka)^i 



*(*) = ( h ) ^j^P^cost^-^-^+u 



where we have multiplied and divided by Uo, the average velocity of 
the surface of the sphere. The air velocity near the sphere is, of 
course, not in phase with the pressure, nor is it entirely radial; but 
far from the sphere the velocity is radial and in phase with the pressure. 
When kr is very large, the intensity at the point (r,#) and the 
total power radiated are 



Vn.27] RADIATION FROM SPHERES 321 

F 'W = (jzrr) 1 2 ^#^(cos^)P n (cost?) • . / 

\ kaU °/ TO ^o ^"^ > (27.18) 

• cos [5 TO — 8 n + ir(m — n)] 

n= fd* r r 2 Tsint? ^ = pg!_/^ ffl 

Jo Jo 2™ 2 -£J (2m + l)D 2 m 

The function /?,(#) is called the angle-distribution function for radiation 
from a sphere. 

When ka = (2ira/\) is quite small, all but the first terms in these 
series can be neglected, and, using Eq. (27.17), we have for the long- 
wavelength limit 

p ca - ia ( -~ J (4ra 2 Uo)e ik( - r - c ^ (27. 19) 

which is identical with Eq. (27.4) for the simple source, if Q , the 
strength of the source, is set equal to (4wa 2 U )- 

Radiation from a Point Source on a Sphere. — To show graphically 
how the radiation changes from symmetrical to directional as the 
frequency of the radiated sound increases, we shall work out the 
details of the radiation problem for two cases. 

The first case is that of a point source at the point # = on the 
surface; i.e., the surface velocity of the sphere is zero except for a 
small circular area of radius A around & = 0. The definition of 
U(&) is 

77<W = i Uo (0 ^ * < A/a > 
K) \0 (A/a<#<C7r) 

and the coefficients U m are 

U m = (m + i)u f^ a P m (coad) sm&dfr — >$(m + i)u (A/a) 2 

J0 A-+0 

since P m (l) = 1. The intensity and power radiated can be obtained 
fromEqs. (27.18): 

• cos[3 w — S„ -f- -j7r(m — n)] 



322. THE RADIATION AND SCATTERING OF SOUND [VII. 27 

, / A 4 \ 4tt ^O 2m + 1 /27ra\ 

At very low frequencies, only the first- terms in these series are 
important, and the pressure wave again has the familiar form for the 
simple source 



V 



'--^(^) 



(7rA 2 Wo)e ifc(r-c0 



with a strength equal to the velocity w times (xA 2 ), the area of the 
radiating element. 

' Figure 66 gives curves ;f or the distribution in angle of the intensity 
radiated from a point source on a sphere for different ratios of wave- 




"0 90° 180° 

Angle from Spherical Axis ,9 
Fig. 66. — Distribution in angle of intensity radiated from. a point source set in the 
surface of a sphere of radius a, for different values of /x = (2tto/X). Curves also give 
mean-squared pressure at point (a, #) on surface of sphere due to incident plane wave 
traveling in negative x direction; 



VH.27] RADIATION FROM SPHERES 323 

length to sphere circumference. We see again the gradual change 
from radiation in all directions to a sharply directional pattern as the 
frequency is increased. These curves are of particular interest 
because of their dual role, as required by the principle of reciprocity, 
which will be discussed in Sec. 29. As computed, the curves give 
intensity or pressure amplitude squared at a point (r,#) , a considerable 
distance r from the sphere (when kr » 1, \p\ = \/2pcT) at an angle 
# with respect to the line from the center of the sphere through the 
radiating element of area at point (a,0). But the principle of reci- 
procity says that the pressure at a point Q due to a unit simple source 
at point P is equal to the pressure at point P due to a unit source at 
point Q. Consequently, the curves of Fig. 66 also represent the 
square of the pressure amplitude at a point (a,0) on the surface of 
the sphere due to a point source of strength (7rA 2 w ) at the point 
(r,#). Therefore the curves are useful as an indication of the direc- 
tional properties of the ear plus head, or of a microphone in a roughly 
spherical housing. 

Radiation from a Pistion Set in a Sphere. — The other example to be 
worked out is that of a piston of radius a sin (# ) set in the side of a 
rigid sphere. As long as # is not too large, this corresponds fairly 
closely to the following distribution of velocity on the surface of the 
sphere : 

The general formulas in Eq. (27.18) can be used, with 
U m = (m -f- i)w f P m (x) dx = %u [P m -i (cos#o) — P m +i (costfo)] 

Jcos&o 

where, for the case m = 0, we consider P-i(x) = 1. 

These expressions for U m can be substituted in Eq. (27.18) to 
give series for intensity and power. Curves for T as function of # 
are given in Fig. 67. When the wavelength is long compared with 
(2xa), the pressure and intensity are those for a simple source of 
strength 47ra 2 Wo sin 2 (t? /2). 

The radiation impedance for the piston set in a sphere can be 
computed by integrating the expression for the pressure at r = a over 
the surface of the piston. Alter quite a little algebraic juggling, we. 
find 



324 



THE RADIATION AND SCATTERING OF SOUND [VII.27 



Z p — E p — iXp = pdira 2 sin 2 f -~ J (0 P — ix P ) 
d = i^ [Pm-i(cost?o) -P TO+ i(costV)] 2 



X P 



= *2 



TO=0 



M 2 (2m + l)Dl 

[P m -l(C0S ff ) - P OT+ i(cos flp)] 2 
(2m + l)D m 



(27.20) 



• [jm(ir) sin (5 m ) — n m (n) cos (5 m )] 

where fi = (2tt va/c) = (%ca/\). 

The quantities 8 P and Xp are the average acoustic resistance and 
reactance ratios at the diaphragm. These are plotted, in Fig. 68, as 




2 ju 4 6 

Fig. 67. — Distribution in angle of intensity and total power radiated from a piston set 

in a sphere. 

functions of the ratio of equivalent diaphragm circumference to 
wavelength, (2ira p /\) = {wa p /c) [a p = 2a sin (#p/2)], for several differ- 
ent values of #o = 2 sin -1 (a p /2a). We notice that the resistive terms 
all are small at low frequencies, increasing as o 2 ; rising to a value of 
approximately unity at wavelengths a little smaller than one- third 



VH.27] 



RADIATION FROM SPHERES 



325 



times the equivalent diaphragm circumference; and then, for very 
high frequencies, approaching the usual limit of 1 (z = pc). The 
reactance is always positive, representing a mass load. It first 
increases linearly with frequency, as the reactance due to a constant 




2 4 6 

Ratio Piston Circumference to. Wavelength, (Zircip/A) 

Fia. 68. — Values of radiation, resistance and reactance ratios (R r , X r /Tra p 2 pc), 
where a p = 2a sin (#<,/2), as function of (27ra p /X) = (wa p /c) for different values of t? , 
for radiation of sound from a piston set in a sphere of radius, a. 

mass would, but then it reaches a maximum at X~ 2xa, and for 
higher frequencies it diminishes more or less rapidly. The value of 
the initial slope is responsible for the factor in the end correction 
given in Eq. (23.1) and on page 247. 

These curves indicate the sort of radiation load one would expect 
on the diaphragm of a dynamic loud-speaker, set in a spherical case. 



326 THE RADIATION AND SCATTERING OF SOUND [VII.28 

The diaphragm of tho dynamic speaker is not usually the surface 
of a sphere but often has the shape cf an inverted cone in order to 
increase its mechanical strength. The radiation from such a cone 
would naturally differ from that from a section of a sphere; but it 
turns out that the average radiation impedance on a piston is approxi- 
mately the same, no matter what its shape, as long as its circum- 
ference is not changed, and as long as the volume of the mounting 
case is not changed. Therefore the radiation reaction on the dia- 
phragm of a dynamic speaker of outer circumference 2ira p , set in a 
cabinet of volume V P = 4x« 3 /3, is approximately given by {tca\pc) 
times the curves of Fig. 68 for # = 2 sin _1 [apOr /6 V ',)*], with the fre- 
quency scale equal to (2Trva p /c). 

Not only is the radiation impedance for a diaphragm, set in a 
cabinet of roughly spherical shape, given by the curves of Fig. 68; 
the average specific acoustic impedance at the open end of a tube or 
horn, of mouth circumference (2nra f ), is approximately equal to (pc) 
times the curves of Fig. 68 for # ^ 2 sin -1 (Tra p /2a f ) . This is the 
impedance referred to in Sees. 23 and 24. 

A plane piston, or an open end of a tube, of radius a p , set in an 
infinite plane rigid wall, corresponds to the case of a — » °o*, &o — > 0. 
Fairly simple expressions for the distribution in angle of the radiation 
and for the average specific acoustic impedance at the opening, can 
be obtained for this case by the use of Eq. (27.2). This will be dis- 
cussed in the next section. 

28. RADIATION FROM A PISTON IN A PLANE WALL 

In this section we shall consider a useful example of the way we 
can build up the radiation from an extended source by considering it 
to be an assemblage of simple sources. This method has been used 
for cylindrical waves to obtain Eqs. (26.10); in the present discussion 
we shall build up the radiation out of spherical waves. 

Suppose that a circular flat-topped piston of radius a is set flush 
in an infinite plane wall and vibrates with a velocity u e- 2iri,, \ radiating 
sound out into the space in front of the wall. It does not matter, for 
the purposes of our analysis, whether the piston is an actual one or is 
simply the vibrating layer of air at the open end of a tube or horn; 
the effect on the air outside is the same. Actually, the air in the 
open end does not all vibrate with the same velocity, as the top of a 
piston does, but the velocity is nearly uniform in many cases, and the 
results of the following analysis will be nearly correct when used for 



VH.28] RADIATION FROM A PISTON IN A PLANE WALL 



327 



the open ends of pipes. Later in this section we shall indicate how 
the approximation can be improved. 

Calculation of the Pressure Wave.— The method that we shall 
choose to solve our problem is to consider each element of area of the 
piston as a simple source of sound and to add all the waves from all 
the elements together to obtain the resulting wave. The strength of 
the simple source corresponding to the element dS is Q = 2u dS; the 
factor 2 is used because the amount u dS can radiate only into the 
space to the right of the wall. The effect of the wall can be replaced 
by an image elementary source u dS radiating to the left of the plane 
of the wall. The actual element and its image together form a simple 
source of strength 2w dS. 

The radiation pressure at the point P in Fig. 69, due to the element 
dS, is obtained from Eq. (27.3) : 



dp = 



. fpvuo dS 



■ty-JT^)* 



( h -ct )} k = ( 2 *Vc) 



The particle velocity due to this wavelet is entirely radial, so that 
the velocity along the surface of the 
wall is everywhere parallel to the 
surface, as it must be. 

The quantity h is the distance 
from the element dS to the point P. 
If the point P is far from the piston, 
so that a is much smaller than r, 
then (l/h) is practically equal to 
(1/V), and h is approximately equal to 
r — y sin & cos yj/, where y is the dis- 
tance of dS from the piston center, # 
is the angle between r and the per- 
pendicular from the center cf the 
piston, and \p is the angle between the 
plane defined by this perpendicular and r and the plane defined by the 
perpendicular and y. Therefore, when r is much larger than a, the 
expression for the pressure wave at P due to dS is, approximately, 




Fig. 69. — Radiation from a piston 
set in a plane wall. The wave at P 
is the resultant of all the wavelets 
radiated from all the elements of area 
of the piston. 



dp ~ 



-<(*=*) 



Qik{r— cV)q— iky&ir>-dcos$y £y ^r 



where dS = y dy d\p. 

The total pressure at P due to all the simple sources making up 
the piston is the integral 



328 THE RADIATION AND SCATTERING OF SOUND [VII.28 

p ~ _ ih]^l J e * ( r-c«) J ydy\ e- 11 *™*™* d$ 

= -%riv(^Je ik(r -^ f J (kysm&)ydy 

= -™ P u a 2 -^[ ^^ J < (28.1) 

using Eqs. (19.4) and (19.5). The radial velocity at this large dis- 
tance is (p/pc). 

Distribution of Intensity. — Using the expression developed above 
for p, the radiated intensity at large distances turns out to be 

• 2Jx(MBintf) -j' (2jrtl/x) (28 . 2) 



T = Ipculu 2 



Values of [2J x {x)/x~\ can be obtained from Table V or VIII. It is 
unity when x is zero, remains nearly unity until x is about (x/2), goes 
to zero at x about 1.27T, falls to about —0.13 at a; about l.lir, goes to 
zero again at x about 2.27T, and so on, having a sequence of maxima 
and minima which diminish in size. 

When the wavelength X of the sound radiated is longer than the 
circumference 2wa of the piston, then the value of (n sin &) is less than 
(t/2) even for # = 90 deg (i.e., even along the wall), and the term in 
brackets is practically unity for all values of #. Therefore, long 
wavelength sound spreads out uniformly in all directions from the 
piston, with an intensity four times that due to a simple source of 
strength ira 2 u . If the wall were not present and the "piston" were 
the open end of a pipe, this end would act like a simple source of 
strength ira 2 u for long wavelengths; so that the wall, or baffle plate, 
produces a fourfold increase in intensity. The sound reflected from 
the baffle reinforces the sound radiated outward, thereby doubling 
the amplitude of the wave and thus quadrupling the intensity, which 
depends on the square of the amplitude. Of course, to have the 
baffle give this considerable increase in amplitude, it must be con- 
siderably larger than the wavelength of the sound radiated, so that 
it will act as though it were infinite in extent. 

If X is smaller than 2xa, then the reflected sound still reinforces 
the sound radiated straight ahead, and the intensity has its maximum 
value pv 2 (ira 2 u ) 2 /2cr 2 at # = 0. At points off the axis, however, 
the reflected sound interferes with that radiated directly, and T 
diminishes in value as & increases, falling to zero when sin # is about 
equal to 0.6(X/a), then rises to a secondary maximum (where there 



VH.28] RADIATION FROM A PISTON IN A PLANE WALL 



329 



is a little reinforcement) of about 0.02 times the intensity for # = 0, 
then goes again to zero, and so on. Therefore, high-frequency sound 
is chiefly sent out perpendicular to the wall, with little spread of the 
beam. Most of the intensity is inside a cone whose axis is along the 
axis of the piston and whose angle is about sin -1 (X/2a). This main 
beam is surrounded by diffraction rings, secondary intensity maxima, 
whose magnitude diminishes rapidly as we go from one maximum to 
the next away from the main beam. 

Values of the intensity as a function of # are given in polar diagrams 
in Fig. 70, for different values of ju = (2tt va/c). The increased direc- 
tionality with increase in frequency is apparent. It is this direc- 



\-5a 

^ 



A. -2a 





Fig. 70. — Polar diagrams of distribution in angle of radiated intensity from a piston 
set in a plane wall, for different ratios between the wave length X and the radius of the 
piston a. In the curve for X = o, the small loops at the side are diffraction rings. 
Compare this with Fig. 67. 



tionality that makes it difficult to use a single loud-speaker in outdoor 
public-address systems: although the low-frequency sound is spread 
out in all directions, only the people standing directly in front of the 
loud-speaker will hear the high-frequency sound. In rooms of moder- 
ate dimensions the sound is scattered so much by the walls that the 
directionality does not matter particularly (unless it is very marked), 
and one loud-speaker is usually adequate. 

Effect of Piston Flexure on Directionality. — The equations derived 
on the last few pages have been for a plane rigid piston, which moves 
as a unit, every part of the surface having the same velocity. It is of 
interest to see what effect variations in motion from point to point 
over the piston surface will have on the sound radiation. This will 
be particularly useful when the "piston" in the plane wall turns out 
to be the air in the open end of a tube or horn; for it is then not a very 



330 THE RADIATION AND SCATTERING OF SOUND [VTI.28 

good approximation to assume that the velocity is uniform over the 
open end. 

In a previous section we discussed the transmission of various 
modes down a tube of circular cross section. There we used (for 
the case of rigid walls) the following set of characteristic functions: 

J a{irao n y / a) 



*n(y) 



Jo(iraon) 



^cmn{y,4>) _ COS / ,\ \/ , 2ira mn J m (Tra mn y/a) 



Vsmniyrf) Sin J m (Ta mn ) VWmn) 2 ~ W 2 

dJ m (ra mn ) = ^ Eqg (26 12) and (26 15)] 

f * dty J ^f 2 y dy = ira 2 

to describe the distribution of the waves across the tube. These 
functions will be useful for expressing the motion of any sort of piston, 
with any sort of distribution of velocity u (y,i/)e~ iat over its face: 

00 00 

U0\y } Y) = ?\ U n¥n ~T~ ^, \U cmn™cmn \ U smnXsmn) 
n = m,n — \ 

U " = J ^\ W \ My,t)*n(y)y dy > ( 28 - 3 ) 

ira 2 Jo Jo [ 

Ucmn hi r^ rMy,+)Z cmn (y,<t>)ydy 

™ Jo Jo 



U sm „ -wa 2 Jo Jo ' ^ 



smn 



For instance, we can set this series into Eq. (28.1) and utilize the 
properties of the Bessel functions given in Eqs. (19.4) and (19.5) to 
obtain a series for the radiated pressure wave at considerable distances 
(many wavelengths) from the piston, at the point (r,&,<p) in spherical 
coordinates centered at the center of the piston: 

p ik(r—ct) I <r-> 

p ~ -irvpa* —^- \ >► ^«*».W (28.4) 



'n = 



00 . 

+ ^ i m [U cmn cos (nap) + U smn sin (imp)] 2 **<n — ^(^ I 

S 2 — (Ta m n) 2 J 



m,n= 1 

, 2sJi(s) . , A . _ fJm-lis) - J m+1 

where s = ka sin # = (2ira/\) sin #. 

The term for Uo gives the radiation due to the average motion of 
the piston and turns out, of course, to be equal to the final formula 



VII.28] RADIATION FROM A PISTON IN A PLANE WALL 



331 



of Eq. (28.1), since a o = 0. The other terms give the modification 
of the angular distribution due to nonuniform distribution of velocity 
over the surface of the piston. The more important terms are those 
for m = 0, for the velocity distribution is usually radially symmetric 
(independent of \f/). Curves for some of these are shown in Fig. 71, 
as functions of s. We notice that the functions <£„ for n > add 
very little to the central part of the main beam (# ~ 0) but tend to 
modify the shape of the diffraction bands at the edge of the main 



na 


\ 




















0.6 


\ 






















V$o 


















02 


- 






A^ 






A 

i ^ 






,*6 


n 


- 




















-02 






















-04 




V' 






*h- 








$5-'" 




C 


t 


> L 


\ c 


> i 


s i 


3 1 


2 1' 


\ 1 


3 1 


3 20 



Angle Parameter s = (2ira/Msin0 

Fig. 71. — Angle distribution factor <£>„ for radiation from a piston with velocity 
distribution proportional to the characteristic function Jo(.irao n r/a). <I>o gives dis- 
tribution for uniform velocity (rigid piston). 



beam. The zeros of each function $ n (s) coincide with those of the 
others, except that each function has one zero missing, the nth one; 
$o is not zero at s = 0, $i is not zero at s = 7ra i, and so on. All the 
radially symmetric waves are in phase, unless the U n 's are not in 
phase. The intensity at {r,&,<p) is, of course, (l/2pc)|p| 2 , as long as 
r » X. 

By juggling with the various $„'s it is possible to adjust the shape 
of the radiated beam to any desired distribution (within certain 
limits) and then to work back from the corresponding U n 's to deter- 
mine the distribution of velocity of the piston which will give this 



332 THE RADIATION AND SCATTERING OF SOUND [VII.28 

beam. For instance, the first "side lobe" in the diffraction pattern 
may be suppressed, or the first three or so can be drastically reduced 
in value (which is important if one needs a sharply defined beam). 
The limits of possible variation of the beam distribution are strin- 
gent, however, and can soon be learned by attempting to "tailor 
make" a beam. The most important restriction is that the main 
lobe (0 < sin# < a iX/2a) can never be made narrower by juggling the 
velocity distribution of the piston. The beam can be made broader 
by such juggling, but it can be made narrower (for a given wave- 
length) only by increasing the size of the piston. The minimum 
angular size of the main lobe is therefore given by the inequality 
(valid for X < a/2) 

Angular width of main lobe = (& for first zero) 
is larger than (35X/a) = (1, 200,000/ va) 

where the angle is in degrees, X and a are in centimeters and v is in 
cycles per second. 

Radiation Impedance, Rigid Piston. — Often it is more important 
to compute the reaction of the air back on the piston than it is to com- 
pute the distribution in angle of the radiated sound. The radiation 
impedance of the piston is necessary in calculating the mechanical 
and electrical properties of a loud-speaker, for instance. We have 
already given curves in Fig. 68 for the impedance ratios for a piston 
of effective radius a p = la sin (#o/2) set in a sphere of radius a. The 
limiting case of #o -* corresponds to a plane rigid piston of radius 
o p set in a sphere of infinite radius {i.e., in a plane wall). This case 
can be computed in closed form directly from the analysis developed 
in this section, as will be shown. 

To find the reaction of the air on the piston we must find the 
pressure at the point (r,<t>) on the piston, due to the motion of the 
element of area of the piston at (y,i). From Eq. (28.1) we see that 
this is 

dp = -IPV fy J e mh-ct) y d y fy 

where h = -\A 2 + y 2 — 2ry cos ty — 4). The total pressure at (r,<£) 



is 



p = —ipvu<>e- iu>t 



J>"J>*(f) (28 - 5) 



VII.28] RADIATION FROM A PISTON IN A PLANE WALL 333 

where we are now using a, instead of a p , to denote the radius of the 
piston. To obtain the total force on the piston this must be inte- 
grated over the whole surface of the piston: 



F = —ipvuoe-™' 



i % *r rf *r^f e Kf) 



(28.6) 



The calculation of this integral involves some special mathe- 
matical tricks which will not be elaborated on here. The results give 
the total radiation impedance of a rigid piston of radius a set in a 
plane wall: 

■p 
Z* = — =uTt = Ta 2 pc(0o — ixo) = tci 2 (Ro — iXo) 

d = 1 - (-1/1(10); w = 2ka = (4ra/\) = (2<oa/c) 

Xx/2 
sin (to cos a) sin 2 a da 

U(ka) 2 (ka-*0) f(8kd/ZTr) (ka -> 0) 

"* (1 (ka -* oo ) ; Xo ~~* ((2/ik) (ka ~> oo) 

The properties of. these functions have already been discussed in the 
previous chapter, and their limiting values have been given in Eq. 
(23.14). These are the functions we have used to compute approxi- 
mate values of the average impedance of the open end of a tube. 
Tables of values for different values of w are given at the back of the 
book. Curves of d and xo as 'functions of (w/2) = (27ra/X) are given 
in Fig. 68, labeled # = 0. 

Comparison of the curves in Fig. 68, and of the curves of Figs. 67 
and 70, show that a change from plane to spherical baffle makes a 
large change in distribution in angle of the radiated intensity, but 
makes little change in the average radiation impedance load on the 
piston. Sound is sent out in different directions, but its reaction 
back on the piston depends more on the ratio of piston circumference 
%ca (2wa p in Fig. 68) to wavelength X than it does on baffle shape. 

Distribution of Pressure over the Piston. — In many cases (the cases 
discussed in Chap. VI, for instance) it is sufficient to know the average 
reaction of the air on the rigid piston, which is given by the functions 
do and xo. To go further into the details of the problem, or to study 
the reaction of the air on a nonrigid piston, we must know the depend- 
ence on r of the reaction pressure p given in Eq. (28.5). The integral 
of Eq. (28.5) is a complicated function of r, which can most easily be 
expanded in a series of characteristic functions appropriate to a cir- 



334 THE RADIATION AND SCATTERING OF SOUND [VH.28 

cular piston of radius a. These functions have already been given in 
Eq. (26.17). In terms of them we can set 



n = L 



Jo(ir<XOn) . 



where the coefficients in the expansion for the case of the rigid piston 
are 

= -i I d$ I # \ r , ' r dr -r-ydy (28.7) 

Jo Jo Jo xXoVoCiraon) Jo fr 

The coefficient (do — ixo) has already been discussed. It gives the 
average acoustic impedance ratio over the surface of the piston. 

The coefficients f„ = (0„ — *x?0 for n > represent corrections 
to the distribution of radiation load over the rigid piston. Their 
average effect is zero, since the average value of each of the higher 
characteristic functions 

, Jo (iraonr/a) , m 

Jo{Trao n ) 

over the surface of the piston is zero. Expressions for these imped- 
ances can be given in terms of simple integrals: 

1 J\(u)u du 

On = 



% = 2 M P - 

Jo [u 



2 — (iraon) 2 ] V/* 2 - u 2 

-( M 4 /37r 2 a 2 J 0»->O) 

(7a n/2/i 2 ) (m->°°) 



J\(u)u du 



{, 



(28.8) 



Xn = 2M J„ '[u 2 - (Traon) 2 ] V« 2 " M 2 

_^(-9nH 0*-*0) 

\(2A/i) (m-*«) 

M = (2ira/X) = fca, 

0i = 0.092, 02 = 0.0356, g 3 = 0.0194, £ 4 = 0.0116 • • • 

Some of these functions are plotted in Fig. 72, and values are given 
in Table XII at the back of the book. 

We see that for very small and for very large values of n = ka = 
(<aa/c) both d n and x« (f° r n > 0) vanish, being important only for 
values of ka near ira 0n (i.e., for frequencies near ca 0n /2a). Therefore 



VH.28] RADIATION FROM A PISTON IN A PLANE WALL 335 

at very high and very low frequencies the reaction pressure of the air 
on the front of a uniformly vibrating piston is uniform over the face 
of the piston; only for frequencies such that the wavelength is the 
same general size as the piston dimensions is the pressure markedly 
nonuniform over the piston. 




10 



4 6 8, 10 "0 2 4 6 

Frequency Parameter p.=ka=.(2iraA) 

Fig. 72. — Coupling resistance and reactance ratios between radiation and circular 
piston of radius a in infinite plane wall. Ratio f „ = 6 n - i'x» gives coupling between 
zero (uniform) velocity mode and the nth pressure mode. 

Nonuniform Motion of the Piston. — If the piston is not rigid, the 
analysis of the radiation reaction must be carried still further. We 
can express the velocity of the piston surface in terms of a series of 
characteristic functions of the sort we have been using, ^„(r) = 

Jo(Trao n r/a)/Jo(jra n) ' 

u (r) = ^UnVnir); u n = (~\ I u(r)* n (r)r dr 

n = 

as in Eq. (26.17). By analogy with Eq. (28.7) we can expand the 
reaction pressure of the air at a point on the piston a distance r from 
its center: 

oo oo 

p(r) = pC ^ ^ Untnm^m(r) 

71 = m = 



336 THE RADIATION AND SCATTERING OF SOUND [VII.28 

where the impedance ratio f„ ro = (d nm - i Xnm ) = f«« are obvious 
generalizations of the impedance ratios £« (now written £ n) given in 
Eq. (28.7): 

Un = ^ J d * Jo ** Jo * m(r> dT Jo * n{y) T V dy 

The average pressure over the piston is the coefficient of ^ (r) = 1 
in the series 

Pav = pC 2) U n £ n0 , (f n0 = fn) 
» = 

At low frequencies the quantity fro == f o is larger than any of the 
other f's. Therefore at frequencies small compared with (c/2tto) the 
pressure on the piston is approximately uniform, nearly equal to 
pcfo = pc(6o — ixo) times the average velocity u of the piston. This 
approximation was used in Chap. VI for the calculation of radiation 
out of the open end of a tube. 

For frequencies high compared with (c/2wa) the coefficients f mm are 
approximately equal to unity and the terms £ nm (n ^ m) are quite 
small. Therefore at high frequencies the pressure at r is 

00 

P(r) ~ pc ^ MmSmmtmW ~ pcu{r) (since f mm ~ 1) 

l m = 

which is pc times the piston local velocity at r. This is the result 
mentioned on pages 198 and 311. 

Radiation out of a Circular Tube. — We can now indicate how the 
radiation of sound out of the open end of a circular tube (fitted with 
a wide flange) can be calculated with greater accuracy than was pos- 
sible in Chap. VI. As an example we take the case where the tube 
radius a is smaller than (ca i/2v), so that the velocities (c/T 0n ) [see 
Eq. (26.17)] of all the higher normal modes are imaginary and only 
the plane wave (n = 0) mode is transmitted without attenuation. 
Therefore by the time the wave has gone from the piston to the open 
end, these higher modes have become small, and to the first approxima- 
tion only the plane wave mode S^o survives. 

In this case, to the first approximation, the velocity amplitude of 
the air at the open end is uniform and in a direction parallel to the 
tube axis (perpendicular to the plane of the open end). Suppose 
that this velocity has amplitude u . According to Eq. (28.7) the 



VH.28] RADIATION FROM A PISTON IN A PLANE WALL 337 

pressure in the plane of the open end, a distance r from the tube 
axis, due to the radiation of the plane wave mode into the open, is 

« 



n = 



Vn(r) = JoiTaonr/^/Joiiraon) 

The average pressure due to this radiation is of course pcu (6 — ixo), 
which we have used in the previous chapter for calculating the acoustic 
impedance at the other end of the tube. But we now see that there 
are other, smaller, nonuniform terms in the reaction pressure, which 
cause local irregularities in the sound field near the open end. These 
corrections are very small for low and for high frequencies, so that it 
is usually not worth taking them into account. They are discussed 
here to indicate how these and similar details may be calculated if 
need be. 

The additional terms in the pressure modify the velocity distri- 
bution over the open end, by sending back down the tube waves 
corresponding to the higher modes V n . Since, by our assumption, 
these waves attenuate rapidly, they will not extend back into the 
tube any appreciable distance, but they will cause a modification near 
the open end. We have seen on page 309 that the nth mode of trans- 
mission in the tube has a characteristic impedance 

pc/ron = -iaiap/iraonTn), where T n = y/l - (v/p 0n ) 2 , v^ = (ca 0n /2a) 

Therefore, corresponding to the higher terms in the series for the 
pressure, we have a correction to the velocity distribution at the open 
end. Insteafl of being u it is, to the first approximation in the small 
quantities 0„ and x«, 



1*3=0 ~ Uoe-™ 1 



~ 2 fe) ^k- + **-)*.(r) 



where the sign in front of the summation symbol is minus because 
these higher modes are being sent back into the tube (negative z-axis). 
In turn, each correction term for the velocity gives rise to a second- 
order correction to the pressure. The average value of this correction 



n = l ^ ' 



338 THE RADIATION AND SCATTERING OF SOUND [VII.28 

and therefore the average acoustic impedance of the open end for 
the plane wave mode is, for the case we are considering, 



pc 



{e - ixo + 2 (S) V 1 " {wj [26nXn + i{el " X " )] J 



to the second order of approximation in the small quantities n , Xn- 
Usually the summation can be neglected in comparison with O and xo- 

We can also utilize Eq. (28.4), together with the equation for 
w z =o given above, to obtain a somewhat more accurate expression for 
the distribution in angle of the wave radiated out of an opening in a 
plane wall. 

Transmission Coefficient for a Dynamic Speaker. — We have shown 
in the preceding pages that a dynamic speaker of radius a set in a 
plane wall, moving with velocity Uoe~ io>t , radiates the total energy 

fr-a 2 pcU?,d = bra 2 pcUf } ( 2toa / c ) I 

into the open. The quantity do is therefore the transmission coeffi- 
cient for a piston in a baffle, and its curve should be compared with 
those in Fig. 64 for various horns. As with the earlier cases, the 
coefficient is small at low frequencies, which means that the velocity 
amplitude must be increased at the low frequencies to obtain uniform 
response. This can be done by making the piston mass-controlled 
(which is not difficult) for then the velocity amplitude is inversely 
proportional to v over the frequency range of mass control. 

Since do is also the transmission coefficient for the open end of a 
horn, Fig. 72 shows that the open end of a horn must have its circum- 
ference larger than the longest wavelength that it is wished to trans- 
mit; otherwise, the open end will reflect an appreciable part of the 
sound back into the horn, causing strong resonance. 

Most dynamic speakers are designed so that over the useful fre- 
quency range the inertial reactance of the piston itself is the largest 
part of the total mechanical impedance (i.e., the speaker is mass con- 
trolled). Therefore, above some minimum frequency the velocity 
amplitude of the piston driven by a force of amplitude F is u — 
(F /2t vrrip). Above this minimum frequency the energy radiated 
by the speaker is 



n ~ — 

cm 



\n*Q>)[i-¥**\ (28-9) 



VH.28] RADIATION FROM A PISTON IN A PLANE WALL 



339 



The values of f — 5 ) * <A(w) are plotted in Fig. 73 as a function 

of w = (47ra/X). The curve shows that the response of this type of 
loud-speaker is good for low-frequency sound (as long as the fre- 
quency is above the lower limit of the mass control) but that as soon 
as the wavelength becomes smaller than the circumference of the 
speaker the power radiated diminishes. 

If the speaker is resistance controlled and is driven by a force 
whose amplitude is independent of v, the power radiated will be pro- 
portional to (do), shown in Fig. 72. In this case the higher frequencies 
in the useful range will be radiated best. 



1.0 



8R0 
pew 2 



0.5 



flower limit of mass-con trot 



2 4 6 8 

W 
Fig. 73. — Power radiated by a "dynamic" speaker which is mass controlled and which 
is driven by a force with amplitude independent of the frequency. 

Design Problems for Dynamic Speakers. — To review some of the 
material we have developed in this section, and to show how it can 
be applied in the solution of practical problems, we shall discuss the 
design of a dynamic speaker. We cannot go into very many of the 
tricks of design used to improve the behavior of present-day dynamic 
speakers; for it is not our aim to discuss engineering practice. 1 We 
shall discuss some of the major problems encountered, however, to 
bring out the methods of analyzing the problems and the way in 
which the analytic techniques we have developed can help us in the 
design of acoustical equipment. 

We of course wish the dynamic speaker to reproduce passably 
well sound of frequency between 80 and 4,000 cps, a range of about six 

1 For a discussion of the engineering aspects of this problem, see Olsen and 
Massa, "Applied Acoustics," The Blakiston Company, Philadelphia, 1934, for 
instance; or M'Lachlan, "Loud Speaker Design," Oxford University Press, 
Oxford, England, 1934. 



340 THE RADIATION AND SCATTERING OF SOUND [VH.28 

octaves; and we wish it to produce 10 6 ergs of sound energy per second 
if need be. (A person speaking with an average conversational tone 
produces about 100 ergs per sec and when shouting produces about 
10,000 ergs per sec.) We shall assume that we can obtain a magnetic 
field as large as 10,000 gauss at the driving coil. 

We must first decide on the size of the loud-speaker cone, or piston. 
Since the piston is to be mass controlled (see the comments on page 
338), we shall expect the output of the speaker to be more or less 
constant as long as the radiation resistance is increasing with the 
square of the frequency and to fall off when the radiation resistance 
becomes constant. Figure 73 shows that this output is greater than 
half its maximum value as long as w is less than 4 or as long as (va) is 
less than 10,000. For this limit to be at v = 4,000 we should have 
a = 2.5, or about 1 in. However, a piston of 1 in. radius would have 
to vibrate with an amplitude of more than a centimeter at v = 200 to 
radiate 10 6 ergs per sec from its small surface, so that we must sacrifice 
some of the intensity at the higher frequencies to obtain a piston 
large enough to radiate efficiently. We shall choose a to be 10 cm; in 
this case the intensity at c = 4,000 is one-sixteenth of the intensity 
at v = 1,000, a considerable diminution but perhaps not too great. 

The next problem is the driving coil. Increasing the number of 
turns on the coil increases the electromagnetic coupling constant G, 
but it also increases the mass of* the moving parts and the resistance 
of the coil. We choose the coil to be 1 cm in radius and to have 160 
turns of wire of such a size as to make the resistance of the coil 5 ohms. 
In such a case it may be possible to make the mass m p of the coil- 
piston system as small as 30 g. If the magnetic field on the coil is 
10,000 gauss, the electromagnetic coupling constants are D — 10 6 , 
G = 10 5 . It may be possible to keep the self-inductance of the coil as 
low as 0.2 mh. 

The next question is what to do about the space in back of the 
piston. We can, of course, let the back radiate into another room, 
but this is not usually done. If we enclose the back of the speaker, 
the enclosure will form a resonator that will add stiffness to the piston. 
If the inside of the enclosure is made of hard material, standing waves 
can be set up, and the enclosure can have a large number of resonance 
frequencies; so we shall line the interior with absorbing material to 
destroy most of the resonance due to standing waves. This will not 
appreciably alter the resonance of the Helmholtz type (see page 235) 
however, so we must make the volume of the enclosure large enough 
so that this resonance frequency is below the range that we wish to 



VTI.28] RADIATION FROM A PISTON IN A PLANE WALL 341 

reproduce. The stiffness constant of the piston due to an enclosure 
of volume V is given in Eq. (23.3) : K p = (tci 2 ) 2 (pc 2 /V). If we choose 
V to be 20,000 cc, th e natural frequency of the resonator-piston 
system (l/27r) y/K p /m p will be 76 cps, which is just below the lower 
limit that we have set. This resonance will increase somewhat the 
response at lower frequencies. It will be assumed that the frictional 
forces on the piston, other than radiation resistance, are negligible 
(although in actual pistons the frictional resistance is usually larger 
than the radiation resistance). 

The constants assumed for the loud-speaker are therefore 

m p = 30, K p = 6.9 X 10 6 , a = 10, V = 2 X 10 4 \ 
D = 10 6 , G = 10 6 , Re = 5, Lc = 2 X 10~ 4 / 

/a \ / \ ( (28.10) 

™v = i.3x 10 <, «> = (¥) = (m) I 

These are not recommended values of the constants, nor are all of them 
typical values. They have been chosen to show, as clearly as possible, 
how each property of the loud-speaker enters into the radiation process. 
Behavior of the Loud-speaker. — To compute the behavior of the 
speaker we first obtain its mechanical impedance: 

\Z m \ = s/Rl + XI 

R m = Tra^o = 1.3 X lO^Tl - 2^1 



-*^a_, + 1rtlfXo > (28.11) 

11 y 106 
' A + 190 v + 1.3 X 10W (w) 



X m = -jr — ? + 2rvm p + ira 2 X 
Zttv 



where R , X , and M are given in Eq. (28.5). Curves of R m and Z m 
are given in Fig. 74. We notice that over most of the range of fre- 
quency (for v larger than 200) Z m is practically equal to the reactance 
due to the mass of the piston Z m ~ 190j>. This is, of course, what is 
meant by saying that the loud-speaker is mass controlled. The 
sudden drop in Z m on the low-frequency side is due to the resonance 
of the vessel enclosing the back of the piston. 
The electrical impedance of the coil is 

\Z,\ = VfiJ + XI Rb ~ 5 



342 



THE RADIATION AND SCATTERING OF SOUND 



[VII.28 



Curves of these three quantities are shown in Fig. 75. Since the 
resistance of the coil is fairly large, Z E stays fairly constant over most 
of the specified range of frequency. The motional reactance due to 
the piston just cancels that due to the impedance of the coil at about 
600 cps. If the coil resistance were lower, this would cause a pro- 
nounced resonance peak in the response. The impedance rises on 




10,000 



Fig. 74. — Total mechanical resistance and impedance of a "dynamic" loud-speaker 
as a function of the driving frequency. The constants of the loud-speaker are given 
inEqs. (28.10). 

the high-frequency side owing to the inductance of the coil and on 
the low-frequency side because of the mechanical resonance of the 
loud-speaker plus enclosure. This rise in electrical impedance due to 
mechanical resonance is more than enough to cancel out the corre- 
sponding peak in the piston response (see the discussion on page 38). 
We shall assume that an emf of \0e- Mvt volts is applied across the ; 
coil. In actuality, the voltage amplitude will also vary with fre- 
quency, but we must cease adding complications somewhere and so 



VH.28] RADIATION FROM A PISTON IN A PLANE WALL 



343 



shall assume that the voltage amplitude is constant. The current 
through the coil is then (10/|Z*|), the force on the piston is(10D/\Z*$, 
and the velocity amplitude of the piston is u p = (10D/\Z E \\Z m \): The 
power radiated is \u\R m . Curves of u p and of n, the power radiated, 
are given in Fig. 76. They show that II is reasonably constant over the 
range 100 < v < 1,000 and that u p is inversely proportional to v over 
most of the frequency range. Below the mechanical resonance at low 
frequencies the curves for u p and II drop off. The falling off of u p \ 
below the straight line of the (1/v) curve, for high frequencies, is due 
to the inductance of the coil. The drop in II for high frequencies' is 




5000 

Fig. 75. — Total electrical resistance, reactance, and impedance of the driving coil of the 
loud-speaker whose constants are given in Eqs. (28.10). 

due partly to the inductance but is also due to the fact that the radi- 
ation resistance is not rising any more, so that it cannot make up for 
the decrease of u p . In actual loud-speakers the drop is not so rapid, 
because at high frequencies only the central portion of the piston 
vibrates, and the effective mass is reduced. Over the useful range 
of frequencies the over-all efficiency of the loud-speaker is about 
1 per cent. 

At v = 100 the amplitude of motion of the piston (u p /2rv) is 
about 0.2 cm, so that the mechanism should be designed for an ampli- 
tude of about 0.4 Cm if it is to radiate 0.1 watt without rattling. 

The efficiency of the loud-speaker could be increased by decreasing 
the coil resistance, by increasing the magnetic field, by increasing the 



344 



THE RADIATION AND SCATTERING OF SOUND JVIL28 



number of turns on the coil, or by decreasing the mass of the moving 
system. If we decrease the coil resistance much, the electrical imped- 
ance will not be constant over the useful range, and the resonance 
between piston mass and coil inductance will become prominent. If 
we increase the number of turns on the coil, we increase the resistance 
and also the mass, which is not desirable. It would be quite difficult 
to increase the magnetic field to any extent, though if this could be 
done the efficiency could be increased without any concomitant ill 
effects. If we decrease the mass of the system much, it will no longer 
be purely mass controlled, and the response will not be so uniform. 




-io 5 n 



- 100 



1000 



10,000 



Fig. 76. 



-Velocity amplitude u p and power radiated II when the loud-speaker of Eqs. 
(28.10) is driven by an emf of 10 volts amplitude. 



The curves of Fig. 76 are for the loud-speaker out in the open with 
no obstructions near by. The effect of the resonance of a room on 
the response curves will be discussed in the next chapter. 

Transient Radiation from a Piston. — The general equation (27.1) 
for radiation from an elementary source can be used to compute the 
transient pressure wave from a piston in a plane wall. Suppose that 
the velocity of the piston is U(f) and its acceleration is A(t) = dU/dt. 
Study of Fig. 77 shows that a part of the pressure wave arriving at 
the point (r,#) at the time t is that due to the acceleration of the strip 
of length d = 2 \/a 2 — y 2 and width dy a time (l/c)(r — y sin#) 



VII.28] RADIATION FROM A PISTON IN A PLANE WALL 



345 



earlier (if r is much larger than a). The amplitude of this component 
of the pressure pulse is, from Eq. (27.1), 

dp ~^r a/o ' ~ y 2 A\t - -(r - ysm&)\dy 
The total pressure is obtained by integrating this over the surface 

.d1 




Fig. 77. — Radiat ion of a transient from a piston. Sound reaching point P came 
from strip d = 2 \/a 2 - V 2 a time (l/c)(r - 2/ sin #) earlier. Lower curve shows 
pressure fluctuation at P due to a velocity pulse U = d(t) of the piston. 

of the piston, from -a to +a. Change of integration variable and 
integration by parts yields the general formulas 



pc 
p = H 



2irr sin 2 # _ 
,2 Cf 



_ pc l 



J Vo 2 sin 2 ?? - (ct + r - ct) 2 A(r) dr 
(r + ct — ct) U{r) dr 



2irr sin 2 # J e Va 2 sin 2 ?? - (r + ct — ct) 2 



(28.12) 



where the limits of integration are e = t - (l/c)(r + a sin/?) and 
f = t - (l/c)(r - asintf). 

This is a very interesting formula, for it shows, perhaps more 
clearly than Eq. (28.1), the dependence of the pressure wave on the 



346 THE RADIATION AND SCATTERING OF SOUND [VII.29 

angle #. Directly ahead (# = 0) the pressure wave reproduces the 
piston acceleration exactly: 



»=te>H> ( *= o) 



The force on a microphone diaphragm directly ahead of the piston is 
therefore proportional to the piston acceleration; and if the diaphragm 
is mass controlled, its acceleration is proportional to the force, so 
that the accelerations, velocities, and displacements of piston and 
diaphragm are proportional. As & is increased, however, the integral 
covers a larger and larger interval of time t, so that more and more 
of the piston motion gets blurred together in the pressure wave 
arriving at P. 

A simple example of this is for the case when the piston suddenly 
moves outward a distance A. In this case the displacement of the 
piston is Au(t), where u{t) is the step function defined in Eq. (6.9); 
the velocity is A 8(0, proportional to the impulse function; and the 
acceleration A8'(t) is formally proportional to the derivative of the 
impulse function, a "pathological" function going first to plus infinity 
and then to minus infinity in an infinitesimal period of time. Its inte- 
gral properties are V ^ h'{r - a)j{j) dr = ~[(df(a)/da)u(t - a)]. The 
resulting pressure at (r,&) is 

!0 (ct < r — a sin#) 

pc2A r - ct , _ asiiit? <ct<r + asm #) 

27rrsm 2 # v 7 a 2 sm 2# _ ( r _ ct y 
(c/ > r '+ a sin #) 

This pulse, shown in Fig. 77, is a stretched out version of the 8' func- 
tion; the greater the angle &, the greater the stretch. Only directly 
ahead of the piston, at # = 0, is the pressure pulse as instantaneous 
as the piston pulse. 

29. THE SCATTERING OF SOUND 

f When a sound wave encounters an obstacle, some of the wave is 
deflected from its original course. It is usual to define the difference 
between the actual wave and the undisturbed wave, which would be 
present if the obstacle were not there, as the scattered wave. When a 
plane wave, for instance, strikes a body in its path, in addition to the 
undisturbed plane wave there is a scattered wave, spreading out from 
the obstacle in all directions, distorting and interfering with the plane 



VH.29] THE SCATTERING OF SOUND 347 

wave. If the obstacle is very large compared with the wavelength 
(as it usually is for light waves and very seldom is for sound), half of 
this scattered wave spreads out more or less uniformly in all directions 
from the scatterer, and the other half is concentrated behind the 
obstacle in such a manner as to interfere destructively with the 
unchanged plane wave behind the obstacle, creating a sharp-edged 
shadow there. This is the case of geometrical optics; in this case the 
half of the scattered wave spreading out uniformly is called the 
reflected wave, and the half responsible for the shadow is called the inter- 
fering wave. If the obstacle is very small compared with the wave- 
length (as it often is for sound waves), then all the scattered wave 
is sent out uniformly in all directions, and there is no sharp-edged 
shadow. In the intermediate cases, where the obstacle is about the 
same size as the wavelength, a variety of curious interference phe- 
nomena can occur. 

In the present chapter, since we are studying sound waves, we 
shall be interested in the second and third cases, where the wavelength 
is longer or at least the same' size as the obstacle. We shall not 
encounter or discuss sharply defined shadows. So much of the 
scattered wave will travel in a different direction from the plane wave 
that destructive interference will be unimportant, and we shall, be, 
able to separate all the scattered wave from the undisturbed plane 
wave. We shall be interested in the total amount of the wave that 
is scattered, in the distribution in angle of this wave, and in the effect 
of this scattered wave on the pressure at various points on the surface 
of the obstacle. 

Scattering from a Cylinder. — Let us first compute the scattering, 
by a cylinder of radius a, of a plane wave traveling in a direction per- 
pendicular to the cylinder's axis. If the plane wave has intensity 
To, the pressure wave, if the cylinder were not present, would be 

Vv = P e ik ^~^ = p„e*(«- «»*-««>, p = V2pcT^ k = — ; 

A 

where the direction of the plane wave has been taken along the positive 
z-axis. 

In Eq. (19.13) we expressed this plane wave in terms of cylindrical 
waves: 



Vv = Poe*< ro "*- rf > = P [J (kr) + 2 j? i™ cos (m0)J«(Jiy)]r w '» (29.1) 

m = \ 



348 THE RADIATION AND SCATTERING OF SOUND [VTL29 

The radial velocity corresponding to this wave is 



w«r — " 



(?j\ iiJ^kr) + 2 i m+1 [Jm+i(kr) - Jailer)] cos(mtf>)l 



, — 2irivt 



When the cylinder is present with its axis at r = 0, the wave 
cannot have the form given by the above series, for the cylinder dis- 
torts the wave. There is present, in addition to the plane wave, a 
scattered outgoing wave of such a size and shape as to make the radial 
velocity of the combination zero at r = a the surface of the cylinder. 
We shall choose the form of this outgoing wave to be the general series 

CO 

p, = 2^ cos (m4>)[J m (M + iN m (kr)]er™" 

m = 

u„ = (L\ iiAo[Ji(Jcr) + iNxikr)] + 1 ^ A ™ cos (m<j>)[J m +i(hr) 

- J m -i(kr) + iN m+1 (kr) - iN m -!(kr)]\ e- 2 ""' 

The combination J + iN has been chosen because it ensures that all 
the scattered wave is outgoing. 

Our first task is to find the values of the coefficients A which 
make the combination u pr + u sr equal zero at r = a. Equating 
u„ to — Upr at r = a term by term, we obtain 



A m = -e^Po^+^-^sinCTm); Po = V^pcjo ) 

Ji(fco) . _ Jm-x(ka) - J m +x(ka) > (29.2) 

tanTo = "" mM)'' tanTm ~ N m+1 (ka) - N m ^(ka) ) 

where eo = 1 and e m = 2 for all values of m larger than unity. These 
phase angles y m have already been defined in Eq. (26.6), in connection 
with the radiation of sound from a cylinder. Values of some of them 
are given in Table X at the back of the book. The behavior of these 
phase angles completely determines the behavior of the scattered wave. 
It is interesting to notice the close connection between the waves 
scattered by a cylinder and the waves radiated by the same cylinder 
when it is vibrating. The quantities needed to compute one are also 
needed to compute the other. 

The pressure and radial velocity of the scattered wave at large 
distances from the cylinder are 



vn.29] 



THE SCATTERING OF SOUND 



349 



4pcT a , , N .,/ « Pa 

H — — $ s (4>)e tk(r - c » ; u s ~ — 

TV pC 



^.(0) = —7= 2 Cm sin (Tm)^- 1 ' 7 " 1 cos (m^) 



TO=0 



The intensity of the scattered part, at the point (r,<f>)(kr^> 1), is, 
therefore, 



\U<f>)\'< 



(29.3) 



A=|ttoi- 



00 

|^«| 2 = j— ^j e m e„ sin7 TO sin7„ cos (y m — y n ) cos (m^) cos (n<f>) 

m,n = 

where e = 1, e m = 2 (m > 0). This intensity is plotted as a function 
of ^ on a polar plot in Fig. 78, for 
different values of p = (2rva/c) = 

(2iroA). 

It is interesting to notice the 
change in directionality of the 
scattered wave as the wavelength 
is changed. For very long wave- 
lengths (n small) but little is scat- 
tered, and this is scattered almost 
uniformly in all the backward 
directions. As the frequency is 
increased, the distribution in 
angle becomes more and more 
complicated, diffraction peaks 
appearing and moving forward, 
until for very short wavelengths 
(much shorter than those shown 
in Fig. 78) one-half of the scat- 
tered wave is concentrated 
straight forward (the interfering 
beam), and the other half is spread 
more or less uniformly over all the 
other directions, giving a polar 
plot which is a cardioid, inter- 
rupted by a sharp very high peak 




Fig. 78. — The scattering of sound 
waves from a rigid cylinder of radius a. 
Polar diagrams show the distribution in 
angle of the intensity of the scattered 
wave, and the lower graph shows the 
dependence of the total scattered intensity 
on fi = 2ira/X. 



in the forward direction, as will be shown in Eq. (29.4). 

For very long wavelengths only the two cylindrical waves corre- 



350 THE RADIATION AND SCATTERING OF SOUND [VTI.29 

sponding to m = and m = 1 are important in the scattered wave. 
As shown in Eq. (26.6), 



(^) 



To ^ — 7i ~ I ~~ 4~ J ^ ~ 72 ' —, V3 ' ' ' > wnen ^« — > 

The first-order approximation for the scattered intensity at long wave- 
lengths is, therefore, 



~ \ 8c 3 r / 



T (l - 2cos<f>) 2 (coa «c) 



Short Wavelength Limit. — For wavelengths very small compared 
with the circumference of the cylinder, the relatively simple approxi- 
mation of "geometrical acoustics" is valid, with the "scattered" 
wave dividing into two parts, the "reflected" and the "shadow- 
forming" waves. However, the process of demonstrating that the 
series of Eqs. (29.3) really does behave in a simple manner involves 
mathematical manipulation of considerable intricacy. 

In the first place the series of Eqs. (29.3), which is in a form useful 
for calculating scattering at longer wavelengths, is not particularly 
suitable for showing in detail the interference between part of the 
"scattered" wave and the primary wave to form the shadow. In 
optics, one differentiates between "Fraunhofer diffraction," where 
intensities are measured at distances so large that the angle subtended 
by the diffracting object is small compared with the ratio (X/2ira) = 
(1/ka), and "Fresnel diffraction," for distances large compared with 
the wavelength but not extremely large compared with {%ra). The 
Fresnel-diffraction formulas show the shadow with its related diffrac- 
tion bands, but at the great distances involved in the Fraunhofer- 
diffraction formulas, the shadow has become blurred out again. 
Series (29.3) is for distances corresponding to Fraunhofer diffraction, 
so that what can be demonstrated is the separation of the "scattered 
wave" into a "reflected wave" and a "shadow-forming wave" but 
not the details of the interference with the incident wave, which are 
characteristic of the Fresnel formulas. 

In the second place, the simplicity of the formulas for very short 
wavelengths only appears as an average; the scattered intensity varies 
rapidly with angle in a complicated sort of way, and only the average 
intensity per degree (or per minute) varies smoothly. This rapid 
fluctuation is seldom measured, however, for any small change of 
frequency or of position of the cylinder will blur it out, leaving only the 
average intensity. Consequently, our calculations should separate 



VII.29] THE SCATTERING OF SOUND 351 

the rapid fluctuations of intensity from the average behavior of the 
"reflected wave." 

When all the necessary manipulations are made, the expression 
for the scattered intensity at short wavelengths is 

T ' ~ Tr sin (l) + SR COt! (l) Sin2 {k " Sin *> 

+ (rapidly fluctuating terms), kr^>ka^>l (29.4) 

The first term of this expression constitutes the "reflected" 
intensity, which, for a cylinder, reflects more in the backward direction 
(# ~ 7r) than in a forward direction (<^ ~ 0). The second term is the 
"shadow-forming" beam, concentrated in the forward direction within 
an angle {j/ka) = (X/2o) which is smaller the smaller X is compared 
with a. The third term contains rapidly fluctuating quantities that 
average to zero, and so may be neglected. 

Total Scattered Power. — The total power scattered by the cylinder 
per unit length is obtained by multiplying T by r and integrating over 
<t> from to 2ir. The cross terms in the sum of Eq. (29.3) disappear, 
owing to the integral properties of the characteristic functions cos (ra0), 
leaving the result 



x / m = 



/ (67r 5 a 4 A 3 ) To (X » 2xa) 

l4aT (X«27ra) 

(29.5) 



The limiting value for total scattered power for very short wave- 
lengths is the power contained in a beam twice as wide as the cylinder, 
(4a). This is due to the fact, discussed above, that the scattered 
wave includes both the reflected and the shadow-forming waves, the 
first and second terms of Eq. (29.4). The integral of r times the first 
term is the total reflected power, which is just 2a T ; and the integral 
of r times the second term is also 2a T , showing that the shadow- 
forming wave has enough power just to cancel the primary wave 
behind the cylinder. 

The quantity (n s /4aT ) is plotted in Fig. 78 as a function of 
ka = (2ra/\). Notice that in spite of the various peculiarities of 
the distribution in angle of the intensity, the total scattered intensity 
turns out to be a fairly smooth function of ka. We also should note 
that II S is not usually measured experimentally, because of the diffi- 
culty of separating primary from scattered waves at small angles of 
scattering. What is usually measured is more nearly the quantity 



352 THE RADIATION AND SCATTERING OF SOUND [VII.29 

n ezp ~ 2 jT' r ~ A % r d<f> 

where A is a small angle, if the experimental conditions are good, but 
it is never zero. It turns out that n exp is very nearly equal to II, for 
the longer wavelengths; but as the wavelength is made smaller, the 
shadow-forming beam (the second term in Eq. 29.4) is less and less 
included in the integral; until for very short wavelengths n exp is equal 
to in 8 . The transition from n s to iU s comes at values of ka near 
(ir/2A), at wave lengths X near 4aA. 

The Force on the Cylinder. — Returning now to the expression 
for the total pressure due to both the undisturbed plane wave and 
the scattered wave, we find, after some involved juggling of terms 
[the use, for instance, of the last of Eqs. (20.1)], that the total pressure 
at the surface of the cylinder at an angle from the x-axis is 

P. = (p, + *)„ = g£) «-«- 2 2^ «' [ " T " +?] (29.6) 



m = 



where the quantities C m are defined in Eq. (26.6). This expression 
is proportional to the expression in Eq. (26.7), giving the pressure at 
some distance from the cylinder due to a vibrating line element on the 
cylinder, when we make the necessary change from <j> to tt — <f> (since 
now = is in the direction opposite to the source). This is an 
example of the principle of reciprocity. The pressure at a point A 
due to a source at point B is equal to the pressure at B due to a source 
at A, everything else being equal. Therefore the polar curves of 
Fig. 65 show the distribution of intensity about a cylinder having a 
line source, and they also show the distribution of the square of the 
pressure on the surface of the cylinder due to a line source at very 
large distances away from the cylinder (distances so large that the 
wave has become a plane wave by the time that the wave strikes the 
cylinder). 

When n is very small (X » a) the expression for the pressure at 
r = a reduces to 

p — ► P (l + 2ika cos 4>)e~ 2Tivt 

M->0 

which approaches in value the pressure P e- 2Tivt of the plane wave 
alone as n goes to zero. 

The net force on the cylinder per unit length is in the direction 
of the plane wave and is 



Vn.28] 



THE SCATTERING OF SOUND 



353 



-f 



(29.7) 



f = a p cos <f>d(f> = 4aP ( j—jr ) e^* - *™***/ 2 

' zco(47T 2 a 2 /c)Poe- iat [« « (c/o)] 

/ 4aX P e i( » /C > («-f')+ix/4 ( X « 27ra) 

This force lags behind the pressure, which the plane wave would have 
at r = if the cylinder were not there, by an angle — 71 + (ir/2). 
The limiting* formula, for co small, has been used on page 149 in dis- 
cussing the forced motion of a string (see Prob. 14). 

The quantity F/P is the net force on the cylinder per unit length 
per unit pressure of the plane wave. This quantity, divided by 2ra, 
is plotted in Fig. 79 as a function of ka = ix. We note that for small 



0.8 r 




Fig. 79. — Amplitude of sideward force per unit length F on a, cylinder of radius o 
due to the passage of a plane wave of pressure amplitude P„, plotted as a function of 
ft = 2ira/\. 

frequencies the force is proportional to the frequency (i.e., to ft) but 
that when /* becomes larger than unity (i.e., when the wavelength 
becomes smaller than the circumference of the cylinder) the linear 
relation breaks down and the force diminishes with increasing fre- 
quency thenceforth. This result is of interest in connection with the 
so-called velocity-ribbon microphone, which consists of a light metal 
strip more or less open to the atmosphere, pushed to and fro by the 
sound wave. The ribbon is in a transverse magnetic field, so that the 
motion induces an emf along the ribbon, which actuates an amplifier. 
The net force on the strip is, of course, not exactly the same function 
of ft as that given in Eq. (29.7) for -the cylinder, but the behavior will 
be the same in general. The force on the strip will be proportional 



354 THE RADIATION AND SCATTERING OF SOUND [VII.29 

to the frequency for small frequencies, but this linear dependence will 
break down when the wavelength becomes smaller than twice the 
width of the strip. 

There is an approximate method of finding the net force on the 
cylinder which gives the correct result for wavelengths longer than 
the circumference of the cylinder. If the pressure in the plane wave 
is P e ik(Xr - ct) , the pressure at the surface of the cylinder due* to the plane 
wave is P^^^-^ ivt . If ^ is small, this can be expanded into p p ~ 
P (l + i/x cos <}>)e- 2irivt . This is the pressure due to the plane wave; 
there is also a scattered wave, enough of a wave to make the radial 
velocity at the surface come out to be zero. This scattered wave con- 
tributes a term ifiPo cos <$> e~ 2 * ivt to this approximation to the pressure, 
so that the net pressure is that given in Eq. (29.6), and the net force 
is the limiting value given in (29.7). We thus see that even for very 
long waves the distortion of the plane wave due to the presence of the 
cylinder contributes a factor 2 to the net force on the cylinder. 

Scattering from a Sphere. — The analysis of the scattering of waves 
from a spherical obstacle follows exactly the same lines as that for the 
cylinder. The expression for a plane wave traveling to the right 
along the polar axis is 

oo 

p p = p oe «(rcostf- c t) = p ^ (2m + \)i m P m {cosd-)j m {kr)e~^ ivt (29.8) 

m = 

where P = \/2p c T , and where the factors P m and j m are defined 
in Eqs. (27.6) and (27.10), respectively. The expression for the wave 
scattered from a sphere of radius a whose center is the polar origin is 

oo 

Vt = -p ^ (2w + l)i m+1 e- iS » sin 5 TO P TO (cost?). 

m = 

• [j m (kr) + m OT (/cr)]e- 2 ™< (29.9) 

where the angles 8 m have been defined in Eqs. (27.17) in connection 
with the radiation from a sphere. The values of some of them are 
given in Table XI at the back of the book. 

The intensity of the scattered wave and the total power scattered 
are 



a 2 To 



00 

1_ ^ (2m+l)(2n + l) 



r 2 k 2 c 

m,n = 

• sin 8 m sin 8 n cos (8 m - 8 n )P m (cos#)P n (cos#) ( (29.10) 
( (16ir 4 i' 4 a 6 To/9c 4 r 2 )(l - 3 cos#) 2 (ka « 1) 

\ (a 2 /4r 2 ) + (a 2 /4r 2 ) cot 2 (&/2)J\(ka sin &) {ka » 1) 



VII.29] 



THE SCATTERING OF SOUND 



355 



(29.11) 



n, = 27ro 2 To(2/fc 2 a 2 ) J; (2m + 1) sin 2 5 m 

m = 

( (2567r 5 a 6 /9X 4 ) T • (X » %ra) 
(27ra 2 To (X « 2wa) 

The discussion concerning the short wavelength limit for the 
scattering from a sphere is similar to that preceding and following 
Eq. (29.4) for the cylindrical case. The total power scattered is that 



X=2jtci 




n/^2 



12 3 4- 

Fig. 80. — Distribution in angle of intensity scattered from a sphere of radius a and 
total power scattered II, per unit incident intensity. 

contained in an area of primary beam equal to twice the cross section 
ira 2 of the sphere. Half of this is reflected equally in all directions 
from the sphere (the first term in the last expression of Eq. 29.10); 
and the other half is concentrated into a narrow beam which tends to 
interfere with the primary beam and cause the shadow (the second 
term in the last expression). If the experimentally measured "total 
scattered power" includes everything from # = A to & = ir — A, 
n exp will equal II 8 for wavelengths longer than (4a/ A) and will approach 
ill s for wavelengths much shorter than (4a/ A). 

Figure 80 shows polar curves of the scattered intensity as a function 



356 



THE RADIATION AND SCATTERING OF SOUND [VH.29 



of the angle of scattering #, for different values of n = ka, and a 
curve n s as function of ix. As with the cylinder, the directionality 
of the scattered wave increases as the frequency increases. 

The Force on the Sphere. — The total pressure at a point on the 
sphere an angle # from the polar axis (note that the point # = is the 
point farthest away from the source of the sound) turns out to be 

oo 

p a = iVr*"«(l/A;a) 2 ^5 ??Ltl p m (cos #)*-*<«■*-»«»> 

m = 

~ (1 + $ika cosifyPoer*-* (ka « 1) (29.12) 

As with the cylinder, this expression is proportional to that for the 
pressure, at large distances, due to a simple source set in the sphere. 
The curves of Fig. 66 therefore show the dependence of pi on &. 



v % 1 




8 



Fig. 81. — Ratio of pressure amplitude at a point on a sphere to pressure amplitude 
of the plane wave striking the sphere, plotted as a function of ix = (2ira/\). Solid line 
is the pressure at a point facing the incident wave; dotted line is the average pressure 
for a circular area around this point with an angular radius of 30 deg. 

The amplitude of the pressure at the point nearest the source of 
sound (# = ir), for a plane wave of unit pressure amplitude, is plotted 
in Fig. 81 as a function of /*. We notice that for wavelengths long 
compared with the circumference of the sphere the pressure at # = t 
equals the pressure of the plane wave but that for shorter wavelengths 
the distortion of the wave due to the presence of the sphere makes p a 
differ from P . This general fact will be true for obstacles of other than 
spherical shape, even though the dependence of p a on n will be some- 
what different from that given in Eq. (29.12). Therefore, a micro- 
phone measures the pressure of the wave striking it only as long as its 
circumference is smaller than the wavelength of the wave. For 



VII.29] 



THE SCATTERING OF SOUND 



357 



smaller wavelengths a correction must be made for the distortion due 
to the presence of the microphone. 

By using Eqs. (27.7) we can find the average value of the pressure 
on that portion of the surface of the sphere contained between the 
angles # = t and # = -k — # : 



Pav = 



j r* 

cos# J*~» V ' 



sinddd 



= Poe-^ 1 



>b i m e- i5m pm-i (cosflp) - P m +i (cos^o) l 
^3 k 2 a 2 D m I P_ x (cos# ) - Pi (cos #„) 

iPoe-^l + £ifca(l + cos do)] (ha « V 



(29.13) 



where P_i(cos# ) = 1. The dotted line in Fig. 81 gives the values 
of (pav/Poe - ^) as function of n = ka for the case # = 30 deg. This 
curve will be useful in discussing the behavior of microphones later 
in this chapter. 

Design of a Condenser Microphone. — As an example of the use of 
the scattering formulas derived above, let us discuss the behavior of a 
condenser microphone in a spherical 
housing. We shall have to make a 
number of simplifications that do not 
correspond to actuality in order to 
avoid confusing complications. We 
shall try to include, however, enough 
of the complications encountered in 
actual practice to show the difficulties 
involved and the relative effect of the 
various complicating factors. 

The first simplification that we 
shall make concerns the container 
for the microphone. We shall 
assume it to be spherical in shape, 
because we have analyzed the effect of a sphere in distorting a plane 
wave, and we have not so analyzed the effect of a cube, for instance. 
The effect of the sphere will be enough like the effect of any other 
object of about the same size so that the behavior that we compute 
will be typical. 

The microphone that we shall use will be a resonator-coupled 
condenser microphone, arranged as shown in Fig. 82. The radius 
of the sphere will be 5 cm and that of the diaphragm will be 2.5 cm, 
so that the angle of the circular opening in the sphere is 30 deg = # . 




Fig. 82. — Simplified cross section 
of the condenser microphone whose 
constants are given in Eqs. (29.14). 



358 THE RADIATION AND SCATTERING OF SOUND [VH.29 

The diaphragm is set back 1.2 cm from the opening; any larger dis- 
tance would make the first cavity resonance come at too low a fre- 
quency. The spacing between the condenser plates will be 0.02 cm. 

We wish the microphone to respond up to 8,000 cps, and, referring 
to Fig. 44, we see that the lowest resonance frequency of the diaphragm 
cannot be less than 4,000. If the diaphragm is aluminum, 0.0015 
cm thick, the membrane density <r is 0.005 g per sq cm, and the tension 
must be 3 X 10 6 dynes per cm for the frequency v \ to be 4,000 [see 
Eq. (20.9) and the paragraph preceding]. 

If we design the air space in back of the diaphragm properly, we 
can make the damping constant 6d [see Eq. (20.8) come out to be 3, 
by using damping material behind the plates. The constants of the 
system are therefore 

a = 5; b = 2.5; # = 30°; 6 d = 3; I = 1.2 
a = 0.005; A = 0.02; v i = 3,800 

ha = (j^) ~ (V1,000); n = (~\ ~ OMa \ (29.14) 

©-0.06^ ■ »-(£*) -*. 

where c a is the velocity of sound in air and Ca that of transverse waves 
in the membrane. 

Behavior of the Microphone. — Our first task is to determine the 
average pressure at the surface of the opening in the sphere. This 
has been done in Fig. 81, for the case when the opening is pointed 
straight at the oncoming plane wave. There will be, of course, a 
different curve if the microphone is pointed in some other direction. 

Next, we must determine the pressure at the diaphragm, which 
is set at the back of' a tube of length I and radius 6. This quantity 
is given by Eq. (23.28) for cavity response and is shown in Fig. 57. 
Equation (23.28) must be modified so as to use expression (27.20) 
for the impedance of a hole in a sphere, instead of expression (28.6) 
for the impedance of a hole in a plane. The modification is not large. 
The ratio of the average pressure at the diaphragm to the pressure 
of the undistorted plane wave is shown in Fig. 83 as a function of 
frequency. The slow rise of the curve is due to the increasingly 
important distorting effect of the sphere on the wave as v is increased; 
the large peak at about 4,000 cps is due to cavity resonance. 

Finally, having obtained the pressure at the diaphragm, we can 
use Eq. (20.12) to determine the electrical response of the microphone. 
This response can be expressed as follows: If the intensity of the sound 



VH.29J 



THE SCATTERING OF SOUND 



359 



in the plane wave is T , and if the microphone plate is charged to a 
potential of E volts, then the amplitude of the emf delivered to the 




10,000 

Fig. 83. — Ratio of pressure amplitude at the diaphragm of the microphone shown in 
Fig. 82 to the pressure amplitude of the incident plane wave. 

2.0r 




500 1000 

Frequency, -v 

Fig. 84. — Response curve for the microphone shown in Fig. 82. 
is given in terms of Hiy) by Eq. (29.15). 

amplifier is 



E = 0.93 X 10- 3 VTo EoH(v) 



5000 10,000 

The output voltage 

(29.15) 



where H(v) is plotted against v in Fig. 84. A resonance peak for the 
diaphragm-resonator system has merged with the one for cavity 
resonance at the upper part of the useful range. The sudden drop 
in response above this is due to the fact that the waves on the dia- 



360 THE RADIATION AND SCATTERING OF SOUND [VII.30 

phragm become short enough so that the motion of one portion cancels 
that of another, as was discussed on page 202. Compare this with 
the second curve of Fig. 44. 

The response could be made larger by decreasing the resonance 
frequency of the diaphragm, by increasing E , by decreasing the 
spacing A, or by reducing the diaphragm density a. The diaphragm 
has already been made as thin as it can reasonably be, so that <x cannot 
be much reduced. Presumably, E is made as large as it can be for 
the spacing A. Decreasing the resonance frequency voi would mean 
reducing the useful range of frequencies of the microphone, which is 
undesirable. 

The diaphragm cannot be set any farther back in the sphere, for 
then the cavity resonance frequency would intrude on the useful 
range. The sphere should not be made any larger, for it would then 
distort the plane wave to a greater extent. Perhaps a slight advan- 
tage would be gained by reducing the values of a and I below those 
chosen. 

30. THE ABSORPTION OF SOUND AT A SURFACE 

In the previous discussions of scattering we have assumed that the 
scattering surfaces were rigid, an approximation that is not always 
good enough. It is now time for us to take up the task of discussing 
just what a surface does when it is acted on by a sound wave and of 
computing what effect this behavior has on the sound wave. 

Surface Impedance. — Of course different sorts of materials react 
differently to sound, depending on their structure. Nonporous mate- 
rials yield slightly, since they are never perfectly rigid, and porous 
materials also allow some air to penetrate below the surface, producing 
an additional effective motion of the surface. In any of these cases 
we can express the reaction of the surface in terms of a specific acoustic 
impedance, a ratio between pressure at the surface and the normal 
velocity of the surface. This impedance in general depends on the 
nature of the surface material, on the frequency of the wave, and on 
its angle of incidence (we at first consider only plane simple harmonic 
waves; later we shall build up more complex waves out of these). 

Just as occurred with the problem of the reaction of the air on a 
membrane, discussed on page 335, we find that the amount of depend- 
ence of the impedance on the angle of incidence of the wave depends 
on how well wave motion can travel, in the surface material, parallel to 
the surface. If such wave motion is rapidly attenuated or is con- 
siderably slower than that of sound in air, then the impedance of the 



VII.30] THE ABSORPTION OF SOUND AT A SURFACE 361 

surface is nearly independent of angle of incidence, for in this case 
one part of the surface is not aware of the motion of another part, and 
the reaction of one part of the surface is proportional to the local 
pressure at that point. Such surfaces can be called locally reacting 
surfaces. On the other hand if the wave motion in the surface is 
not attenuated, and is as fast as or faster than that in the air, then 
the reaction of one part of the surface will depend on the motion of 
other parts of the surface and the surface impedance will depend on 
the angle of incidence of the wave. Such surfaces can be called 
surfaces of extended reaction. 

We shall consider a few typical cases in order to illustrate these 
general statements. 

Unsupported Panel.— One of the simplest sorts of surface is an 
impervious plane panel of dimensions that are large compared with 
the wavelength in air. We assume that its outer surface at equilibrium 
is in the (?/,2)-plane and that the panel supports are far enough apart 
so that they do not play a part in the reaction between panel and air. 
According to Eq. (21.1) the equation of transverse motion of the 
panel is 

«.£--<">, + 9, m ;:^t /3(1 _ s2)] CO.!) 

where p is the pressure of the air, Q and p the modulus of elasticity 
and density of the material, 2h its thickness, and v its displacement 
away from the air (in the negative ^-direction). The constant g is 
called the flexural rigidity of the panel, and m s is its mass per unit 
area. The velocity of simple harmonic transverse waves in the panel 
is c a = Vco (g/m.)*. 

Now suppose that the wave in air has an angle of incidence $, so 
that the pressure has the form 

V = 2Poe*<«"> <»■•»*-*> cos [7— ) cos* + *] 

The air velocity perpendicular to the surface is 

u x = i —^ c *(./e)c»-«^ef) sin \h£\ cos* + *1 

The value of the phase angle ^ is determined by solving Eq. (30.1) 
for n and then setting - (dv/dt) equal to u x at x = 0. 
The solution for — (dij/dt) is 



362 THE RADIATION AND SCATTERING OF SOUND [VII.30 

_dy _ / 2Po cos * \ c i(a/c) (^^^ 
dt \ z* ) 

z s = ^ = -ia>m s + ig (^-)sin 4 $ 
and the solution for the phase angle is 

tan 1 ^ = il — J = *'(«• ~ **») (30.2) 

where z s is the specific acoustic impedance of the panel for the angle 
of incidence <S>, and k 3 and a s are its acoustic conductance and sus- 
ceptance ratios. 

The equation giving z s in terms of a, m„ g, and * shows that this 
impedance is purely reactive and that at high frequencies it depends 
very strongly on the angle of incidence. At frequencies low enough 
so that c s , the speed of transverse waves in the panel, is considerably 
smaller than c, the speed of sound in air, the specific acoustic imped- 
ance of the panel becomes 

z s ~ -i(am s (« « c 2 y/m,/g) 

a pure mass reactance, due to the mass of the panel. This illustrates 
the general statement made at the beginning, that if the speed of wave 
motion in the surface is much slower than that of sound in air the 
surface has local reaction and the impedance is independent of angle 
of incidence. When the impervious panel is quite thin (as with the 
fabric lining of an airplane cabin, for instance), the flexural rigidity 
is extremely small and z s c^ -i<am 8 over the whole of the useful fre- 
quency range. 

For some laminated panels the flexural rigidity factor is complex, 
g = g s — ir s , so that the impedance has a real part 



= -wrn, + (ig. + r.) I ^-jsin 4 * 



and energy is lost by motion of the surface. The amount of energy 
lost is strongly dependent on frequency and on angle of incidence, 
however. For such a panel, with very little support, the impedance 
becomes very small at very low frequencies. 

Supported Panel.— This first example, which has just been dis- 
cussed, is one seldom encountered in practice; it is more usual to have 
the surface material supported by a heavier structure, with distances 



VII.30] THE ABSORPTION OF SOUND AT A SURFACE 363 

between the points of support of the same size or smaller than a wave- 
length. In this case, transverse waves cannot be propagated far 
along the surface without being stopped by the supporting structure, 
so that the panel is, on the average, a locally reacting surface. Each 
portion of panel between supports is a plate with effective specific 
admittance (l/s P ) equal to the ratio of average velocity u x , averaged 
over the portion of panel, to the driving pressure. At the higher 
frequencies this impedance will show a mass reactance z s ~ -t'ww B , but 
at low frequencies the stiffness due to the support structures becomes 
important and z s ~ iK s /a>. There is usually a resistive term. 

Consequently, the specific acoustic impedance of a supported, 
impervious panel can be represented by an equivalent circuit of 
inductance ra s , resistance R s , and capacitance C s = 1/K S in series. 

Porous Material.— A case often encountered is that of a surface 
which is porous enough so that the air motion normal to the surface 
is due more to air motion into and out of the pores than to motion of 
the panel as a whole. Here we must take into account the wave 
motion of the air in the pores. The properties of the pore system can 
be expressed in terms of three quantities: flow resistivity r p , porosity 
P p , and effective air density pm p . 

Under steady-state conditions a pressure drop (Ap) across a thick- 
ness d of the material will force a flow of air through the pores of 
u x = r p d(Ap) cc per sec per sq cm of surface of the material. This 
defines the flow resistivity r p . 

If the inner side of the material is made impervious, then forcing 
air into the material will raise the pressure of the air in the pores. 
If the air were compressed adiabatically, as would happen in the 
open, the rise in pressure would be (pc*/P p d)% [see (Eq. 23.3)] where £ 
is the displacement of the air into the outer surface (in cubic centi- 
meters per square centimeter of surface) andP p is the fraction of the 
volume of the material that is available for the air to flow into (this 
defines the porosity P p ). In many cases the pores are so small that 
the air loses its heat to the pore walls, so that the expansion is iso- 
thermal instead of adiabatic. Then the rise in pressure would be 
(pc 2 /ycPpd)%, where y c = 1.4 for air. 

Finally, the air in the pores may have an effective density greater 
than that in free space because some filaments of the pore material 
may move with the air, so that m p may be greater than unity. 

The equation of motion of the air in the pores is therefore equiva- 
lent to that for electric current in a transmission line, with line resist- 
ance r p per unit length, line inductance P m p per unit length, and shunt 



364 THE RADIATION AND SCATTERING OF SOUND [VII.30 

capacitance (ycP p /pc 2 ) per unit length. The equations of motion of 
the air in the pores, for simple harmonic waves, are 

—iupnipU + r p u = — grad p | 

„ , .ycP pP o>, . , n > (30.3) 

V 2 p + i - — ~- (r p — ia)m p p)p = I 
pc ) 

and the characteristic impedance of the material is 



= P Cyj 



m p + i(r p /po) 
7cP P 



which is a complex quantity, indicating attenuation of the wave. . 

Equivalent Circuits for Thin Structures. — If the flow resistance 
r p is fairly large, wave motion will attenuate rapidly in the material 
and the pores will exhibit local reaction to the sound waves. This is 
the case usually encountered in practice, although for some materials 
(such as hair felt) extended reaction is not negligible. When local 
reaction predominates, so that the impedance is independent of angle 
of incidence, and when the thickness of the porous material is small 
compared with the wavelength, then the specific acoustic impedance 
of the porous material can be expressed in terms of equivalent circuits 
for different structures. 

Figure 85 shows some equivalent circuits for a few structures. In 
the first case the porous material is mounted on a relatively rigid and 
impervious wall, so that the air motion in the pores is responsible for 
the acoustic impedance. The important part of the impedance is 
the capacitative reactance of the pore volume. The effective resist- 
ance and inertia are reduced by a factor i, because the air motion in 
the pores nearest the backing wall is constrained to move less than 
the air in the pores near the outside. 

In the second example shown in Fig. 85 the panel is held away 
from the impervious backing wall by a framework, which prevents 
flexural waves from traveling along the panel, and which also dis- 
courages lateral wave motion in the air space behind the panel. Here 
the panel can move by flexure, as well as allowing air to move through 
the pores, so that the equivalent circuit shows three parallel paths, 
one for flexure, one for transmission through the pores, and the third 
due to the stiffness of the air in the pores. 

The third example shows that laminated structures are equivalent 
to filter networks and that the equivalent circuit can be used to calcu- 



VH.30] THE ABSORPTION OF SOUND AT A SURFACE 



365 



late transmission of sound through wall structures as well as absorp- 
tion of sound by the structure. 



Porous Rigid 

material { backing 



o°<y 
°oo° ° 



° ' 
O oo 0< 



--d 



V//; 



J 



-AWVAAr- 






YcPpd 
pc* 





mppd 



^S 



S^ o 



Impervious 
septum a 
mass ma 



■I- 

i 

Impervious 

septum b 

mass m/, 



r p d m p pd 



jL, 
pc 2 ' 



^AAAAArJpTinT 1 -' B 



m b 



"pc 2 



r 



Fig. 85. — Equivalent electrical circuits giving specific acoustical impedance of 
various wall structures. Equivalence is valid as long as wavelength is long compared 
to t, I or d. 

Formulas for Thick Panels.— When the thickness of porous mate- 
rial is greater than a wavelength of sound in the material, the analysis 
becomes more difficult. The wave equation for motion of air in the 
pores can be written in the form 



te . + <» + *>'(!) , i»-o 



d 2 p 



(n + i q y = y J> p [m p + i(^] 



366 TEE RADIATION AND SCATTERING OF SOUND [VII.30 

and the solution, for the case of material of thickness d f backed by a 
wall (or air space) of characteristic impedance z b (at x = — d) 

p = A cosh I- J {q - in)(x + d) + ^ 

- - feL - * n -^f *-*[© ( * - in)d + *] (m4) 

^ = tanh-^To^^/pc^ + n)\ 

When the frequency is high enough, or the thickness of the material 
is great enough, the wave effectively dies out before it is reflected 
from the back of the material, the hyperbolic tangent is approxi- 
mately equal to unity, and the wall impedance becomes the character- 
istic impedance of the material, z pc . For lower frequencies, Plates I 
and II can be used to compute the impedance. 

The quantity n can be called the index of refraction of the material 
for sound waves, and q can be called its attenuation index. When n 
is larger than unity, the wave travels more slowly in the medium 
than in air; and when q is large, the wave attenuates rapidly. In 
either case the material would be (more or less) locally reactive, and 
the wall impedance would be (approximately) independent of angle 
of incidence. 

Reflection of Plane Wave from Absorbing Wall.— We must next 
investigate the behavior of sound waves that strike absorbing surfaces. 
We have seen from the preceding discussion that in a great many 
cases the surface reacts locally, so that its specific acoustic impedance 
is practically independent of angle of incidence. We shall make our 
calculations for this case, because the results will be valid for most 
types of surface material, and also because the calculations are easier. 
In the rest of this chapter and in the next chapter, therefore, we 
assume that the acoustic properties of a surface are given by its specific 
acoustic impedance, the ratio between the pressure at a point on the 
surface and the normal velocity, into the wall, of the air at the surface, 






pet = pcir!*-** 



The ratio of the impedance z to the characteristic impedance of air 
pc will be called the acoustic impedance ratio of the surface, and its 
reciprocal will be called the acoustic admittance ratio of the surface. 



fe)-' — * (?) 



v = K — ia (30.5) 



VII.30] 



THE ABSORPTION OF SOUND AT A SURFACE 



367 



where 0, x, *, and o" are the acoustic resistance, reactance, conductance, 
and susceptance ratios of the material, respectively. A thin mem- 
brane has a specific reactance coM and susceptance — (1/coilf), where 
M is the mass of the membrane per unit area; a stiff supported panel 
of small mass has specific reactance —(K/u>) and susceptance {<u/K), 
and so on. 

Now suppose again that the surface is at the (?/,z)-plane, with air 
on the positive x side. An incident wave comes in from the right at 
an angle of incidence * and is reflected from the wall. The equation 
for the pressure wave is 

sq = p . (>i(fi>/c) (y sin*— x cos*— ct) p g— i(w/c)(y s in*+a; cos *— ct) 

where the ratio between the reflected pressure and the incident pres- 
sure is e -2*(«-*»(p r = p t . e -2 1 r(a-i^)) an( j th e ra ^ between the reflected 
and incident intensity is e -4 ™. The pressure and air velocity can be 
expressed in terms of hyperbolic functions : 



u x — 



p = -2Poe~ ,r(a_ ^ ) e i(a,/c)(j ' 8in *~ c0 sinh<ir (-?) cos* - a + \> 

2 e -ir(a-*0) e »(<o/c) („ sin *- ct ) cog $ g^ J ^ U ^ J CQ g $ _ a _|_ ^ K 



The values of a and are determined by the impedance of the 
surface 

P — P 

tanh[7r(a - iff)] = * r = f cos* = (0 - i%) cos* (30.6) 

±i ~~\~ ± r 

so that, if the specific acoustic resistance and reactance of the surface 
are known, the values of a and /3 for a given angle of incidence * can 
be computed from Plates I or II at the back of the book. 
The ratio of reflected to incident energy is 



— o — iva = 



1 — f COS* 



1 + £ cos* 



2 = (1 — 0cos*) 2 + x 2 cos 2 * f „ 
(1 + cos*) 2 + x 2 cos 2 * ^ ' 



Curves giving values of this ratio, the fraction of incident intensity 
which is reflected, as function of |f| cos*, for different phase angles <p 
of the wall impedance, are shown in Fig. 86. We note that the amount 
reflected has a minimum value (and therefore the fraction of incident 
energy absorbed has a maximum) when |f| cos* = 1. The larger 
the power factor cos <p for the absorbing surface, the greater the frac- 
tion of energy absorbed, or the smaller the fraction reflected. When 
the power factor is zero (<p = ±90 deg) , no energy is absorbed. 



368 



THE RADIATION AND SCATTERING OF SOUND 



[VH.30 



The curves also show that the fraction of energy reflected 
approaches unity (fraction absorbed approaches zero) as |f| cos<I> 
approaches zero, i.e., as the angle of incidence approaches 90 deg. 
In fact Eq. (30.7) indicates that the surface would not absorb any 
energy from a wave traveling parallel to the surface, no matter what 
the value of z. This seems to be a contradiction of terms, for the 
pressure fluctuations in a wave parallel to the surface would cause 
motion of the surface in a direction perpendicular to the assumed 
direction of the wave. The fact of the matter is that a plane wave 




5 
ISkos$ 

Fig. 86. — Reflection coefficient for plane waves incident at angle of incidence $ on a 
plane surface of acoustic impedance 2 = pc|f \e — iip. 

cannot travel parallel to an infinite plane surface of noninfinite acoustic 
impedance. This will be shown in more detail' in the next section. 
It can also be seen from the equations for p and u x . We notice that 
both of these go to zero when $ = 0, so that a solution in terms of 
plane waves alone is not possible. 

31. SOUND TRANSMISSION THROUGH DUCTS 

Before we discuss the behavior of sound waves in rooms, we should 
return once more to the problem of the transmission of sound through 
ducts, in order to investigate the effect of the impedance of the duct 
wall on the sound wave. This investigation was begun in Sec. 26, 
where Eq. (26.14) gave an approximate expression for the transmis- 
sion of sound in a cylindrical duct. Now we wish to carry the analysis 
further. In recent years the problem of sound transmission in venti- 
lation ducts has become important, so it will be useful to show how to 
compute the relation between sound attenuation and duct wall imped- 
ance, in order to be able to design duct linings to absorb undesirable 



VII.31J SOUND TRANSMISSION THROUGH DUCTS 369 

sound. Since most ventilating ducts are rectangular in cross section, 
we shall confine ourselves to this simple type in the present section.' 
Boundary Conditions.— We assume that the duct cross section is 
rectangular, of width l v in the ^-direction and I, in the z-direction. 
We also assume that the impedance of the duct walls is uniform, the 
acoustic impedance ratio being f for each wall. The pressure, for a 
wave traveling to the right along the duct, is given by the equation 

cosh /2^\ cosh/2*^ x _ ct) 

smh \ ly / smh \ h / 

where r 2 = 1 + (27rc/a>) 2 [(&A)2 + fo./Z,) 2 ], and where the ^'s are 
determined by the boundary conditions at y = ± (l v /2) and z = ± (l z /2) 
(we are taking the central axis of the duct to be the origin of the lateral 
coordinates y and z). 

The air velocity into the duct wall at y = +(l y /2) must be equal 
to the specific acoustic admittance (77/pc) = (1/pcf) times the pres- 
sure at the wall. 

,, _ (2*yv\ a sinft / \ COSh/^TIYkzX ., . ,, 
U " ~ WJ A c 0S h <■*> sink (,-177 6 

(r}\ . cosh, cosh (%cqj\ .. . ., 

This produces the equation for determining g y in terms of r\ : 

Toth ^> =4„)ft) ; X = (WU) (3L1 ' 

where the hyperbolic tangent is used for the waves that are symmetric 
about the center plane y = . (p proportional to cosh(2jr^ tf y/^», and 
the tangent is used for the antisymmetric modes (p proportional to 
the sinh). 

This equation also ensures that the boundary condition at y = 
— (lv/2) is satisfied, for everything is symmetrical (or antisymmetrical) 
about the center plane. 

The boundary condition for the surfaces z = ± (l z /z) is a similar 
equation 



tanh 
coth 



to*) = *-(^(k); x = (2Wco) 



The exact solution of these equations is a complicated process, so 
we shall first obtain an approximate solution, valid for values of wall 
admittance ratio 17 small compared with (\/l) (large wall impedance). 



370 THE RADIATION AND SCATTERING OF SOUND [VII.31 

Approximate Solution. — When |i?|(^/X) is small compared with 
unity, Eq. (31.1) can be solved approximately. There are a large 
number of solutions, each with a characteristic value of g. They can 
be put in order of increasing size and labeled with an index number 
n v = 0,1,2 • • • . For reasons that will be apparent shortly, the 
solutions using the hyperbolic tangent will be labeled with the even 
values of %, 0,2,4 . . . , and the solutions using the cotangent will be 
labeled with the odd values, 1,3,5 .... The properties of the 
functions tanh and coth which are used to obtain the approximate 
solutions are 

tanh (w + iirn) ~ w; coth (w + iirn + iwr) c^ w, w « 1 

Using these expressions, we find that the approximate solutions 
for the characteristic values of g„ and g e are 

***-+(^)-*^ +0 '- < ' ) fe) ( "-" 1A8 ' ■ ' 

with the even values of the n's going with the cosh factor in the expres- 
sion for the pressure and the odd values going with the sinh factor. 
The solution for n y = n z = is the "plane wave" or principal wave, 
which usually has the smallest attenuation and carries the greatest 
part of the power. 

The equation for the transmission coefiicient t gives, to the first 
approximation in the small quantity rj, 

t(0,0) c* 1 + ^g (<r + in), n v = n e = 

n* = 



(31.2) 



r(n y ,0) cr (n y ,0) + ^fcfi^th + * 1 ^ + «)■ 



r(0,n.) ^ r (0,n 2 ) + ^y ^ (h + 2l y ){a + «), n y = ^ ^^ 

r{n v ,n z ) ^ro(n y ,n g ) + ^-^ ~ (<r + «) 

(ny > 0, n* > 0) 



ro(n,,n.) = [l - Qg) ~ (^) J 



VII.31] SOUND TRANSMISSION THROUGH DUVTS 371 

where S = l y l z is the cross-sectional area and L = 2(l y + l z ) is. the 
perimeter of the duct cross section. We note that when n u = the 
duct walls perpendicular to the y-axis (l z ) are half as effective in chang- 
ing r as is the case for n y > 0, and similarly for n z and l y . Since the 
case n y = corresponds to waves traveling "parallel" to the sides l a 
and the cases n y > correspond to waves being reflected back and 
forth between these walls, we can reword our remark to say that the 
walls have (to this approximation) half as much effect on waves travel- 
ing "parallel" to them as they do on waves that are obliquely incident. 

The reason we have put quotation marks around the word parallel 
is that the case n y = does not correspond to waves exactly parallel 
to the I, walls. In Eq. (30.7) we showed that, if a plane wave could 
move parallel to a plane wall, the wall impedance would have no effect 
on the wave. We also showed that this was impossible, and the 
present results reinforce this, for we find that when the wave is as 
"parallel" to the wall as we can make it (n„ = 0) the wall impedance 
still makes itself felt; only half as much as for oblique waves, but not 
zero times as much. 

Principal Wave. — This property of waves and nonrigid walls can 
be clarified to some extent by studying the behavior of the principal 
wave (n y = n z = 0). Remembering that rj is small and that the 
expansion of cosh (u) for u small is 1 + %u 2 , we have 

V ^ A [l + ijL (a + iK)y^ \l +q(? + i*> 2 

. e <V2S)(t<r-«)* e i( M /e)(*-rf) > (jl y = Uz = Q) (31.4) 

In the first place, we see that the wave is not a perfect plane wave, 
for the pressure amplitude depends on y and z and therefore there is 
some air motion perpendicular to the #-axis (just enough to take into 
account the effect due to the sideward motion of the duct walls). In 
addition the wave is damped, as indicated by the factor e-(^««/ss) j by 
an amount proportional to the acoustic conductance ratio k of the 
walls, proportional to the circumference of the duct, and inversely 
proportional to its cross-sectional area S. 

The attenuation of the principal wave in decibels per centimeter is 
therefore approximately (4.34Lk/£) (if L is measured in feet and S in 
square feet, the attenuation will be 4.34Lk/£ db per ft). We note that 
the greater the wall area is, compared with the duct volume (L is, 
compared with S), the greater the attenuation. 

The attenuation of the higher modes is larger than this, being 
approximately (SMLK/Sroiny,^)) for n y > 0, n z > 0; and being 



372 THE RADIATION AND SCATTERING OF SOUND [VTI-31 

(4.34/c//STo(n„,0)) (2l y + l z ) for n g = 0, etc., as long as n y , n z are small 
enough so that to is real. For all modes above certain values of 
n y , n z , to is imaginary for a given frequency, and no true wave motion 
can occur; the attenuation for all higher modes being very much 
greater than for the ones below the cutoff (see page 308). 

The characteristic acoustic impedance of the principal wave is 






u x i(a)/c) — (Lrj/2s) 



(i _ _£_r 

\pc 2icoSzp/ 



where z v is the specific acoustic impedance of the duct walls. There- 
fore the characteristic wave impedance of the principal wave is anal- 
ogous to a circuit with resistance pc in parallel with an impedance 
—ica(2Sz p /Lc). For instance, if the duct walls are covered with 
porous material of thickness d, flow resistivity r p , porosity P p , and 
negligible density m p , the equivalent circuit giving the characteristic 
impedance of the principal wave has one parallel arm a pure resistance 
pc and the other parallel arm a resistance pc(2S/jcPpLd) in series with 
an inductance (2Sr p d/3Lc). 

Transient Waves. — We can use the results of the preceding discus- 
sion to obtain a first approximation to the transient behavior of sound 
in a long duct. Suppose that the average air velocity at the input end 
is the arbitrary function u x0 (t). Then the pressure in the principal 
wave, a distance x along the tube (assuming no reflection from the 
far end), is 

*»H£j>( i+ 5' ,+ *' , X i -&r 

t* 00 

. e -(L V /2s)x+i(w/c) X -iwt j w a . (r)e iuT dr 

In several cases, where the wall admittance 77 has a simple behavior, 
this integral can be evaluated. 

For instance, if the side walls are stiffness controlled, 17 = — iwe 
where e is the specific elastance of the walls, we use the equations, 
derived from Eqs. (2.19) and (2.20), 

.. /» * /» 00 

f(t — a) = -=- I e iaa ~ iat du I f(T)e iaT dT 
2ir J — 00 J — * 

1 f°° f °° d 2 

J- I ^ w 2 e--* dco j_J(r)e^dr = - -^f{t) 



VH.31] SOUND TRANSMISSION THROUGH DUCTS 373 

(to be used with due regard for convergence;) to obtain 

P(t) ~ 1 + (Lce/2S) f'° [' ~ X (~c + M)] ' 

-;8+d£-['-6 + S)]} 

It is not very difficult to show that if the wall impedance is a pure 
resistance, independent of 00(77 = k, a constant), then 

p(t) ~pcj- t <~ I e-wwuxo (t -~)dw 

-K£ + d^"*-M)} (31 - 5) 

The proof of this formula will be left to the reader. 

The transient waves in the higher modes can be computed in a 
similar manner, though the calculations are more difficult. 

The Exact Solution. — In a number of cases, the specific acoustic 
impedance ratio of the wall, f , is small enough so that the approxi- 
mations used in Eqs. (31.2) to (31.4) are not valid. Since these cases 
are also of practical importance, we must set about finding an exact 
solution of the equations 

O*)-**; ,-«+>; k-& -^ (31 . 6 ) 

where z p = Izpler** = (pc/rj) is the specific acoustic impedance of the 
duct wall. In the approximate solutions, n turned out to be approxi- 
mately half an integer and | was small; we cannot expect this to be 
true for the general solution. We shall find a series of solutions, 
however, with a series of allowed values for n, which can be arranged in 
order of increasing size, and labeled /i , in, m • • • . Corresponding 
to each ju„ is a solution for £„, and therefore a g(n) = £„ + ifi n , which 
is the nth characteristic value. The corresponding characteristic 

posh 
function is $ n (y) = ^^ l^rg y (n y )y/l v ] where the cosh function is used 

for the even values of n y [which are obtained by solving Eq. (31.6) with 
tanh] and where the sinh is used for odd values of n y [for which coth 
is used in Eq. (31.6)]. 

The easiest way to solve Eqs. (31.6) is by graphical methods. 
The solutions are displayed on Plate V at the back of the book, in 
terms of contours for constant values of h and <p drawn on the (£,/x)- 
plane. We see that the solutions approach the integer and half- 



374 THE RADIATION AND SCATTERING OF SOUND [VH.31 

integer limits on the imaginary axis, as h gets small, as demanded by 
the approximate solution. The heavy dashed lines separate the modes 
one from the other. For each pair of values of h and <p there 
is a point within each modal region. The point in the lowermost 
region in the "tanh" chart corresponds to go (cosh is used in the 
expression for p); the point in the lowest region in the "coth" chart 
corresponds to gi (sinh is used for p) ; the point between the first and 
second dashed lines (second sheet) of the coth chart corresponds to g 2 
(cosh is used for p) ; and so on. 

The dependence on x is by means of the exponential e i( - WTX/c) , where 



= 1 + 



WW] +m) 

(v)"0 



= 1 _ ( ^ j I My(^y) - kl(n y ) n\(n,) - £(n y ) 



2j M»(Wy) €»(%) _ 2l - /*«(»«) €«(»». 



'] 



The real and imaginary parts of tn (N stands for the pair of numbers 
n Vf n e ) correspond to the "index of refraction" and the damping 
index for each wave in the sequence. This equation also cannot be 
solved by approximate means when r is large. The final solutions 

= cosh\ 2Tg v (n y )y ~\ cosh f 2irflr«(n,)g l g(Wc)(7VC _ rf) (31 7) 
Y sinh L l v J sinh \_ U \ 

are characteristic functions that can be used to fit initial conditions 
and to compute the effect of perturbations, as was discussed in Sees. 
11 and 12. 

It is obvious that the behavior of the sound waves, even of the 
principal wave, is quite complicated when the admittance ratio of the 
walls is not small compared with unity. If one adds to this the fact 
that many porous acoustic materials have impedances that vary 
considerably with change in frequency, it becomes apparent that very 
few sweeping generalities can be made concerning the behavior of 
sound in ducts with highly absorbent walls. An examination of 
Plate V shows that for negative phase angles (stiffness reactance) and 
for large values of h (small values of impedance and/or high fre- 
quencies) the value of £ can become quite large, and therefore even 
the principal wave can be highly damped. Further examination 
shows that in these cases the principal wave is far from a uniform 
plane wave, the negative reactance of the walls having in some way 
pulled most of the energy of the wave away from the center of the 



VH.31] 



SOUND TRANSMISSION THROUGH DUCTS 



375 



duct to the periphery, where it is more quickly absorbed as it travels 
along. A positive phase angle (mass reactance) has the opposite 
effect. The change of acoustic behavior with frequency is greatest 




Frequency Parameter, (1/a) 



Frequency Para meter, (l/a) 



Fig. 87. — Transmission of sound through square duct of width I covered with 
acoustic material of acoustic impedance pc(6 — i%), for principal (0,0) and a higher 
(1,1) model wave. Quantities proportional to attenuation per unit length and phase 
velocity are plotted against frequency. Sudden rise of quantities for (1,1) mode illus- 
trates fact that higher modes cannot be transmitted at low frequencies. 

when h and # are close to one of the "branch points" of Eq. (31.6), 
shown as circles on the dashed lines separating the modes. 

An Example. — As an example, we show in Fig. 87 transmission data 
for a case of a porous wall material with specific acoustic impedance 



376 THE RADIATION AND SCATTERING OF SOUND [VII.31 

equivalent to a resistance and capacitance in series. To simplify the 
problem we set l y = l z = I and choose for a frequency parameter 
(l/\) = (cdZ/27rc). We choose the constants of our acoustic material 
so that f = 1.5 -f 1.5i(\/l), which is stiffness controlled at very low 
frequencies, but is fairly ""soft" at (X/Z) = 1. The first curves in 
Fig. 87 show the variation of acoustic impedance and admittance 
ratios with frequency. They are fairly typical curves. The next 
curves show the variation of the characteristic values /io, m, £o, £i for 
the principal wave and the first higher mode. Finally we show curves 
for the attenuation of two actual waves [the principal wave (0,0) 
and the (1,1) wave] in decibels per centimeter and the ratio between 
their phase velocity and that of sound in the open, l/(real part of t). 
We note that the attenuation of the principal mode is less, than the 
higher mode for all frequencies shown (this occurs in nearly every 
case encountered in practice). We note also the very sudden and 
large increase in attenuation of the (1,1) mode as the frequency is 
lowered below its cutoff frequency. Few other generalizations can 
be drawn from these curves. Slight changes in impedance and in 
size of duct can produce very considerable changes in the shape of the 
curves of £ and n and of all the quantities derived from them. 

Problems 

1. A flexible wire 50 cm long is stretched at a tension of 10 9 dynes between 
rigid supports. The wire is cylindrical, having a radius of 0.1 cm, and weighs 
0.4 g per cm length. The mid-point of the wire is pulled aside 0.5 cm and then 
let go at t = 0. What is the expression for the total power radiated by the string 
in the form of sound? What is its value at t = and at t = 1? What is the ratio 
between the intensity of the fundamental and that of the first harmonic at t = 
and at t = 1? 

2. A long cylindrical tube of radius 10 cm has a long slit 1 mm wide in its 
wall parallel to the tube axis. Air is forced back and forth through the slit at a 
velocity of 10 e' 2 ™. Plot the total power radiated by the slit cylinder per centi- 
meter of its length, as a function of v from v = to v = 1,000. Plot on a polar 
diagram the distribution in angle of the radiated intensity for v = 547.5, v = 876. 

3. The portion of the surface of a long cylinder of radius a, which is between 
—4>a and +<£o, vibrates with velocity normal to the surface, while the rest of the 
cylinder is rigid. Obtain a series analogous to Eqs. (27.20) for the average acoustic 
impedance over the vibrating surface. 

4. Compute the distribution in angle of the pressure ]^(<£)j radiated by a line 
source on a cylinder of radius a, for a wavelength X = ira [see Eqs. (26.7)]. 

5. A hollow cylindrical tube of inner radius 54 cm is lined with acoustic 
material of specific acoustic impedance pc[10 + (2,000/o>)i]. Compute the cutoff 
frequencies and plot the characteristic impedances as function of w for « from to 
10,000, for the first three symmetric modes of wave transmission along the tube, 



VU.31] SOUND TRANSMISSION THROUGH DUCTS 377 

6. A hollow cylindrical tube of inner radius a, lined with material of specific 
acoustic impedance z, is closed at x = I with a rigid plane plate. The other end 
of the tube, x = 0, is driven by a plane piston. Derive the formulas for the pres- 
sure amplitude at any point x along the tube and the formula for the radiation 
impedance on the driving piston, including the effects of the impedance z to the 
first order of approximation in (pc/z). Show how it is possible to measure the 
wall resistance and reactance by measuring the standing wave. What measure- 
ments must be made, and what formulas must be used? 

7. The surface of a sphere of radius a vibrates in such a manner that the 
radial component of velocity at the surface is 

u a = iC/" (3cos2i> + l) e -* ri '* 

Show that when X is large compared with a the radiated intensity and power 
radiated are 

T = Hpo -^- t/*[P 2 (cos#)p, n = iUpo—Jr Ul 

Such a source of sound is called a quadrupole source. Plot the distribution in 
angle of the intensity on a polar plot. 

8. A piston of radius 10 cm is set in the surface of a sphere 20 cm in radius. 
It is vibrating with a velocity \Qe~ 2irivt . Plot the total power radiated as a func- 
tion of v from v = to v = 1,000. Plot on a polar diagram the distribution in 
angle of the radiated sound intensity at v = 164.2 and v = 438. 

9. The mechanical constants of the piston in Prob. 8 are to = 10, R - 1,000, 
K = 1,000. Plot the total mechanical impedance of the piston as a function of 
v from v = 100 to v = 1,000. 

10. The piston of Prob. 9 is driven by a force of lOjOOOe -211 "^' dynes. Plot 
the total power radiated by the piston-sphere system as a function of v from 
v = 100 to v = 1,000. 

11. A dynamic loud-speaker of radius 27.4 cm is set flush in a large flat wall. 
Plot the transmission coefficient of the speaker as a function of driving frequency 
from v = to v = 1,000. When the speaker is oscillating with a certain velocity 
amplitude at a frequency of 400 cps, the sound intensity at a point 500 cm away 
from the wall, straight out from the center of the piston, is 100 ergs per sec per 
sq cm. What is the intensity at a point close to the wall, 500 cm from the center 
of the piston? If the diaphragm vibrates with the same velocity amplitude as 
before but at a frequency of 200 cps, what will be the intensity of the sound at 
these two points? At 800 cps? 

12. A cylindrical tube of radius 5.47 cm and length 34.4 cm has its open end 
set flush in a large plane wall (the tube is inside the wall with its axis perpendicular 
to the surface) . The other end of the tube is provided with a piston that oscillates 
with a velocity of lOOe-^"'. Plot the total energy radiated out of the tube as a 
function of v from v = to v = 5,000, and plot the distribution in angle of the 
radiated intensity at v = 500, 1,000, 2,000. 

13. A piston of radius 5.47 cm, set flush in a large plane wall, is vibrating with 
a velocity lOOe- 2 ™'"'. What are the total power radiated and the distribution in 
angle of the intensity at v = 200, 500, 1,000, 2,000? If the same piston is set in 
the small end of an exponential horn of constant x„ = 34.4 cm, whose open end 



378 THE RADIATION AND SCATTERING OF SOUND [VII.31 

has a radius of 54.7 cm and is set flush in the wall, what are the total power radi- 
ated and the distribution in angle of the intensity at v = 200, 500, 1,000, 2,000? 
Assume that the open end is large enough so that all the energy is radiated out 
of it. What do these results indicate about the directional properties of horns? 

14. Suppose that the pressure at the mouthpiece end of a wind instrument is 
given approximately by the Fourier series 



po = Po 2* cos [2(2n + l)ir Vl t) 



n = 

where the frequency of the driving mechanism has adjusted itself so that the 
impedance at the fundamental vi is maximum [see Eq. (23.17)]. Compute the 
air velocity u e at the open end of the instrument. Assuming that this open end 
acts as a simple source of sound, compute the sound pressure p r a large distance r 
from the open end. Assuming reasonable values for the constants involved, plot 
u e and p r as functions of time. 

15. A velocity-ribbon microphone consists of a conducting ribbon of mass m 
per unit length and of width 2a, suspended so that it moves freely at right angles 
to a magnetic field H and at right angles to its surface {i.e., the mechanical imped- 
ance of the ribbon per unit length is approximately 2wivm over the useful range 
of v). The motion of the ribbon induces an emf in it of magnitude Hlu X 10— 
volt. Assuming that the force on the ribbon due to a plane wave of sound falling 
vertically on it is equal to the corresponding force given in Eq. (29.7) for a cylinder, 
show that the emf induced in the ribbon is 

4a%m -p. x 10-— > 2 -^*!p x io- 



mcy?C\ ft-*o mc 

where P is the pressure amplitude of the incident plane wave. 

16. Using the results of Prob. 15, design a velocity-ribbon microphone and 
plot its response curve. The pole pieces of the magnet are 5 cm long and 2 cm 
apart, and a field of 1,000 gauss can be maintained between them. An aluminum 
(density 2.7) ribbon can be used, of thickness 0.001 cm. What width ribbon 
should be used to have a response constant to within 10 per cent for frequencies 
below 5,000 cps. Plot this response for the range < v < 10,000. 

17. A circular membrane of radius a is set under tension, in a plane wall that 
is otherwise rigid. It is set into vibration at its fundamental frequency v a i. Plot 
the distribution in angle of the radiated pressure, and compute the total power 
radiated for (2iravoi/c) = 10 and for suitable values of the other constants (c 
here is the speed of sound in air, not the speed of waves on the membrane). Use 
the curves in Fig. 71 to compute the angle distribution. 

18. The distribution in angle of radiated pressure given by the following 
equation: 



-(*) 



e iHr-ct) [$„ _ 0.45*1 - 0.1*2] 



[where the functions *„ are defined in Eq. (28.4) and plotted in Fig. 71] has very 
small "side lobes" for radiation at frequencies such that ka = 10. Plot the dis- 



VH.S1] SOUND TRANSMISSION THROUGH DUCTS 379 

tribution in angle of the radiated intensity as function of &, and plot the velocity 
amplitude of the piston, as function of y, which will produce this radiation. 

19. Use the curves of Fig. 72 and the formula of Eq. (28.8) to compute the 
pressure on a flat piston of radius a, set in a rigid plane, vibrating with frequency 
v. Plot this pressure as function of r for (2wav/c) = ka = 0.2, 1, 4. 

20. Suppose that the condenser microphone whose constants are given in 
Eqs. (29.14) were set in a sphere of twice the radius, a = 10. Plot the altered 
response curve of the microphone from v = to v = 10,000, and compare the 
curve with that of Fig. 84. 

21. A ribbon of width 0.5 cm and length 5 cm is set in the side of a cylinder 
2.19 cm in diameter, with its length parallel to the axis of the tube. The interior 
of the tube is so designed that the effective mechanical constants of the ribbon 
per unit length are w = 0.01, R = 20, K = 50. The ribbon is in a magnetic 
field of 1,000 gauss, perpendicular to the axis of the cylinder and parallel to the 
face of the ribbon. Plot the response curve of this "pressure-ribbon microphone" 
for the frequency range v = to v — 10,000. 

22. Discuss the directional properties of the condenser microphone whose 
constants are given in Eqs. (29.14). Plot the response curve for the microphone 
for the axis of the diaphragm pointing at right angles to the direction of incident 
sound and the curve for the axis pointing away from the source of sound. Com- 
pare these curves with that of Fig. 82. 

23. A plane wave of sound of the form 

[15 sin (z) - 10 sin (3z) + 3 sin (5z)] 

where z = l,000n- It 1» strikes the microphone whose constants are given in 

Eqs. (29.14). Plot the emf output of the microphone for one cycle, and compare 
it with the pressure wave in free space. 

24. The pressure variation from a "warble-tone" generator corresponds to 
the expression 

p s y|g— 2t»j> «— t(Ai'/j'„) cos (2xj>„0 

By using the last of Eqs. (19.4) show that this expression is equal to the Fourier 
series 



p - ie- 2T<v o' V J n l — ) g-2Tt>„»i-<(»v*) 



X '•(£) 



Compute the values of the amplitudes for the component frequencies, for A = 10, 
v = 250, Av = 50, v„ = 10. Plot the curve for the real part of p for the time 
range t — to t = 0.1. 

25. Obtain the formula for the ratio of reflected to incident sound energy as 
function of angle of incidence, for a plane wave falling on a plexiglass window of 
density m 8 g per sq cm and of flexural rigidity g s - ir„. Discuss the difference 
between these results and those shown in Fig. 86. 

26. The flow resistivity of a certain acoustic material is r p = 50 cc per dyne-sec 
cm. Its porosity is P p = 0.7, its effective mass is m p = 5, and its thickness is 
2 cm. Plot the specific acoustic impedance of the material when its back surface 



380 THE RADIATION AND SCATTERING OF SOUND [VII.31 

is glued to a rigid wall and also when the panel is spaced out 4 cm from the rigid 
wall (assume that the effect of panel flexure is negligible). Plot resistance and 
reactance for both cases as a function of frequency v from v = to v = 2,000 cps. 
Assume that the equivalent circuit is valid over this range. 

27. A plane wave falls normally on the acoustic material of Prob. 26. Plot, 
as a function of p.(0 < v < 2,000), the fraction of incident power absorbed by the 
material for the two cases mentioned. 

28. A ventilating duct 100 cm square in cross section is to be lined with mate- 
rial to attenuate sound of 1,000 cps most effectively. The acoustic material 
described in Prob. 26 is to be used with the material fastened directly to the duct 
walls (which are assumed rigid) and with the thickness d of material to be chosen 
for optimum results. How thick should the material be to give the greatest 
attenuation per foot for 1,000 cps sound? 

29. The outlet for a propellor test stand is a honeycomb of ducts, each of 6 
in square cross section, lined with material of resistivity 84, porosity 0.7, effective 
mass m p = 2, and thickness 5 cm. Compute the attenuation of the principal 
wave for 200 and 1,000 cps by the approximate formula, and also by use of Plate 
V. How long must the duct be to attenuate the 200-cps sound by 60 db? How 
much will this length attenuate 1,000-cps sound? 

30. Derive Eq. (31.5). Plot curves illustrating the properties of the equation. 

31. A duct has a square cross section 34.4 cm on a side. Use Plate V to 
calculate the optimum wall impedance to give maximum attenuation per length 
of duct at 400 cps for the least attenuated mode. Repeat the calculations for 
1,000 cps. 

32. A dynamic loudspeaker has a conical diaphragm with outer radius a, 
set in a large plane baffle. When the diaphragm is vibrating with frequency 
(u/2ir) and velocity amplitude V , the air in the plane of the baffle has a normal 
velocity approximately equal to 



F [*o(rO +*( f f)*i( r )] (r<a) 



where 6 is a constant related to the "height" of the cone and where the functions 
¥ are given in Eq. (28.3). Show that the distribution in angle of the radiated 
intensity at great distances from the diaphragm is 



T ^ - pcuVln 2 ' 



* + 



(?H 



where n = (2xo/X) and where the functions * are given in Eq. (28.4). Plot this 
distribution as a function of # f or n = 10 and (2irb/X) = 2. 



CHAPTER VIII 

STANDING WAVES OF SOUND 
32. NORMAL MODES OF VIBRATION 

In the preceding chapter we studied sound that is radiated into 
open space, tacitly assuming that there were no obstacles opposing its 
free flow outward. In most cases, however, sound generators are in 
rooms of size small enough so that the waves produced are reflected 
back and forth many times a second. When this occurs we cannot 
say that the waves generated are all radiating outward from the 
source; rather we must say that the source sets into motion one or 
more of the normal modes of vibration of the air in the room. 

This consideration makes it necessary to alter considerably our 
picture of the distribution of intensity about a source of sound. We 
cannot expect, for instance, that the intensity will vary inversely as the 
square of the distance from the source; in some rooms the intensity 
at some point far from the source may be considerably greater 
than at intermediate points. Nor can we expect that the intensity of 
sound in a room is simply related to the power radiated by the sound 
generator. The mechanical behavior of a loudspeaker, its mechanical 
and electrical impedance and total power radiated, will be practically 
unaltered by the properties of the room, but the intensity of the sound 
produced, and the distribution of this intensity, will be greatly altered. 

Room Resonance. — Our point of view with respect to the problem 
of sound in a room can be stated as follows: We look on the air in the 
room as an assemblage of resonators, standing waves that can be set 
into vibration by a source and that will die out exponentially when 
the source is stopped. When the source is started there will be set 
up a steady-state vibration, having the frequency of the source, and a 
transient free vibration, having the frequencies of the normal modes, 
which will die out. The steady-state vibration may be considered to 
be made up of a large number of the standing waves (just as the 
forced motion of a string can be built up out of a Fourier series) whose 
amplitudes depend on the frequency of the source, the "impedance" 
of the standing wave in question, and the position of the source in the 
room. The transient vibration will have the form necessary to satisfy 

381 



382 STANDING WAVES OF SOUND' [VHI.32 

the initial conditions in the room when the source is started and will 
therefore also be made up of many standing waves; but each normal 
mode of the transient vibration will vibrate with its own natural fre- 
quency. We shall study these frequencies in the present section. 

After the transient has died out, the steady-state vibration remain- 
ing will have only the frequency of the source. We shall study the 
shape of this vibration, how it differs from the free radiation from a 
source in the open, and how it can be built up out of the standing 
waves, in Sec. 34. When the source is turned off, these standing 
waves remain, only now they have their own natural frequencies, 
damping out exponentially according to their free vibration properties, 
and perhaps interfering with each other (making beat notes) as they 
do so. The damping of these free vibrations, which is called the 
reverberation, will be discussed in Sec. 33. 

From another point of view the room is a transmitter of sound from 
the speaker to the listener; a generalized horn, so to speak. As such, 
it ought to have more the qualities of the horn than of a musical 
instrument. In other words, it ought to transmit all frequencies 
equally well, and its transient characteristics must not distort the 
sound wave noticeably. This of course could be done by arranging 
to have the walls almost perfect absorbers of sound, so that they 
would not reflect sound back into the room. If this is done, however, 
we lose the effect of the walls in enhancing the sound level by reflection 
and require considerably greater output power in order to be heard 
throughout the room. In the open, sound intensity from a simple 
source diminishes as the square of the distance. In a well-designed 
room, however, the sound level 100 ft away from the speaker may be 
only 5 db below the level 10 ft away, in contrast to the 20-db drop 
which would occur in the open. The engineering problem in room 
acoustics is to design the shape and acoustic impedance of the walls 
so that the room is as uniform a sound transmission system as possible, 
without losing completely the reinforcing effect of the wall reflections. 

Statistical Analysis for High Frequencies. — We shall see later in 
this section that at low frequencies there are few resonances and at 
high frequencies there are many resonances in any given band width. 
Consequently, a sound of average wavelength about the same size as 
the room will excite only a few standing waves in the room; whereas 
a high-frequency sound, with average wavelength small compared 
with the room dimensions, will excite hundreds of standing waves. 
Obviously, the methods of calculation that are easy to use for the 
low-frequency case will be difficult for the high-frequency case, and 



Vin.32] NORMAL MODES OF VIBRATION 383 

vice versa. The situation is analogous to the difference between the 
methods of statistical mechanics, which deal with the average behavior 
of a large number of bodies, and those of ordinary mechanics, which 
deal with the detailed motions of one or two bodies. 

The statistical case, appropriate for high frequencies, is the simplest 
to analyze and is the one usually most useful in practical problems. 
Here hundreds of normal modes of the room are excited by the source; 
the sound is usually fairly uniformly distributed throughout the room 
and is traveling in all directions. When the steady state is reached, 
the sound at any place in the room can be represented as an assem- 
blage of a large number of plane waves, each with the frequency of 
the driver but going in all different directions. 

p(x,y,z) = f 2ir d<p fj Afot^e*-*-*"' sin# d& (32.1) 

where k is a vector of magnitude k = (co/c) pointing in the direction 
of the wave, given by the spherical angles (&,<p); vector r connects 
the origin of coordinates with the point (x,y,z) where the pressure is 
measured; and kr is their scalar product (k times r times the cosine 
of the angle & between them). 

The quantity A gives the amplitude and phase of the component 
plane pressure wave in the <p,#-direction at the point (x,y,z). In 
general this is a function of the coordinates (x,y,z) and of the direction 
(<p,&) of the component wave, corresponding to the fact that the 
sound is not usually completely uniformly distributed in position or 
direction. The average sound energy density at (x,y,z) is propor- 
tional to the square of A, averaged over all directions of k [for the 
energy density in a plane wave is (|A| 2 /2pc 2 )] so that 

w{x,y,z) = — 2 J dip J | A (*>,#) | 2 sin #d# 

The sound intensity in this case is denned as the net flow of energy 
per second into a square centimeter of area (it is to be noted that this 
is not identical with our definition of intensity for plane waves). 
Suppose that 3> is the angle of incidence of the plane wave Ae^ T on 
the square centimeter in question. The flow of energy into a square 
centimeter perpendicular to k would be (l/2pc) | A I 2 , and that flowing on 
the square centimeter with angle of incidence <S> is (l/2pc)|A| 2 cos3>, 
so that the net intensity is 

1{x,y,z) = j- c I dtp \ |^(^#)| 2 cos<S>sin<I><f<S> 



384 STANDING WAVES OF SOUND [VHI.32 

which is, in general, dependent on the orientation of the square centi- 
meter in question (since A is a function of direction). 

The criterion for a room that is satisfactory for hearing is usually 
that A be as independent of <p, &, x, y, and z as possible. A non- 
uniform A produces annoying irregularities in sound intensity, both 
as a function of frequency and also of position in the room. It turns 
out that irregularity of wall shape tends to make A more uniform. 
This is because irregularities tend to scatter sound waves in all direc- 
tions, particularly if they are about the size of a wavelength. Smooth 
concave walls produce nonuniform sound distributions, on the other 
hand, for they tend to focus sound and give rise to localized regions 
of high energy density and marked directionality. Even smooth flat 
walls are undesirable, as we shall see later in the chapter. 

Limiting Case of Uniform Distribution. — The simplest possible 
case to analyze is that where the sound is uniform in density and 
intensity throughout the room. This is a case seldom attained in 
practice but one which it is desirable to approximate, as we have indi- 
cated above. We assume that the room is large enough so that the 
sound from the speaker excites many natural modes of oscillation 
of the room, that the wall shape is irregular enough so that sound is 
scattered in every direction throughout the room, and that there is 
enough absorbing material on the walls- so that the sound does not take 
an unreasonable length of time to die out after the power source is 
shut off (how long a time is "unreasonable" will be discussed later). 

In this case the amplitude factor A of the component plane waves 
has a magnitude that is independent of (x,y,z) and of (<p,&), so that the 
energy density is 

w = — Li ergs/cm 3 (32.2) 

which is assumed to be independent of (x,y,z) even in the transient 
state. Likewise, the intensity is 

T = M^J- 2 = fe?\ ergs/cm* sec (32.3) 

independent of direction. 

We now assume that the energy density and intensity stay uni- 
form even during the transient stage. This is very rarely true in 
actual rooms, but it is a condition that is approached in acoustically 
satisfactory rooms, and it allows a great simplification to be made 



VIH.32] NORMAL MODES OF VIBRATION 385 

in calculating the intensity of sound in the room. For it means that 
we do not need to calculate the behavior of the amplitudes A(<p,&), but 
can concentrate our attention on the average quantities T and w. 
Our assumption corresponds to the requirement that w always equals 
(4T/c), even when both are functions of time. 

To determine how w and T depend on time and on the power 
output of any speaker in the room in this simplified case, we set up 
the equation for energy balance in the room. The power input into 
the room is, of course, the power output II (t) of the speaker, which 
may vary with time. Energy is lost by conversion of sound into heat 
in the air and at the walls. At high frequencies (above about 6,000 
cps) the air can absorb a fairly large amount of energy, particularly 
if the humidity of the air is high. But below about 2,500 cps the great 
majority of the energy lost is lost at the walls, and we can neglect the 
absorption in the air. Each portion of the wall will absorb a certain 
fraction of the energy incident upon it ; and since we have assumed that 
the intensity is uniform, we can conclude that the power lost to the 
walls is proportional to the area of wall surface and to the instantaneous 
intensity T(t). 

Absorption Coefficient. — The fraction of incident energy that is 
absorbed by a portion of the wall depends on the physical character- 
istics of the wall (e.g., its acoustic impedance) and on the distribution 
of sound in the room (e.g., the dependence of A on <p and #). When 
A is independent of <p and #, as we have assumed here, the fraction 
of power lost to the wall depends only on the wall and is called the 
absorption coefficient a of the material. The relation between a and 
the specific acoustic impedance of the material will be discussed 
shortly, after we have set up the energy-balance equation. Values 
of a for different materials and different frequencies are given in Table 
XIII at the back of this book. 

The sum of the products of the absorption coefficients a 8 of each 
material composing the walls, floor, and ceiling of the room times the 
exposed areas A 8 of each is called the absorption a of the room 

a = £ <x a A a (32.4) 

s 

It is not hard to see that the total power lost to the walls (when. the 
sound is uniformly distributed) is Ta. The total energy in the room 
at any instant is equal to the volume of the room V times the energy 
density (w = (4T/c)). Consequently, the energy-balance equation (for 
uniformly distributed sound below 5,000 cps) is 






386 STANDING WAVES OF SOUND [Vin.32 

(32.5) 

Reverberation. — The solution of this equation is 

T = (^p. J e~ act/iV I e a °" iV H(t) dr (32.6) 

indicating that the intensity at a given instant depends on the power 
output n for the previous (4F/ac) sec, but depends very little on the 
power output before that time (due to the exponential inside the 
integral). If the power n fluctuates slowly, changing markedly in a 
time long compared with (4V/ac), then the intensity T will be roughly 
proportional to II, and Eq. (32.6) reduces to 

Intensity level ~ 10 log ( — ) -f 90 db 

if II is in ergs per second and a in square centimeters. If n is in watts 
and a in square feet, the equation is 



© 



Intensity level ~ 10 log I -J +. 130 db 

This result is easily seen from Eq. (32.5), for if (dn/dt) is small then 
d(4:VT/c)/dt can be neglected and oT~H. The intensity is thus 
inversely proportional to the room absorption a, so that for steady- 
state intensity to be large, a should be small. 

On the other hand, if II varies widely in a time short compared with 
(4F/ac), then the intensity will not follow the fluctuations of II and the 
resulting sound will be "blurred." If the sound is shut off suddenly 
at t. = 0, for instance, the subsequent intensity will be 

T = %e- actMV (32.8) 

Intensity level = 10 log T + 90 - 4.34 ( |^ J db 

The "blurring" of rapid fluctuations of speaker power is known as 
reverberation. It is related to the fact that the intensity level in the 
room does not immediately drop to zero when the power is shut off, 
but drops off linearly, with a slope — 4.34(ac/4F) db per sec. This 
linear dependence of intensity level on time is typical of rooms with 



Vm.32] NORMAL MODES OF VIBRATION 387 

uniform sound distribution. We shall discuss cases, later in the 
chapter, that have more complex behavior. 

Reverberation Time. — The slope of the decay curve (the intensity 
level plotted against time after the power is shut off) indicates the 
degree of fidelity with which the room follows transient fluctuations 
in speaker output. The length of time for the level to drop 60 db 
is used as a measure of this slope and is called the reverberation time T. 
If lengths are measured in centimeters, this time is 



= 60 \§i ac ) 



sec 



When lengths are measured in feet, and for air at normal conditions 
of pressure and temperature, the reverberation time is 

T- 0.049 g) = <^ sec (32 . 6) 

When the speaker output changes slowly compared with T, then the 
intensity follows the output; but when the speaker output changes 
markedly in a time less than one-tenth of the reverberation time, then 
the fluctuations will not be followed. 

Therefore in order that the room transmit transient sound faith- 
fully, the reverberation time should not be large. For this require- 
ment a should be large, in contradiction to the requirement that a 
be kept small to keep the steady-state intensity large. A compromise 
must be worked out between these opposing requirements, a com- 
promise that varies with the size of the room. For a small room 
(V ~ 10,000 cu ft) T can be as small as 1 sec and the average intensity 
will still be satisfactorily high; but for a large room (V c^ 1,000,000 
cu ft) T may need to be as large as 2 sec for the intensity to be high 
enough throughout the room. If the room is used primarily for speech, 
which fluctuates rapidly, the reverberation time should be about two- 
thirds of this, for if the hall is large the intensity can be increased by a 
public-address system. If the room is used chiefly for music, we can 
allow more reverberation without detriment (in fact the music does 
not sound "natural" unless there is a certain amount of reverberation). 

Thus an analysis of an extremely simplified example of sound in 
a room indicates the sort of compromise between reinforcement and 
absorption that must be reached for any sort of room, even if 
the sound is not uniformly distributed throughout its extent. The 
analysis has also indicated that a useful criterion to indicate the 
degree of uniformity of the sound distribution is the shape of the decay 



388 STANDING WAVES OF SOUND [VIII.32 

curve for the sound after the source is shut off. If this is a straight line 
(on a decibel scale), then the chances are that the sound is fairly evenly- 
spread throughout the room; but if it is a curve, then it is certain that 
the sound is not uniformly distributed, either in space or in direction of 
propagation or both. 

Absorption Coefficient and Acoustic Impedance. — Before we finish 
our discussion of the idealized case of uniform distribution of sound, 
we must compute the relationship between the specific acoustic imped- 
ance of the wall material and the absorption coefficient a. 

As stated above, this quantity is the average fraction of power 
absorbed by the wall when sound is falling on it equally from all 
directions. 

To obtain this average, we go back to our discussion of the ampli- 
tudes A(<p,&). Suppose that we choose <p and t? so that the polar axis 
is perpendicular to the wall (assumed plane) and so that t? is the 
angle of incidence of the wave of amplitude A(<p,&). In the case we 
are at present considering, A is independent of <p and t?, so that the 
power falling on a unit area of wall is 

T = ^- \ d<p |A| 2 cost? sin t?dt? = ^ |A| 2 

2pc Jo Jo pc ' ' 

But from Eq. (30.7) we can show that the fraction of power lost by a 
wave of angle of incidence t?, on reflection from a plane surface of 
specific acoustic impedance z (acoustic admittance ratio rj = % ~ *<0, is 



a(t?) = 1 



77 — cost?) 2 4k cost? 



y] + cost?) (k -+- cost?) 2 + <r 2 



Therefore the average value of a, which is to be used in the case 
of uniform sound distribution, is given in terms of the acoustic con- 
ductance and susceptance ratios of the wall by the formula 

a = T I d<p \ «WI^I 2 cost? sin t?rft? 

(32.10) 

Values of this quantity, in terms of values of acoustic impedance ratios 
f = (1/r?) = 6 — ix = \^\e~ i<p = (z/pc), can be obtained from the 
contour plot in Plate VI at the back of the book. This plot shows 
that the maximum value of absorption coefficient (a = 0.96) comes 



Vin.32] NORMAL MODES OF VIBRATION 389 

when the specific impedance is a pure resistance, a little bit larger than 
pc{$ ^ 1.25). As the impedance is increased or decreased from this 
value, the absorption coefficient diminishes, and at very large values 
of f, a is approximately equal to 8k = 86/(6* + x 2 )- There is no 
set of values of k and <r for which a is unity. A plane wave can be 
completely absorbed by material of proper impedance; a mixture of 
plane waves can never be completely absorbed. 

Therefore when the room is so designed (irregular in shape, high 
frequencies, sufficient absorption) that the sound in it is fairly uni- 
formly distributed throughout its volume, then the acoustical charac- 
teristics of the room are given by Eqs. (32.6) to (32.8) and the slowness 
of response to transient sounds is measured by the reverberation time; 
the absorption coefficient entering into these equations is given in 
terms of the physical properties of the wall material by Eq. (32.10). 

If, however, the sound is not uniformly distributed, then Eqs. 
(32.6) to (32.8) will not be valid, and Eq. (32.10) for the absorption 
coefficient will have no application; in fact the term "absorption 
coefficient" will have no application. To analyze this less idealized 
(and often encountered) case, we must return to our study of the 
individual standing waves in a room. 

Standing Waves in a Rectangular Room. — To commence the study 
we consider a limiting case of another sort, a room with perfectly 
smooth, rigid walls. Here we are sure that the sound will not be 
uniformly distributed. We choose a rectangular room with sides 
l x , ly, hy for simplicity, and note that the boundary condition for a 
rigid wall is that the air velocity perpendicular to the wall is zero at 
the wall. 

The wave equation in rectangular coordinates is 
d 2 p d 2 p , d 2 p _ 1 d 2 p 
dx 2 + 7Jy~ 2 + ~dz 2 ~ ~c-~dt? 

If we choose the origin at the mid-point of the room, the standing 
waves must have a symmetry about the origin. The solution of the 
wave equation is 

cos/ x' 

p= . Itfa;- 

sm\ c, ^ x „ , ^ x „, ^ (32U) 

v = 2^ \/o)l + u>l + <a\ 

where either the cosine or the sine can be used. In either case the 
pressure wave behaves in the same manner at x = l x /2 as at # = — l x /2, 
etc. 



°) eos L«) cos L*) 

:/ sm\ c/ sm\ c/ 



390 STANDING WAVES OF SOUND [VIIL32 

The velocity in the x-direction is 



-am/ *W jAcmAA 

cos\ c/ sm\ "c/sm\ c/ 



This must be zero at a; = ± Z x /2. For the sine function, corresponding 
to the cosine function in the pressure wave, to be zero at x = +l x /2, 
03 x must have the following values: 

03 X = j n x T (n x = 0, 2, 4, 6 • • • ) 

In this case u will also be zero at x = —l x /2. For the cosine function, 
corresponding to the sine function in the pressure wave, to be zero 

at x = ± l x /2, 

<*x = f n x ir {n x = 1, 3, 5 • • • ) 

i x 

Therefore, the characteristic functions for the rectangular room are 
the functions given in Eqs. (32.11) with the following characteristic 
values of the co's and of the frequency: 



(32.12) 



When n x is an even number, cos (co^/c) is used in the expression for 
p; when n x is odd, sin (co*x/c) is used; and similarly for n y and n 2 . 

The normal mode corresponding to any particular set of values of 
n x , n y , and n 2 can be produced by starting a plane wave in the direction 
given by the direction cosines u x /u, «„/«, and co 2 /co and letting it be 
reflected from the various walls until it becomes a standing wave. If 
the values of w x , <a y , co z , and co = 2ttv are related to n x , n y , and n z in the 
manner specified by Eqs. (32.12), then the reflected parts of the wave 
will combine in such a manner that the resulting motion oscillates with 
simple harmonic motion. If we start any other sort of wave, its 
reflected parts will interfere with each other, and the motion will not 
be periodic, so it will not correspond to a normal mode of vibration. 

Distribution in Frequency of the Normal Modes. — The last expres- 
sion in Eqs. (32.11) suggests that v be considered as a vector with 
components (« x /2jt), (ay/tor), and (»,/2jr). The direction of the 
vector gives the direction of the wave producing the standing wave, 
and the length of the vector the frequency. A normal mode of oscil- 



co x 


_ OK 

"X 


CTT 
<t) y — -y- tl y , 

h 


0) z 


C7T 

= j- n z 

"Z 




(n x , 


n y , n z = 0, 1, 


2,3 


• • • ) 


V 


CO 


m+( 


'nA 2 


<m 



VIII.32] 



NORMAL MODES OF VIBRATION 



391 



lation can therefore be considered as a point in "frequency space," 
whose x component is an integral number of unit lengths (c/2l x ), whose 
y component is an integer times (c/2l y ), etc. The length of the line 
joining this point and the origin is the frequency of the normal mode, 
and the direction of this line is the direction of the wave that can be 
used to generate the standing wave. Some of these "characteristic 
points" are shown in Fig. 88, and it can be seen that they correspond to 
the intersections of a rectangular lattice with x, y, and z spacings 
equal to (c/2l x ), (c/2l y ), (c/2l z ), respectively. It can also be seen that 
all the normal modes are included among the points in the octant of 
space between the positive v x , v y , 
and v z axes, for any of the waves 
of frequency v with directions 
corresponding to 



(cn x /2l x , cn y /2ly,cn z /2lz), 

( — cn x /2l x , cn y /2l y , cn z /2l z ), 

(cn x /2l x , —cn y /2l y , cn z /2l z ), etc., 

will generate, by reflection, the 
same standing wave. 

This picture of a lattice of 
characteristic points in frequency 
space is extremely useful in dis- 
cussing the number and type of 
normal modes having frequencies 
within a given frequency range. 
For instance, since there are 
(8F/c 3 ) lattice cells per unit vol- 
ume of frequency space (V = ld y l z ), there will be, on the average, 
(8V/c s )(irv 3 /6) normal modes having frequency equal to or less than 
v (the factor irv 3 /6 being the volume of an eighth of a sphere of radius 
v). The actual number of modes having frequency less than v varies 
in an irregular manner as v increases, being zero until v equals the 
smallest of the three quantities (c/2l x ), (c/2l y ), (c/2l z ), when it suddenly 
jumps to unity, and so on. 

Axial, Tangential, and Oblique Waves. — Referring to Sec. 31, and 
anticipating the results of the next section a little, we note that waves 
traveling "parallel" to a wall are affected by the wall (are absorbed 
by it, for instance) to a lesser extent than waves having oblique inci- 
dence. Therefore we separate our standing waves into three cate- 
gories and seven classes: 




Fig. 88. — Distribution of allowed fre- 
quencies in "frequency space" for a rec- 
tangular room of sides l x , l v , and l z . The 
length of the vector from the origin to one 
of the lattice points is an allowed frequency, 
and the direction of the vector gives the 
direction cosines of the corresponding 
standing wave in the room. 



392 STANDING WAVES OF SOUND [VUL32 

Axial waves (for which two n's are zero) 

x-axial waves, parallel to the x-axis (n y , n z = 0) 
?/-axial waves, parallel to the y-axis (n x , n z = 0) 
2-axial waves, parallel to the z-axis (n x , n y = 0) 

Tangential waves (for which one n is zero) 

y, z-tangential waves, parallel to the y, 2-plane (n x = 0) 
x, 2-tangential waves, parallel to the x, 2-plane (n y = 0) 
x, ^-tangential waves, parallel to the x, y-plane (n z = 0) 

Oblique waves (for which no n is zero) 

It will turn out that, even in the first approximation, waves of 
different classes have different reverberation times and, to the first 

30,- 



2 
"o 

•c 

E 
es 
CO 

.£ 
1 20 



10 



L=220, A=l80O,V=450O 
Bond Width 10 cps 




o jr-rprf 



150 



200 



100 
Frequency, v 

Fig. 89. — Number of standing waves with frequencies between v — 5 and v + 5 
in a room 10 by 15 by 30 ft. Irregular solid line gives exact values; dashed smooth 
curve is plot of Eq. (32.14), giving approximate values of dN. 

approximation, waves of the same class (with v's approximately- 
equal) have the same reverberation time. 

Consequently, it will be quite important to count the number of 
standing waves of a given class having frequency less than v. The 
representation in a lattice system is again useful here; for the axial 



Vm.32] NORMAL MODES OF VIBRATION 393 

waves have their lattice points on the corresponding axis in "frequency 
space" and the tangential waves have their points in the corresponding 
coordinate planes. Again the number of lattice points can be counted, 
or a "smoothed-out" average number can be computed. 

To take an example of practical interest: suppose that the source 
sends out a pulse of sound of frequency v and duration At. According 
to Eq. (22.18), if this pulse is to be transmitted in the room without 
serious distortion of shape there must be a sufficient number (it turns 
out that a "sufficient number" is more than 10) of standing waves 
with frequencies within a frequency band between v — (Aj//2) and 
pq + (Av/2), where Av = (l/At), in order to "carry" the sound. If, 
for instance, we should wish to have the room transmit adequately 
a pulse of length tV sec, we would be interested in counting the number 
of resonance frequencies of the room between *» — 5 and v + 5. If 
this number is less than 10 for a certain value of v , then a pulse of 
frequency v and of duration xtr sec would not be transmitted with 
fidelity in the room. If the number is larger than about 10, and if 
in addition the reverberation time of each the standing waves involved 
is less than about a second, then the pulse will be transmitted with 
reasonable fidelity. Figure 89 shows a curve (solid irregular line) 
representing a count of this sort for a room 10 by 15 by 30 ft. It 
indicates that a pulse of to sec duration would not be reproduced 
adequately unless its frequency were larger than about 150 cps. 

Average Formulas for Numbers of Allowed Frequencies. — It is 
quite tedious to count the individual allowed frequencies less than a 
given frequency or in a given frequency band, so that it is useful to 
obtain "smoothed-out" formulas for average values of the counts. 
This can be done by considering that each lattice point "occupies" 
a rectangular block of dimensions (c/2l x ), (c/2l y ), (c/2l g ) in frequency 
space, with the actual lattice point at the center of the block. Then 
the average number of points can be obtained by dividing the volume 
of frequency space considered by the volume (c 3 /87)(7 = l x l y h) of 
each block. 

As an example we can count up the numbers of different classes 
of waves having frequencies less than v. The average number of 
x-axial waves is just v divided by the lattice spacing in the v x direc- 
tion, (2vl x /c) [i.e., it is the number of blocks in a rod of cross section 
(c 2 /4l y l z ) and length v] and the average number of all axial waves with 
frequencies less than v is 



ZV ax 



© 



394 STANDING WAVES OF SOUND [VIII.32 

where L = 4(^ -f- l v + h) is the sum of the lengths of all the edges 
of the room. 

The average number of y, z-tangential waves is the number of 
blocks in a quarter of a disk, of thickness (c/2l x ) and of radius v, 
minus a correction to allow for the axial waves, which have been 
counted separately. This correction in volume is one half the space 
"occupied" by the y and the z axial lattice points, viz., 

{v C y$>v)(i y + u) 

The factor one-half comes in because only one-half the volume "occu- 
pied" by the axial lattice points is inside the angular sector formed 
between the y- and z-axes, which bounds the quarter disk. Therefore 
the average number of y, z-tangential waves having frequencies less 
than v is 

and the average number of all tangential waves with frequencies less 
than v is 



N --(cw)-'(€) 



where A = 2(1 J y + l x l z + l y l z ) is the total wall area. We are neglect- 
ing the corrections for the overlapping regions at the origin, v = 0, for 
they are independent of v and are small in magnitude. 

The volume "occupied" by the lattice points for the oblique waves 
of frequency less than v is the volume of one-eighth sphere minus the 
volume already counted for the other classes of wave : 

where V = l x l y L is the volume of air in the room. Therefore the 
total number of standing waves of all classes which have frequencies 
less than v is 

The correct value for JV fluctuates above and below this average 
value but is seldom more than one or two units away, unless the room 
is too symmetrical; this will be discussed later. 



VIII.32] NORMAL MODES OF VIBRATION 395 

Average Number of Frequencies in Band. — The number of stand- 
ing waves with frequencies in a band of width dv is obtained by differ- 
entiating the formulas given above: 

dN ax ,x — (2^/c) dv, etc. 

dN ax ~ (L/2c) dv 

dN ta , yz ~ [(2tv/c 2 )I v I z - (l/c)(ly + I,)] dv, etc. 

dN ia ~ [(ttM/c 2 ) - (L/2c)] dv } (32.14) 

dN ob ~ [(4™ 2 7/c 3 ) -■ (7rM/2c 2 ) + (L/8c)] dv 

dN ~ [(4rv 2 V/c») + (7r^/2c 2 ) + (L/8c)] dv 

£ = 4(Z a + ^ + Z,); A = 2(l x l u + ZJ, + l y l z ); V = l x l y l z 

The value of dN, obtained from this formula, for dv = 10 and for 
appropriate values of the other constants is shown as the dashed line 
in Fig. 89. It is seen that this curve is a good " smoothed-out " 
approximation to the correct step curve. At very high frequencies 
just the term proportional to v 2 is important. 

We notice that the average number of allowed frequencies in a band 
increases with the square of the frequency at the higher frequencies. 
If we assume that the average intensity of sound in a room (for a con- 
stant output source) is proportional to the number of standing waves 
that carry the sound (i.e., the number with frequency inside the band 
characteristic of the driver), then the intensity in the room increases 
as the square of the frequency, for high frequencies, according to 
Eqs. (32.14). This is very interesting, because the power output 
into free space from a simple source is proportional to v 2 , according 
to Eq. (27.4). Therefore the power transmitted from source to 
receiver in a room varies, on the average, with frequency as it does in 
the open; but superimposed on the smooth rise are fluctuations (as 
shown in Fig. 89) due to the fluctuations of the number of standing 
waves in the frequency band of the driver. These irregularities of 
response are more pronounced, the more symmetric is the shape of the 
room, or the narrower is the frequency band of the sound source. 

When dN, as given by Eqs. (32.14), becomes two or less, then the 
fluctuations become so large that they appreciably reduce the fidelity 
of transmission. For the room referred to in Fig. 89 this lower limit, 
for a band width of 10 cps, is about 50 cps; for a band width of 5 cps, 
it is about 100 cps; etc. 

The Effect of Room Symmetry. — We have mentioned several times 
in the preceding pages that the response curve of a room, as evidenced 
by the exact curves for dN, is more irregular when the room is more 
symmetrical. This is due to the increase in the number of degenerate 



396 



STANDING WAVES OF SOUND 



[VIII.32 



modes, standing waves with different n's that have the same frequency. 
As an example of the effect of degeneracies we can consider the sequence 
of rooms of dimensions l x , l y = ql x , l z = L/q, so chosen that the volume 

Table 4. — Characteristic Frequency Parameters (2lv/c) and Corresponding 

Quantum Numbers for Standing Waves in a Cubical Room of Side I 

and in a Room of Dimensions I, I \/2> Z/a/2 





Cubical 


room 




Non cubical room 


(2lv/c) 


n x 


n y 


n, 


(2lv/c) 


n x 


n y 


71* 




fl 








0.707 





1 





1.000 . 


° 


1 





1.000 


1 










k 





1 


1.225 


1 


1 





1.414 


(i 


1 






1 


1.414 


{o 



2 


1 






lo 


1 ' 


1 


1.581 





1 


1 


1.732 


1 


1 


1 


1.732 


{\ 





1 




f 2 










2 





2.000 





2 





1.871 


1 


1 


1 




lo 





2 




lo 










/2 


1 





2.000 


2 


1 




ll 


2 







{I 

1 


1 





2.236 


) 2 
)l 






1 
2 


2.121 
2.236 


3 
2 




1 




1° 


2 


1 


2.345 


1 


3 







\0 


1 


2 




{;. 





1 




f 2 


1 


1 


2.449 


2 





2.449 


1 


2 


1 




{o 


1 


1 




u 


1 


2 


2.550 


3 


1 




f 2 


2 





2.739 


i 


3 


1 


2.828 


2 





2 




ii 





2 




lo 


2 


2 


2.828 


4 







/3 










\i 


1 


2 




lo 


3 





2.915 


3 





3.000 


Jo 

)2 



2 


3 

1 


3.000 


P 






2 





I 2 


1 


2 




li 


4 







\1 


2 


2 











V = (l x ) z remains the same but the relative dimensions change as we 
change q. The natural frequencies are 



-&)Mf) 



+ q 2 n 2 t 



VUI.32] NORMAL MODES OF VIBRATION 397 

which change in spacing as we change q. We shall write down the 
lowest allowed frequencies for two rooms of this sequence: one for 
q 2 = 1, a cubical room, the most symmetric; and one for q 2 = 2, which 
is not symmetric. 

Table 4 gives the allowed frequencies and the combinations of 
integers (n x ,n y ,n z ) which label the corresponding characteristic func- 
tions for these two cases. We notice immediately the tendency of all 
the characteristic frequencies to "clump together" in the cubical- 
room case. Threefold and even sixfold degeneracies are common 
even at these low frequencies (for instance 2.236 and 3.000). These 
result in large ranges of frequency within which there is no character- 
istic value, so that the response is very irregular. In contrast, the 
case of q 2 = 2 never gives more than twofold degeneracies in the 
frequency range considered, and the allowed frequencies are therefore 
more evenly spaced along the scale. We note that, because the room 
volumes are equal, there are approximately the same number of 
frequencies equal to (3c/2Z) or less (28 in one case, 27 in the other) 
but the particular values of (n x ,n y ,n z ) included, and their order on the 
frequency scale differs. If we had picked an incommensurate value 
for q 2 (the cube root of 5 for instance), we would have had no degener- 
acies at all, and the allowed frequencies would have been still more 
evenly spaced along the scale. We can never get absolutely uniform 
spacing with a rectangular room, of course, because the lattice in 
frequency space is always rectangular. A room with irregular walls 
would correspond to a more random arrangement of lattice points in 
frequency space and, perhaps, to a more uniform response. 

Nonrectangular Rooms. — Our analysis of standing waves has 
depended to some extent on the fact that we have chosen to study 
rectangular rooms. This is not a serious limitation, for most rooms 
approximate a rectangular form. Nevertheless, it would be more 
satisfactory if it could be shown that Eqs. (32.13) and (32.14) hold 
for all room shapes. This cannot be done for several reasons. In 
the first place, although it is not difficult to generalize the quantities V 
(room volume) and A (area of walls) to rooms of other shape, the 
quantity L (total length of edge) becomes a problem. (For example, 
if L for a cylindrical room is just 4irR, what is L for a room of octagonal 
floor plan, or with a floor plan that is a polygon of a large number of 
sides — approaching a circular form — and what is L for a spherical 
enclosure?) 

In the second place it becomes progressively more difficult to 
define axial and tangential waves as the room shape is made more com- 



398 STANDING WAVES OF SOUND [VHI.32 

plex (this of course is another aspect of the reciprocal relationship 
between uniformity of room shape and uniformity of wave behaviors- 
all waves are oblique waves if the room is irregular enough). 

As an example of how these questions can be answered in one 
other case, we shall outline the solution for a cylindrical room, with 
circular floor plan of radius a and of height I. The solution of the 
wave equation in cylindrical coordinates is 



cos 



<t)'-(?) 



V = gin (m<p) cos^j J»yf) <r M «, " = ^ Va>i + «? (32.15) 

where z = is at one of the end "walls," the floor. The properties 
of the Bessel functions J m have been given in Eqs. (19.4) and (19.5). 
In order to have the z component of the particle velocity zero at 
z = and z = I, we must have the derivative of p with respect to z 
zero at these points. The derivative is zero at z = 0, because we 
have already chosen the cosine function. For it to be zero at z = I, 
we must have (wj/c) = n z ir, (n z = 0,1,2,3 • • • )• I n order to have 
the radial particle velocity zero at the cylinder walls, we must have 
(dJ m /dr) = at r — a. For this to be true, we must have ((o r a/c) = 
Tra mn where a mn is a solution of the equation [dJ ' m {ira) / 'da] = 0. The 
characteristic functions for the cylindrical room are therefore those 
given in Eq. (32.15), and the characteristic values are 

-j- J (n, = 0,1,2 • • • ) 

co r = I — ^_ J (where a mn is given in Table 5) 



c 
" = 2 



V(t)' + ( s S b )' ™ 



The quantities <x mn have already been mentioned in Eqs. (26.12) and 
(26.15), in connection with the cylindrical pipe. 

If I is not longer than 1.71a, the mode with the lowest frequency 
is a transverse one, with one diametric mode, the air "sloshing" back 
and forth across the cylinder. 

The next problem is to label the "axial" and "tangential" waves. 
The 2-axial waves are obvious ones; they correspond to those for which 
m = n = 0, where the motion is parallel to the z-axis. Similarly, 
the waves for which n z = are obviously parallel to the floor and 
ceiling and should, probably, be called <p, r-tangential waves, by 
analogy with the rectangular case. The waves for which motion is 



VIIL32] 



NORMAL MODES OF VIBRATION 



399 



entirely radial (parallel to r) for which n z = m — are those which 
focus the sound along the cylinder axis. They can perhaps be called 
r-axial waves, though they are not parallel to any wall. The waves 
for which n z = n = are those which travel close to the curved walls 
and have little motion near the cylindrical axis. These can be called 
<p-axial waves, though the air motion is not entirely in the ^-direction 
(perpendicular to r and to z) . We shall see in the next section that the 
r-axial waves are least absorbed by acoustic material on the curved 
walls and the ^>-axial waves are very strongly absorbed. 



Table 5. — Characteristic Values a mn for the Cylindrical Room Solutions 

OP dJ m (ira) /da = 








1 


2 


3 


4 





0.0000 


1.2197 


2.2331 


3.2383 


4.2411 


1 


0.5861 


1.6970 


2.7140 


3.7261 


4.7312 


2 


0.9722 


2.1346 


3.1734 


4.1923 


5.2036 


3 


1.3373 


2.5513 


3.6115 


4.6428 


5.6624 


4 


1.6926 


2.9547 


4.0368 


5.0815 


6.1103 


5 


2.0421 


3.3486 


4.4523 


5.5108 


6.5494 


6 


2.3877 


3.7353 


4.8600 


5.9325 


6.9811 


7 


2.7304 


4.1165 


5.2615 


6.3477 


7.4065 


8 


3.0709 


4.4931 


5.6576 


6.7574 


7.8264 



«m0 — (w/tt) 

OCmn ^ n + \m + \ 



(m » 1) 

{n ^> 1, n > m) 



Except for m = 0, each mode is doubly degenerate, corresponding 



cos 



to the duality . (m<p) in the characteristic function. 

J sin v 



This can be 



represented formally by allowing m to take on negative as well as 
positive values, with negative m corresponding to the cos(m^>) case 
and positive m to the sin(m<p) case (where, of course, a-^ n = a mn ). 

Frequency Distribution for Cylindrical Room. — A certain amount of 
arbitrariness comes in when we set up the lattice in frequency space 
for computing allowed frequencies. Here the standing wave is made 
up of waves with components in all directions perpendicular to the 
z-axis, so that there is no obvious direction that corresponds to the 
v m , n ,n z vector, as there was for the rectangular room. 

To be strictly accurate, we should represent each wave by a portion 
of a circle in frequency space, corresponding to the fact that a cylin- 
drical wave is made up of a distribution of plane waves in a variety 
of directions. Formally, however, we can set up arbitrarily a sym- 



400 



STANDING WAVES OF SOUND 



[VIII.32 



metrical grid of points on the (v x ,v y ) plane, each the proper distance 
from the origin, which can represent all the allowed frequencies. 

A possible representation is given in Fig. 90, where the r-axial 
waves (w = 0) lie along the main diagonal and the doubly degenerate 

^-axial waves (n = 0) lie along 
the v x and i^-axes. This type 
of representation is useful for, 
as we shall see later, the ^>-axial 
waves depart most radically from 
the average reverberation behav- 
ior, being much more rapidly 
absorbed than the rest. In fact 
it is not unreasonable to expect 
waves tangential to any concave 
curved wall to be absorbed more 
rapidly than oblique waves, and 
waves tangential to a flat or a 
convex curved wall to be ab- 
sorbed less rapidly than the rest. 
The third dimension of the 
lattice (the ^-direction) is ob- 
tained by adding similar two- 
dimensional arrays, one above 
the other, spaced a distance (c/2l) apart. The process of counting 
the number of standing waves with frequencies less than v is now 
carried out as before. The "smoothed-out" formula is somewhat 
more difficult to calculate, for the asymptotic formulas for the a's for 
large m and n must be utilized. The equations analogous to Eqs. 
(32.14) are 




(c/a) 



(Zc/a) 



Fig. 90. — Formal representation of 
allowed frequencies in a cylindrical room. 
Allowed frequencies given by radial dis- 
tance from origin to one of the small circles. 
Lattice is made up of a parallel sequence of 
similar planes, spaced (c/2h) apart. 



dN aXyZ ~ (22/c) dv 
dNax, v ^ (2xa/c) dv 
dN ta ,r<p^[2irv/c 2 )(ira 2 ) - (7ra/c)]dv 
dN ta , z<p ~[(2rv/c 2 )(2T<il) - (2va/c) - (2l/c)]dv 
dN ~ [(±tv 2 V/c s ) + (tvA/2c 2 ) + (L/8c)] dv 
V = iraH; A = 27ra 2 + 2-wal; L = lira + U 



(32.17) 



This corresponds to our definitions of volume V and area A for the 
earlier case; but L has a form that could not easily have been predicted. 
It appears likely that the equation for dN is a general one as far 
as the terms in v 2 and v go, but that the constant term (if it is needed) 
must be worked out for each case. It is seldom that this constant 



Vm.33) DAMPED VIBRATIONS, REVERBERATION 401 

term in the series is needed, however. Indeed, in many cases only the 
first term (4.irv 2 V/c z ) dvis sufficiently accurate. It can be proved that 
this term has the correct form for a room of volume V of any shape 
whatsoever. 

The curve for dN from Eq. (32.17) is roughly similar to Fig. 89, but 
the allowed frequencies in this case are not so evenly spaced as with the 
rectangular room, so that the actual value of dN does not approach the 
average curve for so low a frequency as in the rectangular case. This 
is due to the fact that at the higher frequencies there are many modes 
having nearly the same frequency, the asymptotic form of a showing 
that a m -i t n+\ ^ <w, etc. This "bunching together" of the allowed 
frequencies is essentially due to the symmetry of the enclosure about 
the cylindrical axis, so that a number of standing waves can be set up, 
having different directions but having about the same frequency. 

The distribution in frequency of the standing waves in a spherical 
enclosure is even less regular than that in a cylinder, and still higher 
frequencies must be u3ed before the actual curve for dN is smoothed 
out and approaches the average curve. Such an enclosure would not 
be satisfactory for use as a room, because of the fluctuation in its 
resonating characteristics as the frequency is changed. In fact, we 
can state as a general rule that the more symmetrical an enclosure is, 
the larger will be the range of frequency over which the resonance 
properties fluctuate, and the less desirable will it be for use as an 
auditorium. The curve for dN for a room of the same volume as 
that used for Fig. 89, but having very irregular walls, will approach the 
smooth average curve still more rapidly than the curve shown does. 

Irregular walls also serve to spread out the sound energy more 
or less uniformly over the room. Most of the high-frequency standing 
waves in a rectangular enclosure have an average amplitude that is 
nearly the same everywhere in the room, but many of the standing 
waves in a spherical room have larger amplitudes near the center than 
near the wall. In rooms having smooth concave surfaces, focal points 
of considerable excess intensity may occur to render the room undesir- 
able as an auditorium. 

33- DAMPED VIBRATIONS, REVERBERATION 

The standing waves discussed in the last section do not continue 
in the enclosure forever; they lose energy and damp out. Over the 
range of frequencies useful in acoustics, most of this energy is lost at 
the walls, some of it being transmitted through to the outside, and 
some going to heat the wall. At very high frequencies an appreciable 



402 STANDING WAVES OF SOUND [VIII.33 

portion of the energy is lost in heating the gas itself, but for the fre- 
quencies in which we are interested we can consider that this loss is 
negligible compared with that lost at the walls. 

We have already computed [see Eq. (32.6)] the reverberation in a 
room with sufficiently irregular walls so that the sound is uniformly 
distributed throughout its volume. In this section, we shall discuss 
the reverberation of individual standing waves in a room. Pursuing 
the analogy mentioned at the beginning of Sec. 32, the analysis of this 
section is more analogous to the detailed analysis of ordinary mechanics 
than to the calculations of statistical mechanics. 

Rectangular Room, Approximate Solution. — We start first with the 
rectangular room of sides l x , l v , l z , with walls (and floor and ceiling) 
that are fairly hard (specific acoustic impedance large compared with 
pc). In this case the standing waves are not much different in form 
from those given in Eq. (32.11) for a rectangular room with rigid walls: 



_ cos /im x x\ cos f -Kn y y \ cos / irn z z \ 
* N ~" sin\ l x ) sin\ l y ) sin\ l t ) 

— «vfc) ,+ fey +&)■-*"•' 



(33.1) 



where the origin of coordinates is at the center of the room. Even 
values of n correspond to the use of the cosine in \f/ N , odd values of n 
correspond to the use of the sine. If the wall impedance has a resis- 
tive term, energy is lost and the free vibration of each standing wave 
must contain an exponential factor e~ kt , corresponding to the loss of 
energy. In Eq. (4.8) we related the damping factor k for a simple 
oscillator to the fractional loss in energy of free vibration per second. 
This relationship holds for any oscillation and can be used here to 
obtain a first approximation to the damping constant for a standing 
wave of sound. 

The total energy of the standing wave corresponding to the trio of 
numbers N = n x , n„, n z is [using Eqs. (33.1) and (25.3)] 

^' = 5 J J/ k + ^ "*]*""*! (33 . 2) 

VA-x * _ /i / \ \ 

= K — 2» An ~~ W € »* e »» e »J j 

where e = 1 but d = e 2 = e 3 = • • • =2, and where V = U y l z . 
We notice that the axial waves have more energy content than the 



VIII.33] DAMPED VIBRATIONS, REVERBERATION 403 

tangential, and the tangential have more than the oblique. This is 
because each cosine or sine factor has mean-squared value %, whereas a 
constant factor (n = 0) has average value 1. If there are nodal planes 
in all three directions (oblique waves), the average value will be least. 
The energy lost per second at a square centimeter of wall surface is 
the average value of the pressure p times the velocity normal to the 
wall; this equals (/cp 2 /2pc), where k is the acoustic conductance ratio 
of the wall material. The rate of total power loss at the walls is 
therefore 



= WcSf K 



(S)p 2 (S) dS 

where the integration is over every element of all area for which k is not 
zero. 

Referring to Eq. (4.8), we see that the damping constant k is 
equal to i times the ratio between power loss and total energy of 
vibration. In the present case, we can write the equation for k as 
follows: 

k ^ Ac [average value of tap* over all walls] 
N 2V [average value of ^* over volume of room] ^ ^ 

This expression is correct to the first order in k for standing waves in 
rooms of any shape. A is the wall area and V is the room volume. 
The "Q of the room" for the iVth wave [see Eq. (4.4)] is (o> N /2k N ). 

Wall Coefficients and Wall Absorption.— The exponential decay 
factor for energy or intensity of the standing wave will therefore have a 
term e~ 2k ^ (because energy or intensity is proportional to the square of 
p). Comparison of this with Eq. (32.8) for the statistical case of 
uniformly distributed sound shows that the quantity that takes the 
place of the average absorption coefficient a in the case of a single 
standing wave is the average wall coefficient a N , where 

5 ^ average value of 4k^ over all walls 

average value of ^ over room volume (33.4) 

to the first approximation in the (assumed small) quantity k. In the 
statistical case, a was independent of the sound; in the present case, 
a N depends on the standing wave as well as on the wall materials. 

These formulas show that absorbing material (material with a 
nonzero value of k) should be placed at those parts of the wall where 
the pressure is largest if it is wished to absorb the standing wave most 



404 STANDING WAVES OF SOUND [VTII.33 

rapidly. Material placed at the corners of the room is twice as, effec- 
tive, on the average, as if it were placed at other locations, because all 
standing waves have pressure maxima there. 

One might be tempted to place strips of absorbing material in a 
regular pattern on the wall, corresponding to the maxima of a particu- 
lar \f/ N , in order to damp out the wave having a particular trio of values 
of n x , n y , n z . This would *absorb that particular standing wave 
rapidly, but other waves, with different values of the w's, would not 
have their maxima coincide with the strips of absorbing material, and 
so would be much less rapidly damped. Whenever there is a wide 
variation in the reverberation times of different standing waves, the 
acoustical conditions in the room turn out to be unsatisfactorily 
nonuniform; so any regular pattern of patches of absorbing material 
will usually lead to unsatisfactory acoustics in one or more frequency 
ranges. In order to maintain as much acoustic uniformity as possible, 
it is best to cover several walls completely with material, or else to 
distribute patches of material in a nonuniform manner over all the 
walls. A random arrangement of patches has the additional advan- 
tage of scattering sound and tending to distribute it more uniformly 
throughout the room. This scattering effect is not calculable by first- 
order approximations, but will be discussed briefly later in this section. 

If each wall has a uniform acoustic impedance, or if its impedance 
varies in a completely random manner, then the maxima and minima 
of each standing wave fall with equal likelihood on the most absorbing 
parts of the wall. In this case the average value of (4k^) is equal to 
the product of the average value of (4k) times the average value of 
(^), each averaged separately over the wall in question. In this 
case Eqs. (33.3) and (33.4) can be further simplified. 

Suppose that we multiply the acoustic conductance ratio of each 
wall material by eight times its exposed area and sum over all the area 
of the two walls perpendicular to the z-axis, calling this quantity 

a*~ Y (8k s )A* (33.5) 

x walls 

the absorption (first-order approximation) of the x walls. Comparison 
with Eq. (32.4) shows that, to the first order in k, the quantities (8/c) 
for the rectangular room correspond to the absorption coefficients a 
in irregularly shaped rooms with uniformly distributed sound. The 
quantities a y and a z can similarly be defined. 

Finally, utilizing the properties of the squares of sine and cosine to 
obtain average values for \[/%, we obtain approximate formulas for the 



Vra.33] DAMPED VIBRATIONS, REVERBERATION 405 

damping constant kn and corresponding reverberation time for the 
standing wave specified by N = (n x , n y , n e ): 

0.0497 ^ (33,6) 



T N 



\^n x dx + -$*n Jly + Tt^nflz) 



where Tn is given for dimensions in feet and for air at standard condi- 
tions. These formulas are good only to the first order in the (sup- 
posedly) small quantities k; more accurate formulas will be discussed 
later in the section. 

The factors ^e„ (= ^ for n = 0; = 1 for n > 0) constitute the 
difference between this formula and that of Eq. (32.9) for the room with 
uniform distribution of sound. When n x (for instance) is zero, the x 
walls contribute one-half as much (to this approximation) to the 
absorption of the wave as they do for waves with n x > 0. This is 
due to the properties of the functions \f/ N ; the ratio between the average 
energy in the room and the average intensity on the x walls for n x = 
(waves parallel to the x walls) is one-half what it is for the waves that 
reflect from the x walls (n x > 0), so that for the parallel waves the 
fraction of total energy absorbed by the x walls is one-half the usual 
fraction. 

Reverberation Times for Oblique, Tangential, and Axial Waves. — 
These formulas illustrate some of the effects of room regularity on 
sound absorption. A rectangular room has walls smooth enough so 
that some standing waves move parallel to two or more walls. For 
these standing waves the walls parallel to the motion have less effect 
on the damping (one-half as much, to the first approximation) as they 
do for oblique waves. This behavior is, in general, true for smooth 
plane walls and smooth convex walls, in rooms of other than rectangu- 
lar shape (by "smooth" we mean that the radius of curvature of any 
part of the Wall is considerably larger than a wavelength). For 
smooth concave walls it turns out that the waves "parallel" to its 
surface are more strongly absorbed than are oblique waves. In either 
case, however, the waves moving tangential to one or more walls are 
absorbed at a different rate from the rest. 

As an example, it can be mentioned that the standing waves in a 
room with a floor plan that is an isosceles right triangle also has three 
classes of tangential waves: those tangential to the plane-parallel 
floor and ceiling (for which the floor and ceiling absorptions are 
multiplied by one-half), those "tangential" to the two walls at right 



406 STANDING WAVES OF SOUND [VHI.33 

angles (for which the absorption for these walls is multiplied by 
three-quarters), and those tangential to the diagonal wall (the factor 
for this wall is also three-quarters) . Another example is the cylindrical 
room, which will be worked out later. 

In a rectangular room (or indeed in any room with "smooth" 
walls), if a sound source exciting a number of standing waves is sud- 
denly turned off, the various standing waves will take different lengths 
of time to die out. The oblique waves will damp out most quickly, 
then the tangential waves (if the walls are not concave), and the axial 
wave parallel to the most absorbent walls will linger nearly twice as 
long as the oblique waves. Therefore, the distribution of sound in 
the room will change as it dies out, being at first fairly random, but 
eventually being mostly parallel to the absorbent walls. Experience 
shows that this behavior, if the differences of reverberation time are 
pronounced, is acoustically objectionable. 

Therefore, if a rectangular room cannot be changed in shape by 
introduction of irregularities, and if it is to be used for purposes of 
speech or music, requiring good acoustics, the absorbing materials 
should be placed about equally on all walls, so that a x , a y , and a z are 
nearly equal. This will produce the least difference between rever- 
beration times for a given value of total absorption. Irregularities of 
wall shape, to scatter the axial and tangential waves, would improve 
the acoustical conditions considerably, however, and these should be 
introduced whenever possible. 

Decay Curve for Rectangular Room. — If the source in the room 
emits sound in a frequency band dv (between v — \dv and v + \dv), 
there will be excited, on the average, dN " & oblique waves, dN ta tan- 
gential waves, and dN ax axial waves, where the formulas for the 
quantities dN are given in Eqs. (32.14). If each of these wave groups 
is excited and contributes to the resulting average intensity by an 
amount proportional to the dN's, we can write down a formula for 
the average attenuation of sound in a rectangular room 

T ~ ? D x (t)D v (t)D.(t) 



where a x , a y , a z are defined in Eq. (33.5). To this approximation, 
therefore, the sound-decay curve (the intensity level vs. time) for a 
rectangular room can be expressed in terms of a sum of individual 
terms, 10 log(D x ), etc. Each of these terms starts at t = as a straight 
line with negative slope proportional to a x . A time (8V/ca x ) In (c/2vl x ) 



Vm.33] DAMPED VIBRATIONS, REVERBERATION 407 

later the curve has a break, ending up, beyond this, as a straight line 
with slope proportional to \a x , half the initial slope. Since each of 
the additive terms has it's "break" at a different time (unless the room 
is completely symmetrical), the resulting decay curve is quite far from 
being the straight line that was indicated in Eq. (32.8) for a uniform 
distribution of sound in an irregular room.. 

At high enough frequencies, the "breaks" come late enough so 
that the first 20 or 30 db of the curve is nearly straight, with a slope 
and indicated "reverberation time" corresponding to the oblique 
waves. If, by chance, this result were assumed to correspond to that 
given in Eq. (32.8) for an irregular room with diffuse sound, then the 
quantities (8k) (where k is the wall-conductivity ratio) would be 
presumed to equal the absorption coefficient a. We have seen, how- 
ever, that this is an inaccurate correlation, which may work fairly 
well for very stiff walls [k very small; see the comments on Eq. (32.10)] 
but which fails for more absorptive walls, when the break in slope of 
the decay curve is more pronounced. 

If most of the absorbing material is concentrated on the two 
opposite walls of a room, then those standing waves that do not 
reflect from the absorbing walls will take about twice as long to die 
out as do all other waves. When a sound with a "spread" of fre- 
quency is used to excite a number of standing waves at the same time, 
the dying out of these waves after the source is shut off is a rather 
complicated phenomenon. When only two or three standing waves 
are excited, these waves as they die out may alternately reinforce and 
interfere with each other, owing to the fact that they are of slightly 
different frequency. The intensity will then fluctuate instead of 
decreasing uniformly, the sort of fluctuations obtained depending on 
the position of source and microphone in the room and on the manner of 
starting the sound. 

If more than three standing waves have been excited, these 
fluctuations will be more or less averaged out, and the resulting 
intensity will first diminish uniformly at a rate dependent on the 
reverberation time of the standing waves which are reflected from all 
six walls. After these waves have died out, the rest of the sound, due 
to waves not striking the most absorbent walls, remains and dies out 
more slowly. The intensity level as a function of time approximates a 
broken line, the steeper initial part corresponding to the most of the 
standing waves, and the less steep later part due to the waves that do 
not strike some walls. 

In such cases the term "reverberation time " has a specific meaning 



408 



STANDING WAVES OF SOUND 



[VHI.33 



only in connection with the damping out of single normal modes, i.e., 
in connection with the slopes of the two portions of the broken line of 
intensity level against time. The actual length of time that it takes 
for the intensity level to drop 60 db will depend on the relative amounts 
of energy possessed by the rapidly and the slowly damped standing 



waves. 



Figure 91 illustrates these points; it shows curves of intensity level 
in a room 10 by 15 by 30 ft as a function of the length of time after 
the sound is shut off. The two smallest walls are supposed to be much 
more absorbent than the other four. The solid line is the curve for 




Time- 



Fig. 91. — The decay of sound in a room with two opposite walls more absorbent 
than the rest. Solid curve shows the average decay of a large number of normal 
modes; dotted line shows the interference effects possible when only two normal modes 
have been excited. 

intensity when the room has been excited by a tone with wide enough 
frequency "spread" to excite 10 standing waves about equally, so 
that a smooth decay results, and the "break" in the curve is apparent. 
The dotted line is the curve when only two standing waves are excited, 
the resulting fluctuation, due to interference, masking the exponential 
decay. 

These interference oscillations and breaks in the curve for decay of 
intensity level are present even when all the walls are about equally 
absorbent, but they are less pronounced. 

Cylindrical Room. — We can use the characteristic functions given 
in Eqs. (32.15) and (32.16) to compute the damping constants for the 
different standing waves in a cylindrical room of radius a and height I. 
If the absorbing material is distributed either at random or com- 
pletely uniformly over each wall, the quantity k can be averaged 



Vm.33] DAMPED VIBRATIONS, REVERBERATION 409 

separately as before. The wall behavior is then expressed in terms of 
the absorptions. 

a z = X(Sk s )A s (over flat floor and ceiling) 
a r = 2(8k s )A s (over cylindrical surface) 

The wave properties are evidenced through the averaged values of 
p 2 over the walls and throughout the volume. The mean-square 
value of the Bessel function is 

Therefore the damping constant for the iVth wave (N = m,n,n z ) in 
the cylindrical room is 

*'^[***+ l- ( i/„^, ] (33-7) 

which is to be compared with Eq. (33.6). Values of a mn are given on 
page 399. 

The r, ^-tangential waves (n z = 0) are absorbed half as much by 
the flat floor and ceiling as are the waves for n z > 0, as is the case for 
the flat walls in a rectangular room. The curved cylindrical walls 
absorb most waves less effectively than do flat walls; for the factor 
l/[2 — 2(m/7raw) 2 ] is smaller than unity unless n is small. Most 
waves in a cylindrical room are focused toward the center, away from 
the curved wall, and so are not affected as much by this wall as they 
would if it were flat. 

In contrast, the standing waves for which n = (the ^,2-tangential 
waves) are much more strongly absorbed than if the wall were flat; 
for these waves l/[2 - 2(m/Tra mn ) 2 ] is much larger than unity. These 
(p, z-tangential waves move "parallel" to the curved wall and have 
large amplitude only near this wall, so it is natural that it will absorb 
them strongly. 

When a sound source in a cylindrical room is shut off, the <p,z- 
tangential waves damp out most rapidly and the r-axial waves least. 
Therefore the sound nearest the curved walls vanishes first, eventually 
leaving the radial wave motion, which focuses the sound along the 
axis and which attenuates most slowly. This effect was mentioned on 
page 399, where we discussed the classes and numbers of standing 
waves in a cylindrical room. 

Second-order Approximation.— In the foregoing calculations, we 
took into approximate account the effect of the flow of air into and out 



410 STANDING WAVES OF SOUND [VIII.33 

of the wall surface on the power absorption at the surface, assuming 
that the standing wave shape is not changed by the yielding of the 
walls. Actually, of course, the fact that the wall impedance is not 
infinite does affect the shape of the standing waves, and the second 
approximation to the solution must take this into account. 

How this correction is to be calculated can be seen most easily 
by assuming that we gradually increase the admittance of the walls, 
starting from zero and ending at the actual distribution of admittance 
which is under consideration. When the walls are all rigid, the stand- 
ing waves are given by the characteristic functions ^n given in Eq. 
(33.1) or (32.15), no energy is absorbed by the walls, and the natural 
frequencies are (o)n/2tt), with the characteristic values ca N given in 
Eqs. (32.12) or (32.16). As the wall admittance is increased slightly 
the shape of the characteristic functions is not changed very much at 
first; there is a slight air velocity into the wall, of an amount equal to 
the specific acoustic admittance of the wall (still small) times the 
pressure at the wall surface. This motion of air produces radiation 
out into the room; in other words the original standing wave is modified 
by the scattered waves produced by the reaction of the wall to the 
standing wave. 

Suppose that we start with the standing wave \f/N(P)e~ iaNt [where 
P stands for the point (x,y,z)]. When the wall admittance is not 
zero, the air velocity out from the wall at the point Q = (X, Y,Z) on 
the wall surface is — (l/pc)^w(Q)i?(Q), where rj = (pc/z) is the acoustic 
admittance ratio (small) of the wall at the point Q. The element of 
wall area dS Q at Q therefore acts like a simple source of sound, modify- 
ing the pressure at P by an amount [ivN^ N (Q)ii(Q) dS Q /ch]e i < u « /c '><* r -. et \ 
where h is the distance between P and Q [see the discussion of (Eq. 
28.1)]. Consequently, to the first order in the small quantity rj, the 
modified characteristic function for the pressure at P = (x,y,z) is 



walls 



¥ W (P) ~ MP) + {£%) J J [ Y J ««•"•>»-"> dS Q (33.8) 

walls 

This modified standing wave, vibrating within nonrigid walls, 
does not vibrate with quite the same frequency as does the wave ip N 
inside rigid walls. To find the new natural frequency we set the 
expression of Eq. (33.8) into the wave equation. After a great deal 
of involved integration, which does not need to be gone into here, we 
find that the square of the corrected characteristic value is given 
to the second order in the small quantity 17 by the expression 



Vm.33] DAMPED VIBRATIONS, REVERBERATION 411 

2^ 2 L ■ _ (ic\ average of [r)(P)ty N (P)\f/ N (P)] over all walls \ 
N \ \u>nJ average of [^(P)] over room volume ) 

(33.9) 

where Vn(P) is the function given in Eq. (33.8). 

To put these formulas into a form that can be calculated with 
reasonable ease, we assume that the characteristic functions yj/ N have 
the usual properties of orthogonality 

fff+»(P)MP) dV p = { ° [* I $ ohuo) 

room s ' 

where V is the volume of the room, N and M are trios of numbers 
labeling different standing waves, and Aw is some dimensionless con- 
stant depending on the shape of the function $ N . In terms of these 
functions, the outgoing wave in Eq. (33.8) is given in terms of a series 
[a simplified form of Eq. (34.4)] 

where h is the distance between the point P (which can be anywhere 
in the room) and the point Q (which is somewhere on the wall). The 
function ^ N (P) can also be expanded into a series of characteristic 
functions. As long as 97 is everywhere small, the largest term in the 
series will be i[/ N , the rest being correction terms: 

where the functions 

Y MN = Gmn — iBmn, 7] = K — l(T 

Gmn ' k(Q) 

= / / ^ (Q) *"( Q) ds<1 

Bmn walls <r(Q) 

are the transfer admittances of the walls, coupling the Nth and Mth 
standing waves. They have the dimensions of an area, as do the 
absorption constants a x , given in Eq. (33.5). The prime on the sum- 
mation symbol 2 indicates that the sum does not include the term 
M = N, for this term has been written down separately as the first 
and largest term. 



412 STANDING WAVES OF SOUND [Vm.33 

Inserting Eq. (33.11) into Eq. (33.9) gives us the natural fre- 
quency of the iVth wave. The time dependence of the free vibration 
of the Nth wave is given, to the second order of approximation, by 
the exponential 



exp — icon — 



cY NN . c 2 Y* NN 



2FAtf SV 2 usA% 

"^|AS#^jJ f (33 ' 13) 

The real part of the exponent gives the damping constant kx, to 
the second order in the small quantities Y. 

, cGnN , C 2 COjV GnnBnN , ^0' AGmnBmN /„q -.is 

The first term in this, of course, corresponds to the first-order 
expression given in Eq. (33.3). The second term is the correction due 
to the distortion of the standing wave by the wall admittance. 

The imaginary part of the exponent also carries a correction to the 
unperturbed angular frequency un. The first-order correction is 
— (cBnn/2VAn). While the effect of the wall conductance (to the 
first approximation) is to attenuate the free vibrations, the effect of 
the wall susceptance is to change the frequency of free vibration. The 
second-order correction is of the same general form as that in Eq. 
(33.14) for k N , except that the quantities i((? 2 — J5 2 ) appear instead 
oiQB. 

Scattering Effect of Absorbing Patches. — We can now begin to see 
the effect of wall irregularity on the distribution of each standing wave. 
As long as the admittances Ymn in Eq. (33.12) are small, the standing 
wave ^n differs only slightly from the symmetrical wave ^r for smooth 
rigid walls. But as the walls are made softer some of the F's increase 
in size; and if the absorbing material is properly placed so that many 
of the F's get large, each standing wave becomes a more or less random 
mixture of a large number of ^'s, corresponding to wave motion in 
many directions. If enough F's are large enough, each standing wave 
is a random mixture of waves, all waves of about the same frequency 
have about the same value of damping constant, and we arrive at 
the uniform sound-distribution case discussed at the beginning of this 
chapter. 

Thus a placement of absorbing material so that as many Ymn's 
as possible are as large as possible will cause the acoustic conditions to 



Vin.34] FORCED VIBRATIONS 413 

approach the uniform acoustic-distribution case as closely as possible. 
Uniform distribution of absorbing material over any wall will not pro- 
duce this effect, for if t)(Q) is constant over any wall (the yz wall for 

instance) a part of Ymn will equal r\ | I ^m(Q)^n(Q) dy dz, which is 

x = 

zero unless m y = n y and m z = n z , so that in this case many of the F's 
would be zero. 

A study of the integrals defining Ymn shows that, for as many of 
them as possible to differ from zero, the absorbing material should be 
placed on the walls in patches, of dimensions neither large nor small 
compared with the wavelength, the patches being placed in a random 
manner over all walls. In this way, the absorbing material will scatter 
the waves as effectively as possible. 

A study of the scattering effect of small modifications of shape of 
the walls (which will not be computed here) shows that shape irregu- 
larities, if they are about the size of a wavelength and are irregularly 
placed, are even more effective in scattering than are irregularities of 
admittance (though, of course, they do not cause absorption). A 
combination of "bumps" and patches of absorbing material, irregu- 
larly placed, produces the best acoustic effect. 

34. FORCED VIBRATIONS 

We are now in a position to calculate the excitation of the standing 
waves in a room by a sound generator and thus obtain an idea of the 
effect of the acoustic properties of the room on the quality of the sound 
produced by the generator. We shall see that in a moderately absorb- 
ing room the shape of the room will have very little effect on the output 
of the generated sound at high frequencies. The intensity will be 
greater and will be more uniformly distributed over the volume than 
is the case when the source is in free space; but the dependence of this 
intensity on frequency will be the same, its value being simply a con- 
stant times the power output of the source for free space. This is due 
to the fact that at high frequencies the energy is "carried" by a large 
number of standing waves, all of nearly the same natural frequency, 
so that the resulting effect is fairly uniform. 

At low frequencies, however, the majority of the energy will be 
carried by one or two standing waves, and large fluctuations in the 
amplitude of these waves can occur as the frequency is changed. The 
variation of output with frequency will depend more on the character- 
istics of the room than on those of the generator at low frequencies. 



414 STANDING WAVES OF SOUND [VIII.34 

Simple Analysis for High Frequencies.— The intensity of high- 
frequency sound due to a source of output power n can be obtained 
in a very simple manner. If it is possible to assume that the intensity 
of the sound T is uniform throughout the room, then the formulas 
derived at the beginning of this chapter can be used. In particular, if 
the source output II changes little during a reverberation time, Eq. 
(32.7) can be used. Close to the source most of the sound is radiating 
outward from the source, and the intensity varies with the inverse 
square of the distance, as from a simple source. At larger distances 
the outward radiation is lost beneath the randomly scattered waves, 
which have more or less uniform intensity everywhere in the room. 
If II is the power output of the source in watts, r the distance from the 
source in feet, a the room absorption in square feet [see Eq. (32.4)], 
then the intensity close to the source in watts per square centimeter is 
(II/4,0007rr 2 ), where the factor 1,000 is approximately the number of 
square centimeters in a square foot. For large values of r, the inten- 
sity is n/l,000a). 

Intensity and Mean-square Pressure. — At this point we must call 
a halt, to point out the difference between our definition of intensity 
and the way sound "intensity" is usually measured. We have 
defined intensity as sound power falling on one side of a square centi- 
meter of area. This can conceivably be measured, but in many cases 
the result may depend on the orientation of the area. Close to the 
source the intensity is all flowing outward, so that we must arrange 
that the square centimeter be placed perpendicular to the radius r, if 
our measurement is to equal (n/4,000jrr 2 ) ; if it were placed parallel tor 
the intensity measured would not be at all as large. On the other 
hand, throughout the rest of the room, according to our assumption, 
the intensity flows equally in all directions, and the intensity-measur- 
ing device need not have any special orientation to measure the 
predicted amount. 

In actual practice, sound intensity is rarely measured directly; 
what is measured is mean-square pressure, as was mentioned in con- 
nection with Eq. (22.15) (p 2 ms = i|pi 2 )- This quantity is simply 
related to the average energy density w by the relation p 2 ^ = pc 2 w, 
but it is not simply related to the sound intensity. If the intensity is 
flowing in only one direction the relation is piL = pcT, but if it is 
flowing equally in all directions the relation is p 2 ^ = 4pcT, as was 
shown in Eq. (32.3). Consequently, the quantity to compute, which 
can be checked directly with measurement, is the mean-square pres- 
sure, rather than the intensity, or else the pressure level (whieh is 
20 log (prms) + 74 = 10 log (w) + 136) rather than the intensity level. 



VHI.S4] FORCED VIBRATIONS 415 

In terms of these quantities the statements made above become 

C 10 log (n) - 20 log (r) + 49 db (r 2 < a/50) 
Pressure level c^ < (34.1) 

( 10 log (n) - 10 log (a) + 66 db (r 2 > a/50) 

for the statistical case, where II is source power in watts, r the distance 
from the source in feet, and a the room absorption in square feet. 
The criterion for range of validity (r 2 vs. a/50) is obtained by equating 
the two formulas. If the power II is measured in ergs per second, r in 
centimeters, and a in square centimeters, the formulas for mean-square 
pressure are 

2 f (pcll/4irr 2 ) (r 2 < a/50) 

Prms — ( (4 pc n/a) (r 2 > a/50) 

a = ^a s A s 

s 

Close to the source, the first expression is larger and is used; far from 
the source the second term, representing the random sound, is larger 
and is valid. 

If the sound generator is a simple source of strength Q Q , then Eq. 
(27.4) shows that the power generated will be (po; 2 Q§/8xc). Conse- 
quently, over most of the volume of the room the mean-square pres- 
sure due to a simple source is 



(pWQ$\ 
\ 2ira J 



A - - DsfV (34 - 2) 

This equation is obtained here, for the case of the room with uniform 
distribution of sound, so as to compare it with the expression we shall 
obtain for a rectangular room, where the assumption of uniform dis- 
tribution is not valid. 

Solution in Series of Characteristic Functions. — The preceding 
equations are satisfactory for high frequencies, where the resonance 
frequencies are close enough together for the response to be fairly uni- 
form, and for rooms of sufficiently irregular shape for the sound to be 
uniformly spread over the room. It certainly will not be valid at low 
frequencies, where the response is far from uniform. To determine 
the range over which Eqs. (34.1) and (34.2) are valid, we must analyze 
the coupling of the source with the individual standing waves. 

For a given room, the characteristic functions, being the solutions 
of the wave equation which satisfy the proper boundary conditions, 
can be represented by the sequence yp N , where N represents a trio of 



416 STANDING WAVES OF SOUND [VHI.34 

integers and the ^'s satisfy the orthogonality condition given in Eq. 
(33.10). The corresponding characteristic values can be written as 
(o) N — ik N ). If the walls are rigid k N = 0, and if the walls have some 
simple shape (rectangular, cylindrical, etc.) the values of ca N can be 
computed, as has been done in Sec. 32. If the walls are not rigid, the 
values of w N and kn can be obtained to the first approximation by Eqs. 
(33.1), (33.3), (33.6), or (33.7), to the second approximation by Eqs. 
(33.13) and (33.14). In any case the functions \f/ N are solutions of the 
equation 



2 Aw + A\vY , 

VVjv = I ) yp N 



Any source of sound may be considered to be an assemblage of 
simple sources, as was indicated in Eqs. (27.2) and (27.3) and as 
was done in Eq. (28.1) in computing the radiation from a piston. 

A source distribution qixa^e-™' can be expressed as a series of 
characteristic functions : 

Qm = (?b) / // q{x',y',z')+ M (x',y',z') dx' dy' dz' ) (34 ' 3) 

room 

The steady-state distribution of sound in the room can likewise be 
represented as a series XA m ^m, with the series satisfying Eq. (27.3), 
which becomes 

V A n [(i0)N + &JV-) 2 + i0 2 ]\pM — lO)pC 2 ^ Qm$M 

Solving this for the coefficients A, we finally obtain a series solution for 
the steady-state pressure 

p — pce ^ 2 ^ An + .^ _ w2) Mx,y 4 z) t<54.4) 

where we have assumed that ## is small enough so that we can neglect 
k% compared with a>%. This corresponds to the series of Eqs. (10.16) or 
(11,6) for the string. 

The equation shows that the steady-state pressure wave at a point 
(x,y,z) is the sum of the waves corresponding to the different normal 
modes of the room, each with amplitude proportional to the values of 



Vm.34] FORCED VIBRATIONS 417 

the standing wave at the source and at (x,y,z) and inversely propor- 
tional to the "impedance" of the standing wave: 

(2o) N kif/o>) — i[u — (co^/co)]. 

Steady-state Response of a Room. — The energy density at any 
point in the room is (p 2 /2pc 2 ), and the intensity, by Eq. (32.3), is 
(p 2 /Spc) in those parts of the room where the energy is fairly uniformly 
distributed (i.e., not too near the source). When k is small the only 
functions yp n that have large coefficients in expression (34.7) are those 
having allowed values of oo n nearly equal to 2rv (i.e., those standing 
waves whose allowed frequency is almost equal to the frequency of 
the source). Even these coefficients may be small if Jjjqrf/ n dv is small 
(i.e., if the source of sound is located near & node of the nth standing 
wave). 

The only important terms in the expression for the intensity at a 
point (x,y,z) are those for waves whose natural frequency is close to the 
frequency of the source and which are so distributed in space that 
neither the source nor the point (x,y,z) is at a node of the wave. At 
low frequencies the allowed frequencies are spaced so widely apart that 
for some frequencies no coefficients A M will be large, and for some 
others only one may be large ; and this will be large only when the source 
is put in particular places in the room. Only for large frequencies, 
when several standing waves have frequencies near enough to that of 
the source to be excited strongly, will the intensity be distributed more 
or less uniformly over the room and will the intensity at any point 
be uniform when the frequency of the source is changed. The 
intensity close to the source will always be larger than that some 
distance away, as a calculation by Eq. (34.4) will show; .but at high 
frequencies the intensity throughout the rest of the room will be fairly 
uniform. 

As mentioned earlier the measurable quantity is not energy density 
or intensity but mean-square pressure. This is obtained by squaring 
the series for p, integrating over the room volume, and dividing by 2V. 
In virtue of the integral properties of \J/ N , given in Eq. (33.10), this 
becomes 

Pma ~ 2V 2 ZA (2u> N k N /u) 2 + \(col/o) - u,) 2 ld4 - D; 



N 



where Ajv is the average value of *//% over the room. If the largest 
dimension of the sound source is less than a half wavelength, the inte- 
gral Jjjq^NdV becomes Qo^n(S), where Qo is the equivalent source 



418 STANDING WAVES OF SOUND [VIIL34 

strength (amplitude of total air outflow from source) and where 
iPn(S) is the value of -^n at the position S of the source. 

Rectangular Room. — If the room under consideration is a rec- 
tangular one of sides l x , l y , U, the characteristic functions are those 
given in Eq. (32.11). To the first approximation the allowed values 
of un are given in Eq. (33.1) and those of the damping constant k N are 
given in Eq. (33.3) or (33.6). The pressure at the point P = (x,y,z) 
due to a simple source Qoe'™* at S = (x ,yo,Zo) is then 

_ PC 2 Qo iut ^ 6n,e w ,6n.^y(^)^(P) ,„, fi , 

P— V e 2j (2coa-W") + *'[K/co) - «1 

JV 

where eo = 1, e 2 = e 3 = • • • =2. The mean-square pressure 
throughout the room is then 

Prms— 2 y 2 ^ (2u, N k N /u) 2 + [ K/co) - 0>] 2 ^ ' ' 

N 

The source factor E N (S) = e^e^e^i/^^) is unity if the source is 
placed at random in the room (or is moved about in the room to average 
out the effect of source position), but is e nx e ny e ni if the source is placed 
at a corner of the room. 

Transmission Response. — Figure 92 shows response curves for a 
room 10 by 15 by 30 ft (the one used for Fig. 89), with a damping 
constant for oblique waves of 1c n = 10 (assumed independent of 
frequency) corresponding to a reverberation time of about 1 sec. The 
source is located at one corner of the room, thereby making \[/n(S) 
unity for all values of N and making the variation in energy depend 
only on the impedance functions for the normal modes and on the 
position of the observer. Curves for the energy density at the center 
of the room, at the center of an opposite wall, and for the average 
density are plotted as functions of v. We see that at low frequencies 
the response is very nonuniform, both as to position and as to fre- 
quency, but that it becomes more uniform at higher frequencies. 

These curves should be compared with Fig. 89, which gives the 
number of allowed frequencies between v — 5 and v + 5 for the same 
room. In Sec. 32 we did not include the damping of the standing 
waves, and we had to assume that the frequency distribution of the 
sound source was "spread out" in order to obtain a uniform response 
even at high frequencies. For a spread of Av we found that the 
response would be fairly uniform for frequencies larger than 
\/c 3 /£tVAv [according to Eq. (32.14)]. In the present section we 
see that absorption at the walls spreads out the response of each 



VHI.34] 



FORCED VIBRATIONS 



419 



standing wave so that, instead of responding "infinitely well" to just 
one frequency and only moderately well to others, there will be a range 
of frequencies over which the response will be about equally large. 
We can show, from Eq. (34.7), that the range of frequency, over which 
the response of a given standing wave is larger than one-half its 
maximum response, is between v n — \/3(A-/27r) and v n + V 3(A-/2r), 




50 100 50 

y v 

Fig. 92. — Response curves for a room 15 by 30 by 10 ft with reverberation time 
1 sec. Dotted lines correspond to approximate formula (34.12), which the actual 
curves approach asymptotically. 

so that the response has a "spread" of Sv = \/3(k/ir) c~ (k/2). This 
means that if the frequency of the source is high enough so that the 
average difference between the allowed frequencies of the standing 
waves that resonate is less than (k/2), then several standing waves will 
have large amplitude, and the response of the room will be uniform 
even if the source sends out a single frequency. 

Utilizing the fact that k ~ (7/T), where T is the reverberation 
time, and using Eq. (32.14), we see that the general rule for the lower 



420 STANDING WAVES OF SOUND [VHI.34 

limit of uniform response is the following: If a source is sending out 
energy over a frequency range from v to v + Av in a room of volume V 
and reverberation time T, the response of the room will be fairly 
uniform in frequency and in distribution over the room for all fre- 
quencies greater than v„n a , where 



1 £f__V 

\4a-7 [A»+ (4/T)]/ 



iovvT (34 - 8) 

for V in cubic feet 



VAv + (4/7 1 ) 

For a pure tone (Av = 0), and for a reverberation time between 1 and 
3 sec, this JWn is approximately equal to 10 4 / \/F. For an auditorium 
as large as 50 by 100 by 200 ft this lower limit is 10 cps, so that no large 
resonance fluctuations will occur in the useful frequency range (30 to 
10,000 cycles). For a room as small as 10 by 12 by 20 ft, however, the 
response will fluctuate considerably for frequencies below 200 cycles, 
unless the spread in frequency of the source is as large as 100 cycles or 
larger. 

The preceding analysis, of course, does not take into account the 
possible focusing effects of the walls. If some of the walls are smooth 
and curved, a considerable concentration of the energy may occur in 
some parts of the room, with detrimental effects on the uniformity of 
response and the reverberation time of the room. 

The Limiting Case of High Frequencies.— As a last example of 
the method of standing-wave analysis of sound in a room, we shall 
show how the standing-wave formula (34.7) for a simple source goes 
over into the simple formula (34.2) when the frequency is high enough 
for uniform response. In such a case the allowed frequencies are close 
enough together so the quantity {(wwA-^/tt^) 2 + [(<a%/%rv) — Zwv] 2 }- 1 , 
when considered as a function of a) N , does not change much as u N 
changes from one of its allowed values to the next. When this is true 
we can change the summation of Eq. (34.7) into an integration over the 
variable u = (a N . We shall have to separate the axial from the tan- 
gential from the oblique waves in performing the integration, since they 
have different values of ku, even to the first order. The number of 
such waves having values of a N between u and u + du can be obtained 
from Eqs. (32.14), by letting u = 2rv. All these terms are obtained by 
modifying the series of Eq. (34.7) from a series over N to a number of 
integrals over dN, where the dN's for different types of waves are 
given in Eqs. (32.14). The largest term of all comes from the chief 
term (AttvW/c 3 ) dv = (u 2 V/2t 2 c s ) du in the expression for dN Q b and is 



VIII.34] 



FORCED VIBRATIONS 



421 



P 2 cQW 
4tt 2 



E (S) 



!. 



u 2 du 



(2uko) 2 + (u 2 - co 2 ) 2 



(CJ >>> ^Vaan) 



where 2£ (£) is unity if the source is not near a wall; is 2 if it is on a wall, 
not near an edge; is 4 if it is on an edge, not near a corner; and is 8 if it 
is near a corner ("near" is less than a wavelength). The quantity k Q 
is given to the first order, according to Eq. (33.6), by the expression 

^° ~ \w)' a ° = a * + ay + a * ~ 2 ^ k ^ a " 

^ ' all 

walls 

where a is the room absorption for the oblique waves at the frequency 
v (k may depend on v) and is the expression that comes closest to the 
absorption a given in Eq. (32.4) for a room with uniform distribution 
of sound. 

The integrand written above is large only when u is nearly equal 
to co (since k is usually much smaller than co). Therefore no great 
error is made if we extend the lower limit of integration from to minus 
infinity. If this is done, the integral can be readily performed. 



/: 



u 2 du 



if. 



dx 



(2uk ) 2 + (w 2 - a> 2 ) 2 — 4 J_ . k 2 + x 2 
Therefore this largest term in the expression for w becomes 

P 2 co 2 Q 2 



7T 
Wo 



T^VE Eo{s) 



2tOo 



Eo(s) 



which is to be compared with the expression for p 2 ^ given in Eqs. (34.2) 
for the statistical case of uniform distribution of sound. 

Approximate Formula for Response. — The other terms in the 
integral, for the oblique, tangential, and axial waves, can be computed 
in a similar manner. The result is, to the second order in (v mia /v), 

J P f «H28 



Pi. 



B x = 



2rrao 



B X B V B* 



^2L 



O-x "f* dy 4" 0>z 
■%0>x -j- CLy 4" Q>z 
dx + CLy + a z 



2 + — 

2i x \j£a x + Oy + a z 



etc. 



(source not 
near an x-wall) 
(source close 
to an x-wall) 



(34.9) 



— 2 (8K a )A B ; a = a x + a y + a z ; X = (2irc/co) 



x-walls 



etc. 



422 STANDING WAVES OF SOUND [VIII.34 

When the room dimensions are large compared with a wavelength, 
this reduces to Eq. (34.2) which was derived for the case of uniform 
sound distribution. 

To recapitulate: at low frequencies, below jw ^ 10 4 s/T/kV, a 
room is an irregular transmitter of sound, having alternate bands of 
good and bad transmission, more or less irregularly placed depending 
on the values of the resonance frequencies of the various standing 
waves (which are widely spread on the average at low frequencies). 
If any of the room dimensions is equal (l x = l y , etc.) or is a simple frac- 
tion of each other (l, = %l y , etc.), then multiple degeneracy will occur 
and the irregularity at low frequency will be still more pronounced. 
At frequencies above jw the individual resonance peaks merge 
together over most of the frequency range and most of the irregularity 
disappears. At high enough frequencies the mean-square pressure is 
practically uniform over the room and equal to the value computed 
statistically, given in Eq. (34.2). 

The same sort of analysis 1 can be carried through for the pressure 
wave itself, given by Eq. (34.6). The analysis is somewhat more 
difficult than that for p^, owing to the fact that when (x,y,z) is 
near the source the phase difference. between *[/ N (S) and \f/ N (x,y,z) must 
be taken into account. The results are that when v is larger than v min 
the amplitude of the pressure wave at a distance r fro m a sou rce in a 
rectangular room is (j>vQ /2r) when r is less than V^o/loV and is 
ap proxima tely constant, equal to 2pvQ y/v/a, when r is larger than 
■x/ao/l&ir. (This holds when neither the source nor the point (x,y,z) is 
very close to a wall.) Close to the source the wave behaves as if the 
source were not in a room, being radiated outward in a form corre- 
sponding to that given by Eq. (27.4) for the pressure wave from a 
simple source in free space. Far from the source the waves are uni- 
formly distributed and are traveling in all directions, though there is 
everywhere a slight preponderance of waves coming from the source 
and a slight diminution in amplitude as r is increased. The more 
highly damped the room is, the greater will be the portion of the room 
filled with radially outgoing waves, and the smaller will be the portion 
having uniform density, as was indicated in Eq. (34.1). Of course, if' 
the room has focusing walls, these results will be considerably altered. 

When either source or observer is close to a wall, interference 
effects will occur at certain wavelengths, owing to the reinforcing 

1 See E. Fermi, Quantum Theory of Radiation, Reviews of Modern Physics, 
4, pp. lOOjf. (1932), for a discussion of the methods used, details of the calculation 
of the integrals involved, etc. 



vm.34] FORCED VIBRATIONS 423 

effect of the reflected wave. When the source is sending out sound of 
frequency v (larger than v min ), the mean-square pressure a small 
distance 8 from a wall will be proportional to cos 2 (2rv8/c), unless the 
walls are very highly absorbent. The phenomenon is the analogue of 
the optical effect utilized in the Lippmann process of color photography. 

Exact Solution. — All the calculations we have made so far have 
been by the use of approximate solutions of the boundary condi- 
tions. In some cases these first- (or second-) order approximations are 
sufficiently accurate ; but if the wall-admittance ratios get large (magni- 
tude larger than about 0.2) the approximate solutions lose their 
validity, and the formulas given in this and the previous section may 
be several hundred per cent in error. In most cases, when this is 
true, the problem cannot be computed, but in a few cases, where the 
wall admittance is constant over each wall, an exact solution can be 
obtained. 

As an example of such a case, and of the exact solution that can 
be obtained, we consider a rectangular room of dimensions l x , l y , l z 
(origin of coordinates at corner of room) with the wall at x = l x having 
an acoustic impedance [acoustic-admittance ratio (pc/z) = t\ = k. — ia] 
constant over the whole wall and with the other five walls perfectly 
rigid. This is a somewhat idealized version of a small sound room set 
up to. measure the absorbing properties of acoustic material. The 
material is sometimes placed on one wall, the other walls being bare 
concrete. 

The general method of solution has already been discussed in Sec. 
31. We set 






(34.10) 



N =? n,n y ,n z 
The boundary condition at x = l x is that p = zu x , or 






tanh(7T0„) = iiP^b X = (2rc/co) (34.11) 

The equation is similar to that of Eq. (31.1) or (31.6) and can be 
solved graphically by means of one part of Plate V, at the back of 
the book. Note, however, that in this case (five walls rigid, one wall 
soft) h = (2|i?|Z/X), an extra factor of 2. The solution is complex, 
g = £ + i\i- We note that the whole set of functions ^n depends on 



424 STANDING WAVES OF SOUND [VIII.34 

the driving frequency {w/%r) through the complex quantity g n . Con- 
sequently, these characteristic functions are useful primarily for expres- 
sing the steady-state driven motion of air in the room in terms of a 
series expansion. From the steady-state response we compute the 
transient behavior by the operational-calculus methods. 

The equation for the forced motion due to a simple source at 
S = £o,2/o,Zo is, from Eq. (27.3), 

where P = x,y,z and An is the average value of \[>% over the room 
volume. The calculation of the series expansion for p is carried 
through in the same manner as for Eq. (34.6) . The result is 

V " V ^J A N (co 2 - un) 

As we have mentioned, the constant un is now a complex quantity. 
The real part co% corresponds to the resonance frequency, and the 
imaginary part k% corresponds to the damping constant. If we can 
neglect (k° N ) 2 compared with (<4) 2 , as we usually can, the quantity 
wj, can be written as «) 2 - 2ua%K- But from the definition of w 2 . 
given in Eq. (34.10), using the notation g N = (fw + iiur (where £ and n 
can be determined from Plate V), we can write 

co- - a** - w [(?y + fey + fey - <*^)] 

or 

7TC 

V I /._ I \ I... I \ i,„ I I 

(34.12) 

/^■V2„„t„\ v 

n w 



vey+ey+ey 

o ~ , / V 2 c v,vgA 



The resulting series for p and for the mean-square pressure (when the 
source is placed at random away from the walls) is 

PW ~ V 2± An o) 2 - « ~ **&) 5 , 

* , \ (34.13) 

Pmu, — 2F 2 ^ (2<^) 2 + [«) 2 - 0> 2 ] 2 

We note that since $n and /in, solutions of Eq. (34.11), are functions 



Vin.34] FORCED VIBRATIONS 425 

of w, all the functions rpx and all the quantities &$ and k% are functions 
of the driving frequency (w/27r). 

The expression for pL, is quite similar to that of Eq. (34.7), and 
the discussion of its behavior would follow the same lines as that given 
earlier. The difference is in the value of k%, which is now 



~w~)-\w)\-~ir-r 



where \ N = (2irc/w° N ) and A a = l y l z , the area of the wall that is covered 
with acoustic material. Equation (33.6) would give k^ as 



(») 



(4e w /c s J, 



to the first order in the conductance ratio k s , for the case under consid- 
eration. Detailed examination of the solutions of Eq. (34.11) shows 
that the quantity 

a „ sW = (4*w^. n = (e^ (34 . 14) 

is approximately equal to (ie„/c s ) when k is small (0.1 or less), but is a 
much more complicated function of k and of <o than this when k is not 
small. 

The Wall Coefficients.— The quantity <x% N is the closest analogue 
to the absorption coefficient for the acoustic material that one can have 
in a rectangular room with at least one "soft" wall. It has been 
called the wall coefficient [see Eq. (33.4)]. We notice that it is not only 
dependent on the acoustic impedance of the material (as is the a for a 
room with uniform sound distribution), but also depends on the size 
of the room and on the particular standing wave that is being damped. 
For hard walls (k small) the variation with room size becomes negli- 
gible, and the only dependence on the type of wave is in the difference 
between tangential and oblique classes. For larger values of k the 
quantity a N (u>) may change appreciably from one value of N to 
the next (and it can be shown to depend also on the impedance of the 
opposite wall). In other words, each standing wave has a different 
reverberation time; and the decay curve for a sound involving several 
standing waves may be far from a straight line. 

At present our analysis is for the steady-state, driven motion of 
the room, so that we cannot yet say that k% is the damping constant 
for the reverberation. As we shall see shortly, the damping constant 



426 STANDING WAVES OF SOUND [VIII.34 

(and the related wall coefficient a) for the free vibration of a standing 
wave is obtained by setting w equal to < - ik° N in solving for £ y and 
hk. Since < also depends on co, this means a series of successive 
approximation calculations to find the correct values of k and a to use 
for the transient decay rate. 

Transient Calculations, Impulse Excitation. — To show how the 
free vibrations of the room can be computed by contour integral 
evaluation from Eqs. (34.13) we shall take as a first example the 
response to a sudden explosion of unit strength at the point S = x ,yo,z. 
This corresponds to a simple source at S of strength 8(t), where 5 is 
the impulse function defined in Eq. (6.11). By an obvious generaliza- 
tion of Eqs. (6.16), the resulting pressure distribution in the room is 



Pi(0 



(« < 0) 



where p(co) is the series given in Eq. (34.13). 

Finding the locations of the poles of each term in the series for p(u) 
is rather complicated. Both < and k N vary as co is varied (since £ 
and n depend on co) ; in fact, k% changes sign when co becomes negative 
(since £ changes sign). However, in principle, a value of «°' and k% 
can be obtained for every value of co, even for complex values. The 
position of a pole is found by varying co over the complex plane until 
it equals <a% - ik% for that same value of co. The value of co at that 
point can be called < - ik r N , the corresponding values of $ and v can 
be called & and » r N , and that of a° xN can be labeled just a xN . 

The other pole turns out to be at - a r N - ik r N , as would be expected 
from the symmetry of the equations. The corresponding values of 
£, fi, and a are just — &, n N , and <x x n. # _ 

The characteristic functions $», which are complex quantities, are 
also functions of co. Their values at the poles can be represented by 
the following symbols: 

For a, = o> N - iky, mp) = r K (P)*"™; A * = W** 

For co = -< - ik N ; MP) = MP)e-^ p) ; A* = A^r^ 

where ¥ N is a real function of x, y, z, the magnitude of the pressure 
wave; and <p N , the phase lag of the JVth wave, also depends on x, y, z. 
The average value of $\ also has its magnitude and phase. 

We can now compute the residues about the poles of the function 
p(co). The resulting expression for the "impulse wave" is 



VHI.84] FORCED VIBRATIONS 427 

^.j^&cop^,. (<<0) (34 , 5) 

\ • cos [tf„t - <p' N (P) - ^(Q) + ^] ( t > 0) 

The individual standing waves damp out exponentially, each with the 
damping constant k r N , given in terms of the wall coefficients <x xN by 
Eq. (34.14), where the parameters n r N , & are computed for the reso- 
nance frequency « = «£ - iV N , as mentioned above. This series 
does not converge, corresponding to the fact that the excitation cor- 
responds to an impulse function. But for a physically possible fluctua- 
tion of source strength Q(t), the corresponding pressure wave 

P = jl „ Q(j)p»(t - r) dr (34.16) 

analogous to Eq. (6.17) is given by a series that does converge. This 
integral is the counterpart, for the exact solution, of Eq. (32.6) for 
the statistical solution. 

Exact Solution for Reverberation.— As a last example we shall 
compute the response of the rectangular room, with the wall at x = l x 
covered with material of admittance ratio v = K - ia and the other 
five walls rigid, for a simple source of strength <? sin (2™*) from 

t oo to t = 0, and of strength zero from t = to t = oo (j e 

the source, of frequency v, is shut off at t = 0). This can be computed 
by the use of Eq. (34.16), or it can be computed by obtaining the 
Laplace transform for the source function and calculating another 
contour integral for the pressure. We choose the latter method to 
provide one further example of its use. ' 

We first compute the steady-state pressure wave in the room if the 
source is not shut off at t = 0. We calculate the parameters * and M 
for the frequency v and from them compute the steady-state functions- 
t or positive values of %rv : For negative values of 2™ : 

< - iK = u f N e- ia »° < + ik° N = «/e"fc- 

MP) - f»(?)^(« MP) = r N (P)e- i ^°^ 

A* = A» e*** An = A >-i**° 

Then, using the series for p(„) given in Eq. (34.13), we have that the 
steady-state wave due to a source Q sin (2wvt) is 



428 STANDING WAVES OF SOUND [Vm.U 



-V 2irV 2i A%G%{2nrv)' 

N 



p.(2r,0 = ^ real part of |^w ^ _ (2 ^ )2 _ (< _ ^ )t j 

cos[2*v* - <p%{P) - <p%(S) + ^ + r^ - (34.17) 

where 

[G° N (2irv)f = (ZkvY + K) 4 - 2(2ttv) 2 (co/) 2 cos (2Q&) 

and 

-«> = [s^)] sinW) 

Next we compute the pressure oscillations in the room for a source 
that is zero for t < and Qo sin (2irv0 for t> 0. To do this we need 
the Laplace transform [see Eq. (6.19)] for this source function 

X27TV 
sin (2rrf)«r* dt = ^^ + g2 

Then the resulting pressure is, using Eq. (34.13) again, 

p u (t) = ^ I *(-iw)p(w) dco 

oc 2 Qo r - 7 _• * "^ * w $n(PWn(Q) 

= ""V" ^ ,2l JW[co 2 - (2™) 2 ][co 2 - (co» - t/#)*J 

J- * AT 

The residues about the poles a> = ±2irv result in the function 
p s (2rv) of Eq. (34.17); whereas the residues about the poles 
w = ± W £ - tA* [by a calculation similar to that for Eq. 34.15)] 
give — p r (t), where, for t > 0, 

m _ P c2 £° >? M p )^( S 
Vrit) ~~ ir 2a A^(2x.) 



pc 2 Qo ^ + r N ( P)MS) p - kN rt. 

• cosK* - <P r N (P) - <&(&) '+ *Sr + r r J (34.18) 



where 

[G* N (%tv)Y = (2rv)» + K - ik r N \ 4 ~ 2(2rv)*K ~ **&!' cos (2^) 
and 

and 



VHI.34] FORCED VIBRATIONS 429 

Finally, the effect due to a source that has been of strength 
Qo sin (2rrvt) from t = — <*> to t = 0, and is turned off at t = 0, must 
be the difference between the steady-state response ps(2irv) and the 
expression p s (?Trv) — Pr(t) which represents the effect if the source is 
zero before t = and is turned on at t = 0. Consequently, the pres- 
sure wave in the room due to a source that has been on for a long time 
and is turned off at t = is 

/ p.(2rv) (t < 0) [see Eq. (34.17)] 
V \ Pr(t) (t > 0) [see Eq. (34.18)] 

The steady-state response p 8 and the reverberation p r are similar 
series. There are differences in the values of the amplitudes of the 
terms, represented by the superscripts and r, owing to the fact that 
the driving frequency (2rv) is used in p s for the calculation of £# and 
hn (and thus w%, k%, \f/%, etc.), whereas the "natural frequency" 
(<a N — ik r N ) is used in p r for the calculation of £ and p. The change in 
amplitude is not large for those standing waves having resonances near 
the driving frequency (a> N nearly equal to 2irv). 

The major difference between the series is in the exponential e~ klfH 
and in the terms 2irvt or w N t in the cosine terms. The reverberation 
series p r damps out in time, whereas the steady-state pressure p 8 has 
a constant amplitude (as, of course, it must). In addition, the indi- 
vidual standing waves in the reverberation each oscillate with their 
own natural frequency co^, whereas every term in the steady-state 
solution oscillates with the driving frequency (2rv). Since there are 
many natural frequencies u r N with nearly the same value, complicated 
"beat-note" effects often occur in the reverberation (see Fig. 91). 

It is a far cry from auditorium acoustics to second quantization; 
yet the methods used above to analyze the behavior of sound in rooms 
are quite similar to the methods Dirac used to predict the existence of 
the positron three years before its experimental discovery. This 
similarity is an interesting example of the unifying influence of theo- 
retical physics. 

Problems 

1. A rectangular office room is 15 ft high, 20 ft wide, and 30 ft long. The 
walls are of plaster, wood, and glass, with absorption coefficient 0.03. The floor is 
covered with a carpet (a = 0.2) and the ceiling with acoustic material (5 = 0.4). 
Ordinarily, six persons are present in the room. What is the reverberation time? 
If four typewriters are going in the room, each producing 1 erg of sound energy per 
second, what will be the intensity level in the room? 



430 STANDING WAVES OF SOUND [VIII.34 

2. An auditorium is 30 ft high, 50 ft wide, and 100 ft long and contains 500 
wooden seats (a = 0.15 apiece). The walls, ceiling, and floor have an absorption 
coefficient of 0.03. What is the reverberation time when the auditorium is empty? 
When it is full? How much acoustic material (a = 0.4) must be placed on the 
ceiling and walls in order that the reverberation time may be 2 sec when the audi- 
torium is empty? What is then the reverberation time when the room is filled? 
What must then be the power output of a public-address system in order to have 
the average intensity level in the filled auditorium 90 db? What will be the 
intensity when the auditorium is empty? 

3. Calculate and plot the absorption coefficient, as a function of frequency of 
the material given in Prob. 23, Chap. VII, for both values of spacing from the wall, 
for < v < 2,000 cps. 

4. A room with cylindrical walls, of radius 5 m, has a flat floor and ceiling, 
4 m apart. Plot the number of allowed frequencies in the room between v and 
v + 5 as a function of v from v = to v = 50. Above what frequency will this 
curve become fairly uniform? 

5. A rectangular corridor is 2 m wide, 3 m high, and 10 m long. Plot the 
number of allowed frequencies in the enclosure between v and v + 5 as a function 
of v from v = to v — 100. Above what frequency will this curve be fairly 
uniform? 

6. A cubical room 5 m on a side has an average absorption coefficient for floor 
and ceiling of 0.2; for the walls, a value of 0.04. What is the reverberation time 
for those waves which strike floor and ceiling? For those waves which do not 
strike floor and ceiling? List all the allowed frequencies between zero and 100 cps 
and give the position of the nodal planes and the reverberation times of each 
corresponding standing wave. 

7. List frequencies, nodes, and reverberation times of the normal modes 
between v = and v = 100 for the room of Prob. 6 when walls, floor, and ceiling 
all have an average absorption coefficient of 0.1. 

8. The air in the room of Prob. 6 is started into vibration so that all the 
normal modes between v = 98 and v = 102 are set into motion with equal initial 
amplitudes. What normal modes are excited? Plot the decay curve of intensity 
level against time after the source is shut off at the mid-point of the room ; at the 
mid-point of one wall; at a point 167 cm out from two walls and 250 cm up from 
the floor. 

9. A room has constants l v = l z = 500, l x = 505, a x = a y = 0.02, a z = 0.2. 
All the allowed frequencies between v = 81 and v = 83 are excited with equal 
amplitude. Plot the decay of intensity level at the mid-point of the room, at the 
mid-point of the floor, at a point in one corner of the room halfway between floor 
and ceiling (assume that at this point the pressure due to the different standing 
waves are in phase when the sound shuts off). Will the decay curve be the same 
for the corresponding points in the other three corners? 

10. The allowed frequencies of the room, discussed in Prob. 9, between v — 66 
and v = 68 are excited with equal amplitude and are in phase at the mid-point of 
the room at the instant when the source is shut off. Plot the decay curves for the 
mid-point of the room; for the mid-point of one of the walls perpendicular to the 
z-axis; for a point in one corner between walls, halfway between floor and ceiling. 
Will the decay curve be the same for the corresponding points in the other three 
corners? 



Vin.34] FORCED VIBRATIONS 431 

11. The room of Prob. 6 has a point source of strength Qo ■» 10 located at its 
mid-point. Plot the average energy density in the room as a function of v from 
v ■= to v = 100. The source is stopped at the instant that its source function 
q is a maximum, when its frequency is 100 cps. Plot the decay curve of sound 
at the mid-po'nt of one wall. 

12 The point source of Prob. 11 is relocated at the mid-point of one of the 
walls. Plot the energy density as a function of v from v = to v = 100 at the 
mid-point of the room; at the mid-point of the wall opposite the source. 

13. A point source of source function 

q = 15 sin(40rf) - 10 sin (120x0 + 3 sin(200ir*) 

is placed at the mid-point of one of the walls of the room of Prob. 6. Plot the 
curves of pressure against time for one cycle of the sound for the mid-point of the 
room; for the mid-point of the opposite wall; for one of the corners of the room 
farthest from the source. 

14. Discuss the reverberating qualities of a cubical room 5 m on a side, having 
two opposite walls with an absorption coefficient 0.8, the other walls with a coeffi- 
cient 0.05. Find the "average reverberation time" from Eq. (33.6), and find the 
reverberation time for the lowest five standing waves by use of Eqs. (34.12). 

15. Calculate the exact values (by Plate V) of the damping constants and the 
corresponding reverberation times for the lowest five standing waves in a room 
8 by 12 by 16 ft with acoustic material on one of the 8 by 12 walls and with the 
other five walls rigid. The specific acoustic impedance of the "soft" wall is 
3pce i(ir/6) (|z| = 3pc, <p =» —30°) for the frequencies of these five standing waves. 



BIBLIOGRAPHY 

This is not intended as an exhaustive classification of all the books on vibrations 
and sound; it is simply a list of the books that the writer has found particularly 
useful as collateral reading or as reference works giving particular problems in 
greater detail than can be given in the present volume. 

A good book on the philosophy of the scientific method, with many examples 
in mechanics and theory of vibrations, is "Foundations of Physics," by Lindsay 
and Margenau. A good general text on theoretical physics is Slater and Frank 
"Introduction to Theoretical Physics." Reference works on theoretical physics 
are Byerly, "Fourier Series and Spherical Harmonics"; Bateman, "Partial Dif- 
ferential Equations of Mathematical Physics"; and Webster "Partial Differential 
Equations of Mathematical Physics." Reference works on the type of mathe- 
matics used in this book are Whittaker and Watson, "Modern Analysis"; Courant 
and Hilbert, " Mathematische Physik"; and Watson, "Theory of Bessel Func- 
tions." A good set of tables of the functions used, more complete than those given 
below, is in Jahnke-Emde, "Tables of Functions." 

In the field of vibrations and sound the standard reference works are Rayleigh, 
"Theory of Sound"; and Helmholtz, "Sensations of Tone." Useful books for 
collateral reading are Lamb, "Dynamical Theory of Sound"; Richardson, "Sound"; 
Watson, "Sound"; Miller, "Science of Musical Sound"; Crandall "Vibrating 
Systems and Sound"; Trendelenberg, "Klange und Gerausche"; Stewart and 
Lindsay, "Acoustics"; and Bergmann and Hatfield, "Ultrasonics." 

Especially useful are Olsen and Massa, "Applied Acoustics," for its discussion 
of recent practical applications; Fletcher, "Speech and Hearing"; and Stevens and 
Davis, "Hearing, Its Psychology and Physiology," for their discussion of the 
physicophysiological aspect of acoustics. 

In the field of architectural acoustics the classical reference is the "Collected 
Papers" of W. C. Sabine. Books for collateral reading are Knudsen, "Archi- 
tectural Acoustics"; Knudsen's article in the Reviews of Modern Physics, 6, 1 (1934) ; 
Watson, "Acoustics of Buildings"; and P. E. Sabine, "Acoustics and Architecture."' 
Of course, the issues of the Journal of the Acoustical Society of America should 
be consulted. Two articles in the Reviews of Modern Physics, one by V. O. Knud- 
sen on page 1, Vol. 6 (1933) and one by Morse and Bolt on page 69, Vol. 16 (1944), 
have material of interest in architectural acoustics. 



433 



34 
320 



GLOSSARY OF SYMBOLS 

The more commonly used symbols are listed below, together with their usual 
meanings and a reference to the page where the symbol is denned (if a definition is 
needed) . Only those symbols used in several sections are listed. 

SymbGl , Meaning p 

a 3 Radius of cylinder or circular diaphragm, in cm 298 

I Absorption constant of a room, in sq ft or sq cm 3g 5 

A 5 Amplitude of oscillation, in cm 9 

} Area of room in sq ft 38 _ 

B \ Mechanical susceptance in sec per g on 

} Magnetic induction, in gauss 34 

c Wave velocity, in cm per sec 73 222 

C Capacitance in farads ' 

C m Radiation amplitude for a cylinder 301 

du Piezoelectric constant 

D Ratio mechanical force to current, in dynes per amp 

D m Radiation amplitude for a sphere 

1 J~ 2 - 71828 * * • , base of natural logarithms 

A Emf, in volts 

< General function 

} Fourier transform of force 
F Force, in dynes 

G Mechanical conductance, in sec per g ,. 

h Half thickness of a plate Jj? 

H Specific response function for a diaphragm 

* " V ?F I \ . To corre spond to the usual electrical-engineering nota- 
tion, the x in this book is equal to minus j ■, , 
/ Current, in amps ^ 
/« Hyperbolic Bessel function .rjj 
3 = ~i- See note on i 

jm Spherical Bessel function J.Z 

J m Cylindrical Bessel function ^Jl 

k (Damping constant (R/2m) „_}!! 

I Wave number („/c) = (2x/X) = (2™/c) ' "* 

A Stiffness constant of spring <T 

I Length of string or tube, in cm ?* 
ln(x) Natural logarithm of x 
log (a;) Logarithm to base 10 of x 
L Inductance, in henrys 

«C Symbol representing Laplace transform ? 

Mass, in g bZ 



201 



M Reactance function for piston in plane wall J 

435 



436 GLOSSARY OF SYMBOLS 

Symbol Meaning Page 

m,n Integers labeling normal modes of vibration 84, 390 

n m Spherical Neumann function 317 

N m Cylindrical Neumann function 196 

p Excess pressure in sound wave, in dynes per sq cm 218 

j Power, in ergs per sec 26 

I Pressure amplitude, in dynes per sq cm 224 

P n Legendre function 315 

q Source density, in sec -1 313 

!= (o} m/R) (The "Q" of a circuit or other system) 25, 403 

Modulus of elasticity, in dynes per sq cm 152 

Strength of simple source, in cc per sec 313 

r Radial distance, in cm 311 

S Electrical resistance, in ohms 36 

I Mechanical resistance, in g per sec 24 

{Elastic modulus of crystal 40 

Poisson's ratio 209 

S Area of cross section, in sq cm 217 

t Time, in sec 2 

I Tension, in dynes m 72 

( Reverberation time, in sec ' 387 

u Unit step function , 47 
u,v,w Components of velocity, in cm per sec 

U Velocity amplitude, in cm per sec 23 

( Potential energy, in ergs 3 

I Volume, in cc or cu ft 394 

w Energy density in sound wave, in ergs per cc 383 

W Energy, in ergs 2 
x,y,z Rectangular coordinates 

( Stress in crystal, in dynes per sq cm 39 

] Mechanical reactance, in g per sec 29 

Displacement of string 71 

Y Mechanical admittance, in sec per g 35 

z Specific acoustic impedance (p/u) in g per sq cm per sec 237 

( Electrical impedance, in ohms 36, 38 

I Mechanical impedance, in g per sec 29, 38 

a Absorption coefficient 385, 403 

a,/3 Acoustic impedance parameters for plane wave 139, 240 
a^ n ,(3„ Characteristic numbers for Bessel functions 189,210,399 

ey\ Wavelength constant for bar and plate 155, 209 

& e Ratio of specific heats = 1.4 for air 221 

y m Phase shift for scattered cylindrical wave 301 

r Electromechanical coupling constant 35 

5 Unit impulse function 47 

/ Change in density of air due to sound wave 218 

S m Phase shift for scattered spherical wave 320 

A Separation of plates in condenser microphone 204 

V 2 Laplace operator, Laplacian 174, 296 

s Mass per unit length of string, in g per cm 72 



X 

y 



v 



GLOSSARY OF SYMBOLS 437 



Symbol Meaning 

«o =* 1, «i = 62 •= es =» • • • = 2 402 

f Acoustic impedance ratio = (z/pc) = — ix 144, 240 

t\ (Displacement of membrane, in cm 173 

\ Acoustic admittance ratio = (pc/z) = k — ia 144, 240 

( Phase angle 31 

I Angle in spherical coordinates 311 

B Acoustic resistance ratio 135, 240 

( Dielectric susceptibility 40 

} Acoustic conductance ratio 144, 240 

X Wavelength, in cm 83 

m = (wa/c) = ka 320 

v Frequency, in cps 6 

j Strain in crystal, in cm per cm length 38 

I Displacement of air in sound wave, in cm 218 

7t = 3.14159 • • • 

(Electric polarization 39 

( Power radiated, in ergs per sec or in watts 229 

\ Density 154 

p / Density of air at standard conditions 218 

pc Characteristic impedance of air for sound = 42 g/cm ! sec 222 

! Piezoelectric coupling constant, in ohms per g 40 

Mass per unit area, in g per sq cm 173 

Acoustic susceptance ratio 144, 240 

2 Summation symbol 85 

\ Ratio propagation velocity to sound velocity c 307 

I Transmission coefficient 273 

T Intensity of sound, in ergs per sec per sq cm 223 

(Phase angle 31 

(Angle in spherical or polar coordinates 296, 311 

4> Angle in cylindrical coordinates 296 

\ Laplace transform of / 52 

( Angle of incidence 367 

X Acoustic reactance ratio 135, 240 

^ Complex reflection phase for plane wave = ir(a — i0) 239 

SP Characteristic function 

<*> = 2irv, angular velocity 21 

Q Mechanical phase angle 201 

Cii Approximately equal to 

oo Infinity 

-+. Approaches as a limit 

> Is greater than 

< Is less than 

J> Contour integral 12 

|Z| Magnitude of Z = IP + X 2 10 

Capital subscripts for Z,R,X, etc., denote electric impedance in ohms, lower- 
case subscripts denote mechanical impedance in grams per second, Greek sub- 
scripts denote analogous impedances (see pages 38 and 237). 



438 



TABLES OF FUNCTIONS 



Table I. — Trigonometric and Hyperbolic Functions 
(See pages 4 and 136) 



X 


sin (x) 


COS (x) 


tan (a;) 


sinh (x) 


cosh (x) 


tanh {x) 


e* 


e~ x 


0.0 


0.0000 


1.0000 


0.0000 


0.0000 


1.0000 


0.0000 


1.0000 


1.0000 


0.2 


0.1987 


0.9801 


0.2127 


0.2013 


1.0201 


0.1974 


1.2214 


0.8187 


0.4 


0.3894 


0.9211 


0.4228 


0.4018 


1.0811 


0.3799 


1.4918 


0.6703 


0.6 


0.5646 


0.8253 


0.6841 


0.6367 


1.1855 


0-5370 


1.8221 


0.5488 


0.8 


0.7174 


0.6967 


1 . 0296 


0.8881 


1.3374 


0.6640 


2.2255 


0.4493 


1.0 


0.8415 


. 5403 


1 . 5574 


1.1752 


1.5431 


0.7616 


2.7183 


0.3679 


1.2 


0.9320 


0.3624 


2 . 5722 


1 . 5095 


1.8106 


. 8337 


3.3201 


0.3012 


1.4 


0.9854 


+0.1700 


+ 5.7979 


1.9043 


2.1509 


0.8854 


4.0552 


0.2466 


1.6 


0.9996 


-0.0292 


-34.233 


2.3756 


2.5775 


0.9217 


4.9530 


0.2019 


1.8 


0.9738 


-0.2272 


-4.2863 


2.9422 


3.1075 


. 9468 


6.0496 


0.1553 


2.0 


0.9093 


-0.4161 


-2.1850 


3.6269 


3.7622 


0.9640 


7.3891 


0.1353 


2.2 


0.8085 


-0.5885 


-1.3738 


4.4571 


4.5679 


0.9757 


9.0250 


0.1108 


2.4 


0.6755 


-0.7374 


-0.9160 


5.4662 


5.5569 


0.9837 


11.023 


0.0907 


2.6 


0.5155 


-0.8569 


-0.6016 


6.6947 


6.7690 


0.9890 


13.464 


0.0742 


2.8 


0.3350 


-0.9422 


-0.3555 


8.1919 


8.2527 


0.9926 


16.445 


0.0608 


3.0 


+0.1411 


-0.9900 


-0.1425 


10.018 


10.068 


0.9951 


20.086 


0.0498 


3.2 


-0.0584 


-0.9983 


+ 0.0585 


12 . 246 


12 . 287 


0.9967 


24.533 


0.0407 


3.4 


-0.2555 


-0.9668 


0.2643 


14.965 


14.999 


0.9978 


29.964 


0.0333 


3.6 


-0.4425 


-0.8968 


0.4935 


18.285 . 


18.313 


0.9985 


36 . 598 


0.0273 


3.8 


-0.6119 


-0.7910 


0.7736 


22.339 


22.362 


0.9990 


44.701 


0.0223 


4.0 


-0.7568 


-0.6536 


1.1578 


27.290 


27.308 


0.9993 


54 . 598 


0.0183 


4.2 


-0.8716 


-0.4903 


1.7778 


33.335 


33.351 


. 9996 


66.686 


0.0150 


4.4 


-0.9516 


-0.3073 


3.0963 


40.719 


40.732 


0.9997 


81.451 


0.0123 


4.6 


-0.9937 


-0.1122 


+8.8602 


49.737 


49.747 


0.9998 


99.484 


0.0100 


4.8 


-0.9962 


+0.0875 


-11.385 


60.751 


60.759 


0.9999 


121.51 


0.0082 


5.0 


-0.9589 


0.2837 


-3.3805 


74 . 203 


74.210 


0.9999 


148.41 


0.0067 


5.2 


-0.8835 


0.4685 


- 1 . 8856 


90.633 


90.639 


0.9999 


181.27 


0.0055 


5.4 


-0.7728 


0.6347 


-1.2175 


110.70 


110.71 


1 . 0000 


221.41 


0.0045 


5.6 


-0.6313 


0.7756 


-0.8139 


135.21 


135.22 


1.0000 


270.43 


0.0037 


5.8 


-0.4646 


0.8855 


-0.5247 


165.15 


165.15 


1 . 0000 


330.30 


0.0030 


6.0 


-0.2794 


0.9602 


-0.2910 


201.71 


201.71 


1 . 0000 


403.43 


0.0025 


6.2 


-0.0831 


. 9965 


-0.0834 


246.37 


246.37 


1 . 0000 


492.75 


0.0020 


6.4 


-+0.1165 


0.9932 


+0.1173 


300.92 


300.92 


1.0000 


601 . 85 


0.0016 


6.6 


0.3115 


0.9502 


0.3279 


367.55 


367 . 55 


1.0000 


735.10 


0.0013 


6.8 


0.4941 


0.8694 


0.5683 


448.92 


448.92 


1 . 0000 


897 . 85 


0.0011 


7.0 


0.6570 


0.7539 


0.8714 


548.32 


548.32 


1 . 0000 


1096.6 


0.0009 


7.2 


0.7937 


0.6084 


1 . 3046 


662 . 72 


662 . 72 


1.0000 


1339.4 


0.0007 


7.4 


0.8987 


0.4385 


2.0493 


817.99 


817.99 


1 . 0000 " 


1636.0 


0.0006 


7-6 


0.9679 


0.2513 


3 . 8523 


999.10 


999.10 


1 . 0000 


1998.2 


0.0005 


7". 8 


0.9985 


+0.0540 


+ 18.507 


1220.3 


1220.3 


1.0000 


2440.6 


0.0004 


8.0 


0.9894 


-0.1455 


-6.7997 


1490.5 


1490.5 


1.0000 


2981.0 


0.0003 



TABLES OF FUNCTIONS 



439: 



Table II. — Trigonometric and Hyperbolic Functions 
(See pages 4 and 136) 



X 


sin (xas) 


cos (-kx) 


tan (trx) 


sinh (irx) 


cosh (irx) 


tanh (rx) 


e xx 


e -rx 


0.00 


0.0000 


1.0000 


0.0000 


0.0000 


1.0000 


0.0000 


1.0000 


1.0000 


0.05 


0.1564 


0.9877 


0.1584 


0.1577 


1.0124 


0.1558 


1.1701 


0.8546 


0.10 


0.3090 


0.9511 


0.3249 


0.3194 


1.0498 


0.3042 


1.3691 


0.7304 


0.15 


0.4540 


0.8910 


0.5095 


0.4889 


1.1131 


0.4392 


1.6019 


0.6242 


0.20 


0.5878 


0.8090 


0.7265 


0.6705 


1.2040 


0.5569 


1.8745 


0.5335 


0.25 


0.7071 


0.7071 


1.0000 


0.8687 


1.3246 


0.6558 


2.1933 


0.4559 


0.30 


0.8090 


0.5878 


1.3764 


1.0883 


1.4780 


0.7363 


2 . 5663 


0.3897 


0.35 


0.8910 


0.4540 


1.9626 


1.3349 


1.6679 


0.8003 


3.0028 


0.3330 


0.40 


0.9511 


0.3090 


3.0777 


1.6145 


1.8991 


0.8502 


3.5136 


0.2846 


0.45 


0.9877 


+0.1564 


6.3137 


1.9340 


2.1772 


0.8883 


4.1111 


0.2432 


0.50 


1.0000 


0.0000 


00 


2.3013 


2.5092 


0.9171 


4.8105 


0.2079 


0.55 


0.9877 


-0.1564 


-6.3137 


2.7255 


2.9032 


0.9388 


5.6287 


0.1777 


0.60 


0.9511 


-0.3090 


-3.0777 


3.2171 


3.3689 


0.9549 


6.5861 


0.1518 


0.65 


0.8910 


-0.4540 


-1.9626 


3.7883 


3.9180 


0.9669 


7.7062 


0.1298 


0.70 


0.8090 


-0.5878 


-1.3764 


- 4.4531 


4.5640 


0.9757 


9.0170 


0.1109 


0.75 


0.7071 


-0.7071 


-1.0000 


5.2280 


5.3228 


0.9822 


10.551 


0.09478 


0.80 


0.5878 


-0.8090 


-0.7265 


6.1321 


6.2131 


0.9870 


12.345 


0.08100 


0.85 


0.4540 


-0.8910 


-0.5095 


7.1879 


7.2572 


0.9905 


14.437 


0.06922 


0.90 


0.3090 


-0.9511 


-0.3249 


8.4214 


8.4806 


0.9930 


16.902 


0.05916 


0.95 


+0.1564 


-0.9877 


-0.1584 


9.8632 


9.8137 


0.9949 


19.777 


0.05056 


1.00 


0.0000 


-1.0000 


0.0000 


11.549 


11.592 


0.9962 


23.141 


0.04321 


1.05 


-0.1564 


-0.9877 


0.1584 


13.520 


13.557 


0.9973 


27.077 


0.03693 


1.10 


-0.3090 


-0.9511 


0.3249 


15.825 


15.857 


0.9980 


31.682 


0.03156 


1.15 


-0.4540 


-0.8910 


0.5095 


18.522 


18.549 


0.9985 


37.070 


0.02697 


1.20 


-0.5878 


-0.8090 


0.7265 


21.677 


21.700 


0.9989 


43.376 


0.02305 


1.25 


-0.7071 


-0.7071 


1.0000 


25.367 


25.387 


0.9992 


50.753 


0.01970 


1.30 


-0.8090 


-0.5878 


1.3764 


29 . 685 


29 . 702 


0.9994 


59.387 


0.01683 


1.35 


-0.8910 


-0.4540 


1.9626 


34.737 


34.751 


0.9996 


69.484 


0.01438 


1.40 


-0.9511 


-0.3090 


3.0777 


40.647 


40.660 


0.9997 


81.307 


0.01230 


1.45 


-0.9877 


-0.1564 


6.3137 


47.563 


47.573 


0.9998 


95.137 


0.01051 


1.50 


-1.0000 


0.0000 


00 


55.654 


55.663 


0.9998 


111.32 


0.00898 


1.55 


-0.9877 


+0.1564 


-6.3137 


65.122 


65.130 


0.9999 


130.25 


0.00767 


1.60 


-0.9511 


0.3090 


-3.0777 


76 . 200 


76.206 


0.9999 


152.41 


0.00656 


1.65 


-0.8910 


0.4540 


-1.9626 


89.161 


89.167 


0.9999 


178.33 


0.00561 


1.70 


-0.8090 


0.5878 


-1.3764 


104.32 


104.33 


1.0000 


208.66 


0.00479 


1.75 


-0.7071 


0.7071 


-1.0000 


122.07 


122 . 08 


1.0000 


244.15 


0.00409 


1.80 


-0.5878 


0.8090 


-0.7265 


142.84 


142.84 


1.0000 


285.68 


0.00350 


1.85 


-0.4540 


0.8910 


-0.5095 


167.13 


167.13 


1.0000 


334.27 


0.00299 


1.90 


-0.3090 


0.9511 


-0.3249 


195.56 


195.56 


1.0000 


391.12 


0.00256 


1.95 


-0.1564 


0.9877 


-0.1584 


228.82 


228.82 


1.0000 


457 . 65 


0.00219 


2.00 


0.0000 


1 . 0000 


0.0000 


267.75 


267.75 


1.0000 


535.49 


0.00187 



MQ 



TABLES OF FUNCTIONS 



Table III. — Hyperbolic Tangent of Complex Quantity 
tanh [x(o - i0)\ - 6 - i x - |f |e~** 





tanh (ia) 


e 


X 


in 


<P 





X 


in 


<f> 


a 
























= 


D.00 






P = 


3.05 




0.0000 


0.00 


0.0000 


0.0000 


0.0000 


0-90° 


0.0000 


0.1584 


0.1584 


90.00° 


0.0159 


0.05 


0.0500 


0.0000 


0.0500 


0.00 


0.0512 


0.1580 


0.1660 


72.03 


0.0319 


0.10 


0.1000 


0.0000 


0.1000 


0.00 


0.1025 


0.1567 


0.1872 


56.82 


0.0481 


0.15 


0.1500 


0.0000 


0.1500 


0.00 


0.1537 


0.1547 


0.2180 


45.20 


0.0645 


0.20 


0.2000 


0.0000 


0.2000 


0.00 


0.2048 


0.1519 


0.2549 


36.56 


0.0813 


0.25 


0.2500 


0.0000 


0.2500 


0.00 


0.2558 


0.1482 


0.2956 


30.09 


0.0985 


0.30 


0.3000 


0.0000 


0.3000 


0.00 


0.3068 


0.1438 


0.3388 


25.12 


0.1163 


0.35 


0.3500 


0.0000 


0.3500 


0.00 


0.3577 


0.1386 


0.3836 


21.18 


0.1349 


0.40 


0.4000 


0.0000 


0.4000 


0.00 


0.4084 


0.1325 


0.4293 


17.98 


0.1543 


0.45 


0.4500 


0.0000 


0.4500 


0.00 


0.4589 


0.1256 


0.4758 


15.32 


0.1748 


0.50 


0.5000 


0.0000 


0.5000 


0.00 


0.5093 


0.1181 


0.5228 


13.05 


0.1968 


0.55 


0.5500 


0.0000 


. 5500 


0.00 


0.5596 


0.1096 


0.5702 


11.08 


0.2207 


0.60 


0.6000 


0.0000 


0.6000 


0.00 


0.6095 


0.1005 


0.6177 


9.36 


0.2468 


0.65 


0.6500 


0.0000 


0.6500 


0.00 


0.6593 


0.0905 


. 6654 


7.82 


0.2761 


0.70 


0.7000 


0.0000 


0.7000 


0.00 


0.7088 


0.0798 


0.7133 


6.43 


0.3097 


0.75 


0.7500 


0.0000 


0.7500 


0.00 


0.7581 


0.0683 


0.7612 


5.15 


0.3497 


0.80 


. 8000 


0.0000 


0.8000 


0.00 


0.8070 


0.0561 


0.8090 


3.97 


0.3999 


0.85 


. 8500 


0.0000 


0.8500 


0.00 


0.8558 


0.0432 


0.8569 


2.88 


0.4686 


0.90 


0.9000 


0.0000 


0.9000 


0.00 


0.9041 


0.0295 


0.9047 


1.87 


0.5831 


0.95 


0.9500 


0.0000 


0.9500 


0.00 


0.9523 


0.0151 


0.9524 


0.91 








= 


0.10 






= 


0.15 




0.0000 


0.00 


0.0000 


0.3249 


0.3249 


90.00° 


0.0000 


0.5095 


0.5095 


90.00° 


0.0159 


0.05 


0.0553 


0.3240 


0.3286 


80.32 


0.0629 


. 5079 


0.5118 


82.93 


0.0319 


0.10 


0.1104 


0.3213 


0.3398 


71.03 


0.1256 


0.5031 


0.5186 


75.98 


0.0481 


0.15 


0.1655 


0.3169 


0.3575 


62.43 


0.1878 


0.4951 


0.5296 


69.22 


0.0645 


0.20 


0.2202 


0.3106 


0.3808 


54.67 


0.2493 


0.4841 


0.5445 


62.75 


0.0813 


0.25 


0.2746 


0.3027 


0.4087 


47.78 


0.3099 


0.4700 


0.5629 


56.61 


0.0985 


0.30 


0.3286 


0.2929 


0.4402 


41.72 


0.3692 


0.4531 


. 5845 


50.82 


0.11,63 


0.35 


0.3820 


0.2815 


0.4745 


36.39 


0.4273 


0.4333 


0.6085 


45.40 


0.1349 


0.40 


0.4349 


0.2684 


0.5110 


31.68 


0.4838 


0.4110 


0.6347 


40.18 


0.1533 


0.45 


0.4871 


0.2537 


0.5492 


27.51 


0.5385 


0.3860 


. 6626 


35.63 


0.1748 


0.50 


0.5386 


0.2374 


0.5886 


23.79 


0.5914 


0.3589 


0.6917 


31.25 


. 1968 


0.55 


0.5893 


0.2196 


0.6289 


20.44 


0.6423 


0.3295 


0.7219 


27.16 


0.2207 


0.60 


0.6390 


0.2003 


0.6697 


17.41 


0.6911 


0.2982 


0.7527 


23.34 


. 2468 


0.65 


0.6880 


0.1796 


0.7110 


14.63 


0.7378 


0.2652 


0.7840 


19.76 


0.2761 


0.70 


0.7358 


0.1576 


0.7525 


12.08 


0.7822 


0.2305 


0.8155 


16.43 


0.3097 


0.75 


0.7827 


0.1342 


0.7941 


9.73 


0.8243 


0.1945 


. 8469 


13.28 


0.3497 


0.80 


0.8285 


0.1096 


0.8375 


7.54 


0.8642 


0.1573 


. 8783 


10.32 


0.3999 


0.85 


0.8731 


0.0838 


0.8771 


5.48 


0.9015 


0.1191 


0.9094 


7.52 


. 4686 


0.90 


0.9166 


0.0569 


0.9184 


3.56 


0.9367 


0.0800 


0.9401 


4.88 


0.5831 


0.95 


0.9589 


0.0289 


0.9594 


1.73 


0.9695 


0.0403 


. 9704 


2.38 



TABLES OF FUNCTIONS 441 

Table III. — Hyperbolic Tangent op Complex Quantity.— (Continued) 





tanh (ira) 


e 


X 


in 


<P 


e 


X 


in 


V 


a 
























/3 = 


0.20 






13 = 0.25 




0.0000 


0.00 


0.0000 


0.7265 


0.7265 


90.00° 


0.0000 


0.0000 


1.0000 


90.00° 


0.0159 


0.05 


0.0763 


0.7238 


0.7278 


83.98 


0.0998 


0.9950 


1.0000 


84.28 


0.0319 


0.10 


0. 1520 


0.7145 


0.7315 


78.01 


0.1980 


0.9802 


1.0000 


78.58 


0.0481 


0.15 


0.2265 


0.7019 


0.7375 


72.12 


0.2934 


0.9560 


1.0000 


72.93 


0.0645 


0.20 


0.2993 


0.6831 


0.7458 


66.34 


0.3846 


0.9230 


1.0000 


(67.38 


0.0813 


0.25 


0.3698 


0.6593 


0.7560 


60.72 


0.4706 


0.8824 


1.0000 


61 .93 


0.0985 


0.30 


0.4376 


0.6312 


0.7680 


55.27 


0.5504 


0.8348 


1.0000 


56.60 


0.1163 


0.35 


0.5023 


0.5989 


0.7816 


50.01 


0.6236 


0.7818 


1.0000 


51.42 


0.1349 


0.40 


0.5635 


0.5627 


0.7964 


44.96 


0.6896 


0.7241 


1.0000 


46.40 


0.1543 


0.45 


0.6212 


0.5235 


0.8123 


40.12 


0.7484 


0.6632 


1.0000 


41.55 


0.1748 


0.50 


0.6749 


0.4814 


0.8290 


35.50 


0.8000 


0.6000 


1.0000 


36.87 


0.1968 


0.55 


0.7247 


0.4370 


0.8462 


31.09 


0.8446 


0.5355 


1.0000 


32.38 


0.2207 


0.60 


0.7703 


0.3907 


0.8637 


26.89 


0.8824 


0.4706 


1.0000 


28.07 


0.2468 


0.65 


0.8120 


0.3430 


0.8815 


22.91 


0.9139 


0.4060 


1 . 0000 


23.95 


0.2761 


0.70 


0.8497 


0.2943 


0.8992 


19.11 


0.9395 


0.3423 


1.0000 


20.02 


0.3097 


0.75 


0.8835 


0.2451 


0.9169 


15.51 


0.9600 


0.2800 


1.0000 


16.27 


0.3497 


0.80 


0.9136 


0.1955 


0.9343 


12.08 


0.9757 


0.2195 


1.0000 


12.68 


0.3999 


0.85 


0.9401 


0.1459 


0.9514 


8.82 


0.9869 


0.1611 


1.0000 


9.27 


0.4686 


0.90 


0.9632 


0.0967 


0.9681 


5.73 


0.9945 


. 1050 


1.0000 


6.03 


0.5831 


0.95 


0.9831 


0.0480 


0.9843 


2.79 


0.9986 


0.0512 


1.0000 


.2.93 








- 


0.30 






/} - 


0.35 




0.0000 


0.00 


0.0000 


1.3764 


1.3764 


90.00° 


0.0000 


1.9626 


1.9626 


90 . 00° 


0.0159 


0.05 


0.1440 


1 . 3664 


1.3740 


83.98 


0.2403 


1.9391 


1.9539 


82.93 


0.0319 


0.10 


0.2841 


1 . 3373 


1.3671 


78.01 


0.4670 


1.8708 


1.9283 


75.98 


0.0481 


0.15 


0.4164 


1 . 2904 


1.3450 


72.12 


0.6699 


1.7653 


1.8882 


69.22 


0.0645 


0.20 


0.5382 


1 . 2282 


1.3408 


66.34 


0.8408 


1 . 6326 


1.8365 


62.75 


0.0813 


0.25 


0.6455 


1.1537 


1.3228 


60.72 


0.9776 


1.4832 


1.7762 


56.61 


0.0985 


0.30 


0.7419 


1.0701 


1.3021 


55.27 


1.0809 


1.3261 


1.7109 


50.82 


0.1163 


0.35 


0.8223 


0.9803 


1.2794 


50.01 


1 . 1540 


1.1701 


1.6432 


45.40 


0.1349 


0.40 


0.8885 


0.8873 


1 . 2556 


44.96 


1.2007 


1.0140 


1.5755 


40.18 


0.1543 


0.45 


0.9413 


0.7933 


1.2311 


40.12 


0.2267 


0.8793 


1 . 5092 


35.63 


0.1748 


0.50 


0.9820 


0.7005 


1.2063 


35.50 


1.2359 


0.7500 


1.4457 


31.25 


0.1968 


0.55 


1.0120 


0.6102 


1.1819 


31.09 


1.2324 


0.6323 


1.3852 


27.16 


0.2207 


0.60 


1.0326 


0.5336 


1.1578 


26.89 


1.2198 


0.5263 


1.3284 


23.34 


0.2468 


0.65 


1.0449 


0.4415 


1 . 1344 


22.91 


1.2002 


0.4313 


1.2755 


19.76 


0.2761 


0.70 


1.0507 


0.3640 


1.1121 


19.11 


1.1762 


0.3467 


1.2262 


16.43 


0.3097 


0.75 


1.0509 


0.2916 


1.0906 


15.51 


1 . 1493 


0.2713 


1 . 1808 


13.28 


0.3497 


0.80 


1.0465 


0.2240 


1.0703 


12.08 


1 . 1202 


0.2039 


1.1386 


10.32 


0.3999 


0.85 


1.0387 


0.1612 


1.0511 


8.82 


1.0902 


0.1440 


1.0996 


7.52 


0.4686 


0.90 


1.0278 


0.1032 


1.0330 


5.73 


1.0599 


0.0905 


1.0637 


4.88 


0.5831 


0.95 


1.0148 


0.0494 


1.0160 


2.79 


1 . 0296 


0.0428 


1.0305 


2.38 



442 TABLES OF FUNCTIONS 

Table III. — Hyperbolic Tangent of Complex Quantity. — {Continued) 





tanh (ia) 


e 


X 


m 


V 


8 


X 


in 


<p 


a 
























= 0.40 






= 0.45 




0.0000 


0.00 


0.0000 


3.0777 


3.0777 


90.00° 


0.0000 


6.3138 


6.3138 


9p.00° 


0.0159 


0.05 


0.5115 


2.9990 


3.0423 


80.32 


1.8580 


5.7272 


6.0211 


72.03 


0.0319 


0.10 


0.9565 


2.7833 


2.9431 


71.03 


2.9217 


4.4691 


5.3394 


56.82 


0.0481 


0.15 


1.2948 


2.4799 


2.7976 


62.43 


3.2313 


3.2535 


4.5855 


45.20 


0.0645 


0.20 


1.5189 


2 . 1427 


2.6265 


54.67 


3.1500 


2 . 3362 


3.9220 


36.56 


0.0813 


0.25 


1.6444 


1 . 8124 


2.4473 


47.78 


2.9260 


1.6953 


3.3820 


30.09 


0.0985 


0.30 


1 . 6960 


1.5119 


2.2720 


41.72 


2.6722 


1.2524 


2.9511 


25.12 


0.1163 


0.35 


1.6966 


1.2501 


2 . 1074 


36.39 


2.4310 


0.9417 


2.6070 


21.18 


0.1349 


0.40 


1.6652 


1.0277 


1.9569 


31.68 


2.2154 


0.7188 


2.3291 


17.98 


0.1543 


0.45 


1.6144 


0.8408 


1 . 8208 


27.51 


2.0269 


0.5550 


2.1015 


15.32 


0.1748 


0.50 


1 . 5547 


0.6853 


1 . 6989 


23.79 


1.8633 


0.4319 


1.9126 


13.05 


0.1968 


0.55 


1.4901 


0.5414 


1.5901 


20.44 


1.7210 


0.3371 


1.7538 


11.08 


0.2207 


0.60 


1 . 4247 


0.4466 


1.4932 


17.41 


1 . 5972 


0.2632 


1.6187 


9.36 


0.2468 


0.65 


1.3609 


0.3553 


1.4065 


14.63 


1.4887 


0.2044 


1.5026 


7.82 


0.2761 


0.70 


1.2994 


0.2781 


1.3289 


12.08 


1.3931 


0.1568 


1.4019 


6.43 


0.3097 


0.75 


1.2412 


0.2129 


1.2593 


9.73 


1.3083 


0.1180 


1.3137 


5.15 


0.3497 


0.80 


1 . 1862 


0.1569 


1.1966 


7.54 


1.2331 


0.0857 


1.2361 


3.97 


0.3999 


0.85 


1 . 1349 


0.1090 


1 . 1401 


5.48 


1.1655 


0.0587 


1.1670 


2.88 


0.4686 


0.90 


1.0867 


0.0674 


1.0888 


3.56 


1 . 1047 


0.0360 


1 . 1053 


1.87 


0.5831 


0.95 


1.0418 


0.0314 


1.0423 


1.73 


1 . 0499 


0.0167 


1 . 0500 


0.91 








0-0 


475 




= 0.50 


0.0000 


0.00 


0.0000 


12.706 


12.706 


90.00° 


00 


0.0000 


00 


0-90° 


0.0159 


0.05 


5.780 


9.030 


10.725 


57.35 


20.000 


0.0000 


20 . 000 


0.00 


0.0319 


0.10 


6.213 


4.811 


7.859 


37.75 


10.000 


0.0000 


10.000 


0.00 


0.0481 


0.15 


5.260 


2.682 


5.905 


27.01 


6 . 6667 


0.0000 


6 . 6667 


0.00 


0.0645 


0.20 


4.356 


1.636 


4.653 


20.58 


5.0000 


0.0000 


5.0000 


0.00 


0.0813 


0.25 


3.662 


1.075 


3.817 


16.35 


4.0000 


0.0000 


4.0000 


0.00 


0.0985 


0.30 


3.138 


0.7445 


3.225 


13.35 


3 . 3333 


0.0000 


3.3333 


0.00 


0.1163 


0.35 


2.736 


0.5367 


2.789 


11.10 


2.8571 


0.0000 


2.8571 


0.00 


0.1349 


0.40 


2.422 


0.3979 


2.454 


9.33 


2 . 5000 


0.0000 


2 . 5000 


0.00 


0.1543 


0.45 


2.170 


0.3008 


2.190 


7.90 


2.2222 


0.0000 


2.2222 


0.00 


0.1748 


0.50 


1.9638 


0.2304 


1.9773 


6.70 


2.0000 


0.0000 


2.0000 


0.00 


0.1968 


0.55 


1.7927 


0.1778 


1.8014 


5.67 


1.8182 


0.0000 


1.8182 


0.00 


0.2207 


0.60 


1.6486 


0.1375 


1.6543 


4.76 


1.6667 


0.0000 


1.6667 


0.00 


0.2468 


0.65 


1.5238 


0.1060 


1 . 5293 


3.97 


1 . 5385 


0.0000 


1 . 5385 


0.00 


0.2761 


0.70 


1.4194 


0.0809 


1.4217 


3.27 


1.4286 


0.0000 


1.4286 


0.00 


0.3097 


0.75 


1.3269 


0.0605 


1.3284 


2.61 


1.3333 


0.0000 


1.3333 


0.00 


0.3497 


0.80 


1.2458 


0.0438 


1.2465 


2.02 


1.2500 


0.0000 


1.2500 


0.00 


0.3999 


0.85 


1 . 1737 


0.0300 


1.1741 


1.46 


1.1765 


0.0000 


1 . 1765 


0.00 


0.4686 


0.90 


1 . 1095 


0.0183 


1 . 1097 


0.94 


1.1111 


0.0000 


1.1111 


0.00 


0.5831 


0.95 


1.0518 


0.0085 


1.0519 


0.46 


1 . 0526 


0.0000 


1 . 0526 


0.00 



TABLES OF FUNCTIONS 



443 



Table IV. — Inverse Hyperbolic Tangent of Complex Quantity 

7r(a - ifi) = tanh -1 (0 - t'x) 

(See page 137) 





a 





a 


P 


« 





a 


/3 


a 





6 


X = 


X = 0.2 


x = 0.4 


X = 0.6 


x = 0.8 


0.0 


0.0000 


0.0000 


0.0000 


0.0628 


0.0000 


0.1211 


.0.0000 


0.1720 


0.0000 


0.2148 


0.2 


0.0645 


0.0000 


0.0619 


0.0664 


0.0552 


0.1250 


0.0468 


0.1762 


. 0386 


0.2186 


0.4 


0.1349 


0.0000 


0. 1281 


0.0796 


0.1118 


0.1379 


0.0931 


0.1894 


0.0760 


0.2302 


0.6 


0.2206 


0.0000 


0.2041 


0.0936 


0.1703 


0.1640 


0.1373 


0.2135 


0.1103 


0.2500 


0.8 


0.3497 


0.0000 


0.2955 


0.1426 


0.2255 


0.2110 


0.1749 


0.2500 


0.1386 


0.2776 


1.0 


00 


0-0.5 


0.3672 


0.2659 


0.2593 


0.2814 


0.1985 


0.2964 


0.1576 


0.3106 


1.2 


0.3816 


0.5000 


0.3271 


0.3894 


0.2562 


0.3524 


0.2041 


0.3436 


0.1661 


0.3445 


1.4 


0.2852 


0.5000 


0.2681 


0.4394 


0.2322 


0.4013 


0.1962 


0.3826 


0.1655 


0.3750 


1.6 


0.2334 


0.5000 


0.2255 


0.4610 


0.2060 


0.4307 


0.1823 


0.4111 


0.1593 


0.3999 


1.8 


0.1994 


0.5000 


0.1950 


0.4723 


0.1832 


0.4488 


0.1678 


0.4312 


0.1505 


0.4198 


2.0 


0.1748 


0.5000 


0.1721 


0.4792 


0.1644 


0.4605 


0.1536 


0.4454 


0.1409 


0.4341 


2.2 


0.1561 


0.5000 


0.1542 


0.4841 


0.1490 


0.4686 


0.1411 


0.4557 


0.1317 


0.4454 


2.4 


0.1412 


0.5000 


0.1399 


0.4868 


0.1361 


0.4743 


0.1302 


0.4634 


0.1230 


0.4541 


2.6 


0.1291 


0.5000 


0.1281 


0.4890 


0.1252 


0.4786 


0.1208 


0.4692 


0.1151 


0.4610 


2.8 


0.1188 


0.5000 


0.1181 


0.4908 


0.1159 


0.4819 


0.1124 


0.4737 


0.1079 


0.4665 


3.0 


0.1103 


0.5000 


0.1097 


0.4921 


0.1080 


0.4845 


0.1052 


0.4773 


0.1016 


0.4709 


3.2 


0.1029 


0.5000 


0.1025 


0.4931 


0.1010 


0.4865 


0.0988 


0.4802 


0.0959 


0.4744 


3.4 


0.0965 


0.5000 


0.0961 


0.4940 


0.0950 


0.4875 


0.0931 


0.4826 


0.0907 


0.4774 


3.6 


0.0908 


0.5000 


0.0905 


0.4947 


0.0895 


0.4892 


0.0880 


0.4845 


0.0860 


0.4799 


3.8 


0.0858 


0.5000 


0.0855 


0.4953 


0.0847 


0.4906 


0.0834 


0.4862 


0.0817 


0.4820 


4.0 


0.0813 


0.5000 


0.0812 


0.4958 


0.0804 


0.4916 


0.0793 


0.4876 


0.0778 


0.4838 




X = 1.0 


X = 1-2 


X = 1-4 


X = 1-6 


X = 2.0 


0.0 


0.0000 


0.2500 


0.0000 


0.2789 


0.0000 


0.3026 


0.0000 


0.3222 


0.0000 


0.3524 


0.2 


0.0316 


0.2532 


0.0259 


0.2814 


0.0213 


0.3046 


0.0178 


0.3238 


0.0127 


0.3534 


0.4 


0.0619 


0.2686 


0.0506 


0.2890 


0.0417 


0.3106 


0.0348 


0.3285 


0.0249 


0.3564 


0.6 


0.0892 


0.2786 


0.0729 


0.3012 


0.0602 


0.3185 


0.0503 


0.3360 


0.0362 


0.3612 


0.8 


0.1118 


0.2993 


0.0916 


0.3173 


0.0760 


0.3326 


0.0639 


0.3459 


0.0464 


0.3675 


1.0 


0.1281 


0.3238 


0.1058 


0.3360 


0.0885 


0.3472 


0.0749 


0.3574 


0.0552 


0.3570 


1.2 


0.1373 


0.3493 


0.1150 


0.3558 


0.0974 


0.3628 


0.0832 


0.3699 


0.0623 


0.3833 


1.4 


0.1403 


0.3734 


0.1197 


0.3750 


0.1028 


0.3783 


0.0890 


0.3826 


0.0679 


0.3920 


1.6 


0.1383 


0.3944 


0.1207 


0.3926 


0.1054 


0.3930 


0.0924 


0.3949 


0.0719 


0.4008 


1.8 


0.1341 


0.4120 


0.1190 


0.4080 


0.1056 


0.4064 


0.0938 


0.4064 


0.0745 


0.4093 


2.0 


0.1281 


0.4262 


0.1157 


0.4211 


0.1042 


0.4182 


0.0937 


0.4169 


0.0760 


0.4174 


2.2 


0.1216 


0.4376 


0.1114 


0.4321 


0.1017 


0.4284 


0.0926 


0.4262 


0.0766 


0.4249 


2.4 


0.1150 


0.4468 


0.1067 


0.4412 


0.0985 


0.4372 


0.0906 


0.4344 


0.0764 


0.4318 


2.6 • 


0.1087 


0.4542 


0.1019 


0.4488 


0.0950 


0.4446 


0.0883 


0.4375 


0.0756 


0.4381 


2.8 


0.1027 


0.4602 


0.0973 


0.4551 


0.0913 


0.4510 


0.0855 


0.4478 


0.0743 


0.4437 


3.0 


0.0974 


0.4652 


0.0927 


0.4604 


0.0878 


0.4564 


0.0828 


0.4531 


0.0729 


0.4488 


3.2 


0.0924 


0.4693 


0.0884 


0.4648 


0.0843 


0.4610 


0.0799 


0.4579 


0.0712 


0.4533 


3.4 


0.0877 


0.4727 


0.0844 


0.4686 


0.0808 


0.4633 


0.0771 


0.4619 


0.0695 


0.4573 


3.6 


0.0835 


0.4756 


0.0807 


0.4718 


0.0776 


0.4684 


0.0743 


0.4655 


0.0676 


0.4609 


3.8 


0. 0796 


0.4781 


0.0773 


0.4746 


0.0745 


0.4714 


0.0717 


0.4686 


0.0657 


0.4641 


4.0 


.0.0760 


0.4802 


0.0739 


0.4789 


0.0716 


0.4740 


0.0691 


0.4713 


0.0639 


0.4668 



444 



TABLES OF FUNCTIONS 



Table V. — Bessel Functions for Cylindrical Coordinates 
Jnix) and N n (x) (see pages 188 and 196) 



X 


J»{x) 


No(x) 


Ji(x) 


iVi(x) 


Jz(x) 


N 2 (x) 


0.0 


1.0000 


GO 


0.0000 


— w 


0.0000 


-CO 


0.1 


0.9975 


-1.5342 


0.0499 


-6.4590 


0.0012 


-127.64 


0.2 


0.9900 


-1.0811 


0.0995 


-3.3238 


0.0050 


-32.157 


0.4 


0.9604 


-0.6060 


. 1960 


-1.7809 


0.0197 


-8.2983 


0.6 


0.9120 


-0.3085 


0.2867 


-1.2604 


0.0437 


-3.8928 


0.8 


0.8463 


-0.0868 


0.3688 


-0.9781 


0.0758 


-2.3586 


1.0 


0.7652 


+0.0883 


0.4401 


-0.7812 


0.1149 


-1.6507 


1.2 


0.6711 


0.2281 


0.4983 


-0.6211 


0.1593 


-1.2633 


1.4 


0. 5669 


0.3379 


0.5419 


-0.4791 


0.2074 


-1.0224 


1.6 


0.4554 


0.4204 


0.5699 


-0.3476 


0.2570 


-0.8549 


1.8 


0.3400 


0.4774 


0.5815 


-0.2237 


0.3061 


-0.7259 


2.0 


0.2239 


0.5104 


0.5767 


-0.1070 


0.3528 


-0.6174 


2.2 


0.1104 


0.5208 


0.5560 


+0.0015 


0.3951 


-0.5194 


2.4 


+0.0025 


0.5104 


0.5202 


0.1005 


0.4310 


-0.4267 


2.6 


-0.0968 


0.4813 


0.4708 


0.1884 


0.4590 


-0.3364 


2.8 


-0.1850 


0.4359 


0.4097 


0.2635 


0.4777 


-0.2477 


3.0 


-0.2601 


0.3768 


0.3391 


0.3247 


0.4861 


-0.1604 


3.2 


-0.3202 


0.3071 


0.2613 


0.3707 


0.4835 


-0.0754 


3.4 


-0.3643 


0.2296 


0.1792 


0.4010 


0.4697 


+0.0063 


3.6 


-0.3918 


0.1477 


0.0955 


0.4154 


0.4448 


0.0831 


3.8 


-0.4026 


+0.0645 


+0.0128 


0.4141 


0.4093 


0.1535 


4.0 


-0.3971 


-0.0169 


-0.0660 


0.3979 


0.3641 


0.2159 


4.2 


-0.3766 


-0.0938 


-0.1386 


. 3680 


0.3105 


0.2690 


4.4 


-0.3423 


-0.1633 


-0.2028 


0.3260 


0.2501 


0.3115 


4.6 


-0.2961 


-0.2235 


-0.2566 


0.2737 


0.1846 


0.3425 


4.8 


-0.2404 


-0.2723 


-0.2985 


0.2136 


0.1161 


0.3613 


5.0 


-0.1776 


-0.3085 


-0.3276 


0.1479 


+0.0466 


0.3677 


5.2 


-0.1103 


-0.3312 


-0.3432 


0.0792 


-0.0217 


0.3617 


5.4 


-0.0412 


-0.3402 


-0.3453 


+0.0101 


-0.0867 


0.3429 


5.6 


+0.0270 


-0.3354 


-0.3343 


-0.0568 


-0.1464 


0.3152 


5.8 


0.0917 


-0.3177 


-0.3110 


-0.1192 


-0.1989 


0.2766 


6.0 


0.1507 


-0.2882 


-0.2767 


-0.1750 


-0.2429 


0.2299 


6.2 


0.2017 


-0.2483 


-0.2329 


-0.2223 


-0.2769 


0.1766 


6.4 


0.2433 


-0.2000 


-0.1816 


-0.2596 


-0.3001 


0.1188 


6.6 


0.2740 


-0.1452 


-0.1250 


-0.2858 


-0.3119 


+0.0586 


6.8 


0.2931 


-0.0864 


-0.0652 


-0.3002 


-0.3123 


-0.0019 


7.0 


0.3001 


-0.0259 


-0.0047 


-0.3027 


-0.3014 


, -0.0605 


7.2 


0.2951 


+0.0339 


+0.0543 


-0.2934 


-0.2800 


' -0.1154 


7.4 


0.2786 


0.0907 


0.1096 


-0.2731 


-0.2487 


-0.1652 


7.6 


0.2516 


0.1424 


0.1592 


-0.2428 


-0.2097 


-0.2063 


7.8 


0.2154 


0.1872 


0.2014 


-0.2039 


-0.1638 


-0.2395 


8.0 


0.1716 


0.2235 


0.2346 


-0.1581 


-0.1130 


-0.2630 



TABLES OF FUNCTIONS 



; *45 



Table VI. — Hyperbolic Bessel Functions 
I m (z) = i~ m J m {iz) (see page 210) 



z 


-Tc(z) 


Ii« 


ItW 


0.0 


1.0000 


0.0000 


0.0000 


0.1 


1.0025 


0.0501 


0.0012 


0.2 


1.0100 


0.1005 


0.0050 


0.4 


1.0404 


0.2040 


0.0203 


0.6 


1.0921 


0.3137 


0.0464 


0.8 


1 . 1665 


0.4329 


0.0843 


1.0 


1.2661 


0.5652 


0.1358 


1.2 


1 . 3937 


0.7147 


0.2026 


1.4 


1.5534 


0.8861 


0.2876 


1.6 


1.7500 


1.0848 


0.3940 


1.8 


1.9895 


1.3172 


0.5260 


2.0 


2.2796 


1.5906 


0.6890 


2.2 


2.6292 


1.9141 


0.8891 


2.4 


3.0492 


2.2981 


1.1111 


2.6 


3.5532 


2.7554 


1.4338 


2.8 


4.1574 


3.3011 


1.7994 


3.0 


4.8808 


3.9534 


2.2452 


3.2 


5.7472 


4.7343 


2.7884 


3.4 


6.7848 


5.6701 


3.4495 


3.6 


8.0278 


6.7926 


4.2538 


3.8 


9.5169 


8.1405 


5.2323 


4.0 


11.302 


9.7594 


6.4224 


4.2 


13.443 


11.705 


7.8683 


4.4 


16.010 


14.046 


9.6259 


4.6 


19.097 


16.863 


11.761 


4.8 


22.794 


20.253 


14.355 


5.0 


27.240 


24.335 


17.505 


5.2 


32.584 


29.254 


21.332 


5.4 


39.010 


35.181 


25.980 


5.6 


46.738 


42.327 


31.621 


5.8 


56.039 


50.945 


38.472 


6.0 


67.235 


61.341 


46.788 


6.2 


80.717 


73.888 


56.882 


6.4 


96.963 


89.025 


69.143 


6.6 


116.54 


107.31 


84.021 


6.8 


140.14 


129.38 


102.08 


7.0 


168.59 


156.04 


124.01 


7.2 


202.92 


188.25 


150.63 


7.4 


244.34 


227.17 


182.94 


7.6 


294.33 


274.22 


222.17 


7.8 


354.68 


331.10 


269.79 


8.0 


427.57 


399.87 


327.60 



446 TABLES OF FUNCTIONS 

Table VII. — Bessel Functions fob Sphekical, Cooedinates 

j«(x) = y/tc/2x J n+ \(x), Unix) = -y/ir /2x N n+i (x) 

(see page 317) 



X 


io(x) 


no(x) 


h(x) 


ni(x) 


h(.x) 


ws(«) 


0.0 


1.0000 


— X 


0.0000 


_» 


0.0000 


— 00 


0.1 


0.9983 


-9.9500 


0.0333 


- 100 . 50 


0.0007 


-3005.0 


0.2 


0.9933 


-4.9003 


0.0664 


-25.495 


0.0027 


-377.52 


0.4 


0.9735 


-2.3027 


0.1312 


-6.7302 


0.0105 


-48.174 


0.6 


0.9411 


-1.3756 


0.1929 


-3.2337 


0.0234 


-14.793 


0.8 


0.8967 


-0.8709 


0.2500 


-1.9853 


0.0408 


-6.5740 


1.0 


0.8415 


-0.5403 


0.3012 


-1.3818 


0.0620 


-3.6050 


1.2 


0.7767 


-0.3020 


0.3453 


- 1 . 0283 


0.0865 


-2.2689 


1.4 


0.7039 


-0.1214 


0.3814 


-0.7906 


0.1133 


-1.5728 


1.6 


0.6247 


+0.0183 


0.4087 


-0.6133 


0.1416 


- 1 . 1682 


1.8 


0.5410 


0.1262 


0.4268 


-0.4709 


0.1703 


-0.9111 


2.0 


0.4546 


0.2081 


0.4354 


-0.3506 


0.1985 


-0.7340 


2.2 


0.3675 


0.2675 


0.4346 


-0.2459 


0.2251 


-0.6028 


2.4 


0.2814 


0.3072 


0.4245 


-0.1534 


0.2492 


-0.4990 


2.6 


0.1983 


0.3296 


0.4058 


-0.0715 


0.2700 


-0.4121 


2.8 


0.1196 


0.3365 


0.3792 


+0.0005 


0.2867 


-0.3359 


3.0 


+0.0470 


0.3300 


0.3457 


0.0630 


0.2986 


-0.2670 


3.2 


-0.0182 


0.3120 


0.3063 


0.1157 


0.3084 


-0.2035 


3.4 


-0.0752 


0.2844 


0.2623 


0.1588 


0.3066 


-0.1442 


3.6 


-0.1229 


0.2491 


0.2150 


0.1921 


0.3021 


-0.0890 


3.8 


-0.1610 


0.2082 


0.1658 


0.2158 


0.2919 


-0.0378 


4.0 


-0.1892 


0.1634 


0.1161 


0.2300 


0.2763 


+0.0091 


4.2 


-0.2075 


0.1167 


0.0673 


0.2353 


0.2556 


0.0514 


4.4 


-0.2163 


0.0699 


+0.0207 


0.2321 


0.2304 


0.0884 


4.6 


-0.2160 


+0.0244 


-0.0226 


0.2213 


0.2013 


0.1200 


4.8 


-0.2075 


-0.0182 


-0.0615 


0.2037 


0.1691 


0.1456 


5.0 


-0.1918 


-0.0567 


-0.0951 


0.1804 


0.1347 


0.1650 


5.2 


-0.1699 


-0.0901 


-0.1228 


0.1526 


0.0991 


0.1871 


5.4 


-0.1431 


-0.1175 


-0.1440 


0.1213 


0.0631 


0.1850 


5.6 


-0.1127 


-0.1385 


-0.1586 


0.0880 


+0.0278 


0.1856 


5.8 


-0.0801 


-0.1527 


-0.1665 


0.0538 


-0.0060 


0.1805 


6.0 


-0.0466 


-0.1600 


-0.1678 


+0.0199 


-0.0373 


0.1700 


6.2 


-0.0134 


-0.1607 


-0.1629 


-0.0124 


-0.0654 


0.1547 


6.4 


+0.0182 


-0.1552 


-0.1523 


-0.0425 


-0.0896 


0.1353 


6.6 


0.0472 


-0.1440 


-0.1368 


-0.0690 


-0.1094 


0.1126 


6.8 


0.0727 


-0.1278 


-0.1172 


-0.0915 


-0.1243 


0.0875 


7.0 


0.0939 


-0.1077 


-0.0943 


-0.1029 


-0.1343 


0.0609 


7.2 


0.1102 


-0.0845 


-0.0692 


-0.1220 


-0.1391 


0.0337 


7.4 


0.1215 


-0.0593 


-0.0429 


-0.1294 


-0.1388 


+0.0068 


7.6 


0.1274 


-0.0331 


-0.0163 


-0.1317 


-0.1338 


-0.0189 


7.8 


0.1280 


-0.0069 


+0.0095 


-0.1289 


-0.1244 


-0.0427 


8.0 


0.1237 


+0.0182 


0.0336 


-0.1214 


-0.1111 


-0.0637 



TABLES OF FUNCTIONS 



447 



Table VIII. — Impedance Functions for Piston in Infinite Plane Wall 
0o - ixo = 1 - (2/w)Ji(w) — iM(w) = tanh[x(« p - ip P )]; w - (4ira/X) 

(see page 333) 



to 


So 


xo 


a p 


fip 


0.0 


0.0000 


0.0000 


0.0000 


0.0000 


0.5 


0.0309 


0.2087 


0.0094 


0.0655 


1.0 


0.1199 


0.3969 


0.0330 


0.1216 


1.5 


0.2561 


0.5471 


0.0628 


0.1663 


2.0 


0.4233 


0.6468 


0.0939 


0.2020 


2.5 


0.6023 


0.6905 


0.1247 


0.2316 


3.0 


0.7740 


0.6801 


0.1552 


0.2572 


3.5 


0.9215 


0.6238 


0.1858 


0.2800 


4.0 


1.0330 


0.5349 


0.2175 


0.3008 


4.5 


1 . 1027 


0.4293 


0.2517 


0.3194 


5.0 


1.1310 


0.3231 


0.2899 


0.3353 


5.5 


1 . 1242 


0.2300 


0.3344 


0.3460 


6.0 


1.0922 


0.1594 


0.3868 


0.3456 


6.5 


1.0473 


0.1159 


0.4450 


0.3207 


7.0 


1.0013 


0.0989 


0.4788 


0.2600 


7.5 


0.9639 


0.1036 


0.4594 


0.2050 


8.0 


0.9413 


0.1220 


0.4241 


0.1887 


8.5 


0.9357 


0.1456 


0.3980 


0.1958 


9.0 


0.9454 


0.1663 


0.3839 


0.2132 


9.5 


0.9661 


0.1782 


0.3799 


0.2344 


10.0 


0.9913 


0.1784 


0.3845 


0.2565 


10.5 


1.0150 


0.1668 


0.3964 


0.2774 


11.0 


1.0321 


0.1464 


0.4153 


0.2958 


11.5 


1 . 0397 


0.1216 


0.4410 


0.3097 


12.0 


1.0372 


0.0973 


0.4734 


0.3158 


12.5 


1.0265 


0.0779 


0.5101 


0.3083 


13.0 


1.0108 


0.0662 


0.5421 


0.2810 


13.5 


0.9944 


0.0631 


0.5490 


0.2409 


14.0 


0.9809 


0.0676 


0.5316 


0.2117 


14.5 


0.9733 


0.0770 


0.5073 


0.2032 


15.0 


0.9727 


0.0881 


0.4877 


0.2092 


15.5 


0.9784 


0.0973 


0.4758 


0.2231 


16.0 


0.9887 


0.1021 


0.4718 


0.2406 


16.5 


1.0007 


0.1013 


0.4750 


0.2591 


17.0 


1.0115 


0.0948 


0.4852 


0.2767 


17.5 


1.0187 


t).0843 


0.5017 


0.2914 


18.0 


1.0209 


0.0719 


0.5247 


0.3007 


18.5 


1.0180 


0.0602 


0.5522 


0.3010 


19.0 


1.0111 


0.0515 


0.5798 


0.2879 


19.5 


1.0021 


0.0470 


0.5968 


0.2610 


20.0 


0.9933 


0.0473 


0.5940 


0.2314 



448 



TABLES OF FUNCTIONS 



Table IX. — Legendre Functions foe Spherical Coordinates 
(See page 315) 



& 


P-i - Pe 


Pi(cost?) 


P2(COS t?) 


Pt(costf) 


P4(cos#) 


0° 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


5 


1 . 0000 


0.9962 


0.9886 


0.9773 


0.9623 


10 


1 . 0000 


0.9848 


0.9548 


0.9106 


0.8352 


15 


1 . 0000 


0.9659 


0.8995 


0.8042 


0.6847 


20 


1 . 0000 


0.9397 


0.8245 


0.6649 


0.4750 


25 


1 . 0000 


0.9063 


0.7321 


0.501-6 


0.2465 


30 


1.0000 


0.8660 


0.6250 


0.3248 


0.0234 


35 


1.0000 


0.8192 


0.5065 


0.1454 


-0.1714 


40 


1.0000 


0.7660 


0.3802 


-0.0252 


-0.3190 


45 


1.0000 


0.7071 


0.2500 


-0.1768 


-0.4063 


50 


1 . 0000 


0.6428 


0.1198 


-0.3002 


-0.4275 


55 


1.0000 


0.5736 


-0.0065 


-0.3886 


-0.3852 


60 


1 . 0000 


0.5000 


-0.1250 


-0.4375 


-0.2891 


65 


1.0000 


0.4226 


-0.2321 


-0.4452 


-0.1552 


70 


1 . 0000 


0.3420 


-0.3245 


-0.4130 


-0.0038 


75 


1 . 0000 


0.2588 


-0.3995 


-0.3449 


+ 0.1434 


80 


1 . 0000 


0.1736 


-0.4548 


-0.2474 


0.2659 


85 


1.0000 


0.0872 


-0.4886 


-0.1291 


0.3468 


90 


1.0000 


0.0000 


-0.5000 


0.0000 


0.3750 


t» 


P 5 (cost?) 


Ps(cos t?) 


P7(cos#) 


P8(cos t5) 


P»(cost») 


0° 


1 . 0000 


1 . 0000 


1.0000 


1.0000 


1.0000 


5 


0.9437 


0.9216 


0.8962 


0.8675 


0.8358 


10 


0.7840 


0.7045 


0.6164 


0.5218 


0.4228 


15 


0.5471 


0.3983 


0.2455 


0.0962 


-0.0428 


20 


0.2715 


0.0719 


-0.1072 


-0.2518 


-0.3517 


25 


0.0009 


-0.2040 


-0.3441 


-0.4062 


-0.3896 


30 


-0.2233 


-0.3740 


-0.4102 


-0.3388 


-0.1896 


35 


-0.3691 


-0.4114 


-0.3096 


-0.1154 


+0.0965 


40 


-0.4197 


-0.3236 


-0.1006 


+0.1386 


0.2900 


45 


-0.3757 


-0.1484 


+0.1271 


0.2983 


0.2855 


50 


-0.2545 


+0.0564 


0.2854 


0.2947 


0.1041 


55 


-0.0868 


0.2297 


0.3191 


0.1422 


-0.1296 


60 


+0.0898 


0.3232 


0.2231 


-0.0763 


-0.2679 


65 


0.2381 


0.3138 


0.0422 


-0.2411 


-0.2300 


70 


0.3281 


0.2089 


-0.1485 


-0.2780 


-0.0476 


75 


0.3427 


0.0431 


-0.2731 


-0.1702 


+0.1595 


80 


0.2810 


-0.1321 


-0.2835 


+0.0233 


0.2596 


85 


0.1577 


-0.2638 


-0.1778 


0.2017 


0.1913 


90 


0.0000 


-0.3125 


0.0000 


0.2734 


0.0000 



TABLES OF FUNCTIONS 



449 



Table X. — Phase Angles and Amplitudes for Radiation and Scattering 

from a Cylinder 
M = ka = (27ra/\) = (wa/c) (see page 301) 
[Dashes indicate values that can be computed by Eq. (26.6)] 



ka 


Co 


7C 


Ci 


71 


C 2 


72 


Cz 


73 


Ct 


74 


0.0 


00 


0.00° 


CO 


0.00° 


00 


0.00° 


00 


0.00° 


00 


0.00° 


0.1 


12.92 


.44 


63.06 


-.45 


2546 


0.00 


— 


0.00 


— 


0.00 


0.2 


6.651 


1.71 


15.55 


-1.82 


318.2 


-0.01 


9565 


0.00 


— 


0.00 


0.4 


3.583 


6.28 


3.875 


-6.97 


39.71 


-0.14 


600.7 


0.00 


— 


0.00 


0.6 


2.585 


12.82 


1.844 


-13.62 


11.72 


-0.69 


119.6 


-0.01 


1595 


0.00 


0.8 


2.091 


20.66 


1.199 


-18.73 


4.922 


-2.09 


38.20 


-0.06 


382.9 


0.00 


1.0 


1.793 


29.39 


0.9283 


-20.50 


2.529 


-4.77 


15.81 


-0.20 


127.3 


0.00 


1.2 


1.593 


38.74 


0.7884 


-18.94 


1.503 


-8.91 


7.712 


-0.57 


52.03 


-0.02 


1.4 


1.447 


48.52 


0.7035 


-14.80 


1.012 


-14.06 


4.212 


-1.35 


24.54 


-0.06 


1.6 


1.335 


58.62 


0.6453 


-8.84 


0.7627 


-19.03 


2.504 


-2.77 


12.85 


-0.16 


1.8 


1.246 


68.96 


0.6019 


-1.61 


0.6309 


-22.49 


1.596 


-5.08 


7.290 


-0.37 


2.0 


1.173 


79.49 


0.5676 


+6.52 


0.5573 


-23.69 


1.086 


-8.44 


4.405 


-0.79 


2.2 


1.112 


90.15 


0.5392 


15.31 


0.5130 


-22.56 


0.7898 


-12.71 


2.801 


-1.55 


2.4 


1.060 


100.93 


0.5152 


24.57 


0.4836 


-19.45 


0.6158 


-17.32 


1.861* 


-2.80 


2.6 


1.014 


111.81 


0.4944 


34.20 


0.4624 


-14.75 


0.5136 


-21.41 


1.287 


-4.73 


2.8 


0.9743 


122.75 


0.4760 


44.11 


0.4457 


- 8.84 


0.4535 


-24.15 


0.9265 


-7.46 


3.0 


0.9389 


133.76 


0.4597 


54.24 


0.4319 


- 1.99 


0.4175 


-25.09 


0.6965 


-11.01 


3.2 


0.9071 


144.82 


0.4450 


64.55 


0.4198 


+ 5.59 


0.3952 


-24.19 


0.5496 


-15.13 


3.4 


0.8785 


155.92 


0.4317 


75.01 


0.4090 


13.73 


0.3804 


-21.64 


0.4566 


-19.29 


3.6 


0.8524 


167.06 


0.4195 


85.58 


0.3992 


2.33 


0.3698 


-17.71 


0.3987 


-22.81 


3.8 


0.8286 


178.23 


0.4084 


96.25 


0.3901 


31.29 


0.3617 


-12.66 


0.3631 


-25.11 


4.0 


C.8067 


189.42 


0.3980 


107.01 


0.3816 


40.55 


0.3549 


-6.72 


0.3412 


-25.90 


4.2 


0.7865 


200.64 


0.3885 


117.83 


0.3737 


50.06 


0.3489 


-0.04 


0.3275 


-25.14 


4.4 


0.7678 


211.88 


0.3796 


128.72 


0.3662 


59.77 


. 3434 


+7.22 


0.3187 


-22.95 


4.6 


0.7503 


223.14 


0.3713 


139.65 


0.3592 


69.66 


0.3383 


14.97 


0.3126 


- 19 .54 


4.8 


0.7341 


234.42 


0.3635 


150.64 


0.3525 


79.70 


0.3334 


23.13 


0.3081 


-15.10 


5.0 


0.7188 


245.71 


0.3562 


161.66 


0.3462 


89.87 


0.3287 


31.62 


0.3044 


- 9.81 


ka 


Ci 


75 


Ce 


76 


Ci 


77 


C 8 


78 


C 9 


79 


2.0 


22.07 


-0.04° 


130.8 


0.00° 


903.5 


0.00° 


7144 


0.00 





0.00 


2.2 


12.82 


-0.10 


68.99 


0.00 


432.1 


0.00 


3099 


0.00 


— 


0.00 


2.4 


7.834 


-0.22 


38.65 


-0.01 


221.4 


0.00 


1452 


0.00 


— 


0.00 


2.6 


4.999 


-0.45 


22.78 


-0.03 


120.2 


0.00 


725.6 


0.00 


4941 


0.00 


2.8 


3.309 


-0.86 


14.03 


-0.06 


68.58 


0.00 


383.4 


0.00 


2418 


o.oo- 


3.0 


2.261 


-1.53 


8.967 


-0.13 


40.86 


-0.01 


212.6 


0.00 


1248 


0.00 


3.2 


1.590 


-2.59 


5.922 


-0.25 


25.27 


-0.02 


122.9 


0.00 


674.6 


0.00 


3.4 


1.149 


-4.18 


4.025 


-0.47 


16.16 


-0.04 


73.80 


0.00 


380.0 


0.00 


3.6 


0.8534 


-6.41 


2.805 


-0.83 


10.65 


-0.07 


45.80 


0.00 


222.1 


0.00 


3.8 


0.6539 


-9.35 


2.000 


-1.41 


7.200 


-0.14 


29.29 


-0.01 


134.1 


0.00 


4.0 


0.5190 


-12.92 


1.456 


- 2.30 


4.985 


-0.26 


19.24 


-0.02 


83.43 


0.00 


4.2 


0.4287 


-16.83 


1.082 


- 3.60 


3.526 


-0.46 


12.95 


-0.04 


53.32 


0.00 


4.4 


0.3693 


-20.62 


0.8211 


- 5.42 


2.542 


-0.77 


8.907 


-0.08 


34.92 


-0.01 


4.6 


0.3312 


-23.74 


0.6374 


- 7.85 


1.865 


-1.26 


6.252 


-0.14 


23.40 


-0.01 


4.8 


0.3071 


-25.76 


0.5081 


-10.88 


1.391 


-2.00 


4.471 


-0.25 


16.00 


-0.02 


5.0 


0.2921 


-26.44 


0.4177 


-14.40 


1.054 


-3.06 


3.251 


-0.42 


11.15 


-0.04 



450 



TABLES OF FUNCTIONS 



Table XI. — Phase Angles and Amplitudes fob Radiation and Scattering 

from a Sphere 

n = ka = (2tto/X) = (ua/c) (see page 320) 

[Dashes indicate values that can be computed by Eq. (27.17)] 



ka 


Do 


So 


Di 


Si 


Di 


Si 


D, 


Si 


Da. 


St 


0.0 


00 


0.00° 


00 


0.00° 


00 


0.00° 


00 


0.00° 


00 


0.00° 


0.1 


100.5 


0.02 


2000 


-0.01 


— 


0.00 


— 


0.00 


— 


0.00 


0.2 


25.50 


0.15 


250.1 


-0.08 


5637 


0.00 


— 


0.00 


— 


0.00 


0.4 


6.731 


1.12 


31.35 


-0.58 


354.6 


-0.01 


5906 


0.00 


— 


0.00 


0.6 


3.239 


3.41 


9.408 


-1.82 


70.73 


-0.06 


785.5 


0.00 


— 


0.00 


0.8 


2.001 


7.18 


4.101 


-3.80 


22.67 


-0.25 


188.9 


- 0.01 


2058 


0.00 


1.0 


1.414 


12.30 


2.236 


-6.14 


9.434 


-0.70 


62.97 


- 0.02 


547.8 


0.00 


1.2 


1.085 


18.56 


1.426 


-8.11 


4.646 


-1.59 


25.82 


- 0.08 


186.9 


0.00 


1.4 


0.8778 


25.75 


1.021 


-8.97 


2.583 


-3.07 


12.22 


- 0.22 


75.74 


-0.01 


1.6 


0.7370 


33.68 


0.7931 


-8.25 


1.584 


-5.19 


6.426 


- 0.51 


34.84 


-0.02 


1.8 


0.6355 


42.19 


0.6529 


-5.87 


1.057 


-7.77 


3.667 


- 1.05 


17.66 


-0.06 


2.0 


0.5590 


51.16 


0.5590 


-1.97 


0.7629 


-10.40 


2.236 


- 1.97 


9.669 


-0.15 


2.2 


0.4993 


60.49 


0.4918 


+ 3.21 


0.5901 


-12.49 


1.444 


- 3.38 


5.635 


-0.32 


2.4 


0.4514 


70.13 


0.4411 


9.44 


0.4837 


-13.51 


0.9823 


- 5.34 


3.459 


-0.64 


2.6 


0.4121 


80.01 


0.4011 


16.50 


0.4148 


-13.14 


0.7036 


- 7.80 


2.220 


-1.18 


2.8 


0.3792 


90.08 


0.3686 


24.23 


0.3676 


-11.31 


0.5308 


-10.54 


1.481 


-2.02 


3.0 


0.3514 


100.32 


0.3415 


32.49 


0.3333 


-8.11 


0.4214 


-13.16 


1.024 


-3.27 


3.2 


0.3274 


110.70 


0.3184 


41.18 


0.3071 


-3.73 


. 3508 


-15.17 


0.7334 


-5.00 


3.4 


0.3066 


121.20 


0.2985 


50.23 


0.2862 


+ 1.65 


0.3042 


-16.17 


0.5443 


-7.23 


3.6 


0.2883 


131.79 


0.2811 


59.57 


0.2688 


7.86 


0.2723 


-15.94 


0.4195 


-9.83 


3.8 


0.2721 


142.47 


0.2657 


69.15 


0.2540 


14.75 


0.2496 


-14.41 


0.3364 


-12.58 


4.0 


0.2577 


153.22 


0.2519 


78.92 


0.2411 


22.20 


. 2326 


-11.67 


0.2807 


-15.10 


4.2 


0.2448 


164.03 


0.2396 


88.88 


0.2296 


30.12 


0.2193 


- 7.84 


0.2432 


-17.00 


4.4 


0.2331 


174.91 


0.2285 


98.97 


0.2194 


38.44 


0.2084 


- 3.08 


0.2174 


-17.95 


4.6 


0.2225 


185.83 


0.2184 


109.20 


0.2101 


47.08 


0.1992 


+ 2.47 


0.1994 


-17.79 


4.8 


0.2128 


196.79 


0.2091 


119.55 


0.2016 


56.00 


0.1912 


8.70 


0.1863 


-16.47 


5.0 


0.2040 


207.79 


0.2006 


129.98 


0.1939 


65.16 


0.1840 


15.48 


. 1764 


-14.04 


ka 


Dt, 


Ss 


-De 


5<s 


Di 


57 


Z>8 


5s 


Dt 


59 


2.0 


51.31 


- 0.01° 


323.7 


0.00° 


2370 


0.00° 


— 


0.00° 


— 


0.00° 


2.2 


27.14 


- 0.02 


155.2 


0.00 


1030 


0.00 


7790 


0.00 


— 


0.00 


2.4 


15.25 


- 0.04 


79.69 


0.00 


483.5 


0.00 


3343 


0.00 


— 


0.00 


2.6 


9.021 


- 0.10 


43.38 


-0.01 


242.2 


0.00 


1541 


0.00 


— 


0.00 


2.8 


5.573 


- 0.20 


24.83 


-0.01 


128.2 


0.00 


755.6 


0.00 


5002 


0.00 


3.0 


3.576 


- 0.37 


14.83 


-0.03 


71.28 


0.00 


390.7 


0.00 


2407 


0.00 


3.2 


2.371 


- 0.68 


9.206 


-0.06 


41.34 


0.00 


211.7 


0.00 


1219 


0.00 


3.4 


1.620 


-1.16 


5.907 


-0.11 


24.88 


-0.01 


119.5 


0.00 


645.8 


0.00 


3.6 


1.137 


- 1.91 


3.904 


-0.21 


15.49 


-0.02 


70.01 


0.00 


356.1 


0.00 


3.8 


0.8183 


- 2.99 


2.649 


-0.38 


9.933 


-0.03 


42.39 


0.00 


203.6 


0.00 


4.0 


0.6043 


- 4.48 


1.841 


-0.66 


6.545 


-0.06 


26.44 


0.00 


120.2 


0.00 


4.2 


0.4583 


- 6.43 


1.308 


-1.09 


4.418 


-0.12 


16.94 


-0.01 


73.09 


0.00 


4.4 


0.3577 


- 8.79 


0.9486 


-1.73 


3.050 


-0.22 


11.13 


-0.02 


45.66 


0.00 


4.6 


0.2881 


-11.45 


0.7015 


-2.65 


2.148 


-0.37 


7.479 


-0.04 


29.24 


0.00 


4.8 


0.2399 


-14.15 


0.5290 


-3.92 


1.542 


-0.61 


5, 130 


-0.07 


19.16 


0.00 


5.0 


0.2065 


-16.56 


0.4072 


-5.58 


1.126 


-0.98 


3.587 


-0.12 


12.82 


-0.01 



TABLES OF FUNCTIONS 



451 



Table XII. — Impedance Functions for Piston in Infinite Plane Wall 

Jo(irao n r/a) 

l^n — ^Xn) 

n = 



p(r) = pcuoe-** V (e n - »x.) /o(7r "°" r/a) ; M = (2*»/A) (see page 334) 



M 


0o 


xo 


0i 


XI 


02 


*2 


03 


X» 


04 


X4 


0.0 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.0000 


0.5 


0.1199 


0.3969 


-0.0014 


-0.0480 


-0.0004 


-0.0180 


-0.0002 


-0.0096 


-0.0001 


-0.0062 


1.0 


0.4233 


0.6468 


-0.0200 


-0.1128 


-0.0058 


-0.0404 


-0.0028 


-0.0214 


-0.0016 


-0.0131 


1.5 


0.7740 


0.6801 


-0.0656 


-0.1806 


-0.0228 


-0.0616 


-0.0106 


-0.0324 


-0.0062 


-0.0204 


2.0 


1.0330 


0.5349 


-0.1966 


-0.2138 


-0.0496 


-0.0750 


-0.0228 


-0.0387 


-0.0132 


-0.0236 


2.5 


1.1310 


0.3231 


-0.3496 


-0.1664 


-0.0788 


-0.0653 


-0.0356 


-0.0392 


-0.0204 


-0.0238 


3.0 


1.0922 


0.1594 


-0.4404 


-0.0340 


-0.0868 


-0.0486 


-0.0386 


-0.0303 


-0.0220 


-0.0210 


3.5 


1.0013 


0.0989 


-0.4458 


+0.1502 


-0.0776 


-0.0408 


-0.0344 


-0.0286 


-0.0196 


-0.0188 


4.0 


0.9413 


0.1220 


-0.3462 


0.3078 


-0.0680 


-0.0604 


-0.0302 


-0.0374 


-0.0172 


-0.0272 


4.5 


0.9454 


0.1663 


-0.1830 


0.3740 


-0.0852 


-0.1016 


-0.0360 


-0.0532 


-0.0202 


-0.0355 


5.0 


0.9913 


0.1784 


-0.0260 


0.3420 


-0.1480 


-0.1348 


-0.0554 


-0.0644 


-0.0302 


-0.0390 


5.5 


1.0321 


0.1464 


+0.0568 


0.2394 


-0.2404 


-0.1202 


-0.0784 


-0.0605 


-0.0414 


-0.0398 


6.0 


1.0372 


0.0973 


0.0646 


0.1389 


-0.3268 


-0.0409 


-0.0920 


-0.0448 


-0.0470 


-0.0325 


3.5 


1.0108 


0.0662 


0.0188 


0.0820 


-0.3516 


+0.0882 


-0.0868 


-0.0286 


-0.0446 


-0.0251 


7.0 


0.9809 


0.0676 


-0.0260 


0.0811 


-0.3006 


0.2170 


-0.0738 


-0.0362 


-0.0382 


-0.0301 


7.5 


0.9727 


0.0881 


-0.0366 


0.1072 


-0.1850 


0.2901 


-0.0756 


-0.0646 


-0.0386 


-0.0423 


8.0 


0.9887 


0.1021 


-0.0140 


0.1246 


-0.0540 


0.2836 


-0.1140 


-0.0970 


-0.0516 


-0.0544 


8.5 


1.0115 


0.0948 


+0.0143 


0.1121 


+0.0296 


0.2272 


-0.1832 


-0.0989 


-0.0716 


-0.0552 


9.0 


1.0209 


0.0719 


0.0264 


0.0828 


0.0538 


0.1370 


-0.2602 


-0.0768 


-0.0870 


-0.0428 


9.5 


1.0111 


0.0515 


0.0130 


0.0568 


0.0264 


0.0795 


-0.2962 


+0.0487 


-0.0864 


-0.0270 


10.0 


0.9933 


. 0473 


-0.0072 


0.0514 


-0.0116 


0.0666 


-0.2710 


0.1609 


-0.0748 


-0.0254 



452 



TABLES OF FUNCTIONS 



Table XIII.- — Absorption Coefficients fob Wall Material* 
(See page 385) 
Based on Bulletin of Acoustic Materials Association, VII, 1940 



Material 



Values of absorption coefficient a 



Frequency, cps 



128 



256 


512 


1,024 


2,048 


0.27 


0.50 


0.88 


0.80 


0.26 


0.79 


0.88 


0.76 


0.40 


0.50 


0.80 


0.55 


0.03 


0.03 


0.04 


0.05 


0.25 


0.37 


0.34 


0.27 


0.27 


0.76 


0.88 


0.60 


0.08 


0.11 


0.25 


0.30 


0.27 


0.50 


0.80 


0.82 


0.01 


0.02 


0.02 


0.02 


0.04 


0.03 


0.03 


0.03 


0.04 


0.03 


0.03 


0.03 


0.04 


0.03 


0.03 


0.02 


0.01 


0.01 


0.02 


0.02 


0.34 


0.74 


0.76 


0.75 


0.02 


0.03 


0.04 


0.04 


0.40 


0.54 


0.52 


0.50 


0.50 


0.40 


0.35 


0.30 


0.07 


0.06 


0.06 


0.06 



4,096 



Acoustex 40R* 

Acoustone D* 

Acoustic plaster 

Brick wall, unpainted . . 

Carpet, lined 

CelotexC3* 

Curtains, light 

heavy drapes 

Floor, concrete 

wood 

linoleum 

Glass 

Marble or glazed tile. . . 

Permacoustic* 

Plaster, smooth on lath 

Temcoustic F2* 

Ventilator grill 

Wood paneling 



Object 



0.16 
0.13 
0.30 
0.02 
0.11 
0.25 
0.04 
0.10 
0.01 
0.05 
0.04 
0.04 
0.01 
0.19 
0.02 
0.33 
0.50 
0.08 



Values of absorption a = aA in sq ft 



Frequency, cps 



128 



Adult person, seated in audience 

Chairs, metal or wood 

Theater chair, wood veneer 

Leather upholstered 

Plush upholstered 



1.5 

0.15 

0.18 

1.0 

1.5 



256 


512 


1,024 


2,048 


2.5 


3.8 


5.0 


4.8 


0.16 


0.17 


0.19 


0.20 


0.20 


0.22 


0.25 


0.25 


1.2 


1.4 


1.6 


1.6 


2.0 


2.5 


2.8 


3.0 



0.70 
0.74 
0.50 
0.05 
0.24 
0.25 
0.30 
0.75 
0.02 
0.03 
0.03 
0.02 
0.01 
0.74 
0.04 
0.42 
0.25 
0.06 



4.5 

0.19 

0.23 

1.5 

3.0 



* Measurements for acoustic material cemented to plaster or concrete. Other means of mount- 
ing give other values for a. 

Acoustical Constants for Air and Water 
Air at 760 mm mercury, 20°C, cgs units: 

P = 0.00121, c = 34,400, P c = 42, P c 2 = 1.42 X 10 6 
y e = 1.40, Po = 1.013 X 10 3 
Water at 20°C, cgs units: 

P = 1.0, c = 146,000, P c = 1.5 X 10 s , pc* = 2.1 X 10 10 



PLATES 



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Resistance Ratio = (R/pc) 



Plate 1. — Bipolar plot of wave-reflection parameters a and /3 against impedance 
ratio f = (z/pc) = — i% = tanh [ir(a — t'/3)j. Standing wave ratio = e~ z * a . See 
pages 137 and 240. 



454 



PLATES 



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.; impedance ratio (z/pc) = f = If \e~ i,p = tan h [ir(a - t/3)]. See pages 137 and 240. 



PLATES 



455 



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See pages 137 and 241. 



456 



PLATES 



Maximum 
Pressure! 

Minimum 
Displacemei 



Maximum 
Displacement 




Scale for Standing Wave Ratio | q. I = I A r /A; 
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Corresponding Values of (A m in/Am<w) 



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(\q\ = standing wave ratio), for different valves of resistance and reactance ratios B 
and X . where (z/oc) = r = 6 - i X . Standing wave ratio is distance from center of 
plot to point 6, x- Angular measure for in units of half-wave-length along tube or 
fttring. See pages 137 and 241. 



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Plate VI. — Contours for absorption coefficient a corresponding to various values 
of magnitude and phase angle of acoustic impedance ratio (z/pc) = |f|e - **' for wall 
material. Only to be used for rooms and frequencies such that sound is uniformly dis- 
tributed in room. See page 388. 



INDEX 



Absorption, of sound, 360, 385 

wall, 403-405 
Absorption coefficient (see Coefficient) 
Absorption constant, 385, 452 
Acoustic admittance ratio, 240 
of horn, 285 

of surface, 306, 366, 369, 410 
Acoustic conductance ratio, 240, 306, 

367, 371, 403, 411 
Acoustic constants of air and water, 

222, 452 
Acoustic filters, 235, 290 
Acoustic impedance, 237 
analogous, 234, 237 
of constriction, 234 
of horn, 272, 285 
of opening, 235 
of tank, 235 
of catenoidal horn, 282 
characteristic, of air, 223, 238, 307, 

452 
of conical horn, 272 
at diaphragm, 200, 333 
of exponential horn, 281 
measurement of, 242, 456 
of piston in sphere, 324 
in wall, 332, 447, 451 
radiation, 237 

of surface, 360, 361, 365, 368, 388 
ratio, 240 

specific, 234, 237, 239, 287 
of surface, 366, 368, 388 
Acoustic power, sources of, 228 

of voice, 228 
Acoustic reactance ratio, 240, 367 
Acoustic resistance (see Acoustic imped- 
ance) 
Acoustic resistance ratio, 240, 367 



Acoustic susceptance ratio, 240, 306, 

367, 411 
Adiabatic compression, 221, 363 
Admittance, acoustic (see Acoustic 
admittance) 
mechanical, of oscillator, 35, 37, 50 

of string support, 144 
transfer, 93, 95, 105 
wave, 91 
Air, acoustic properties of, 222, 452 
load of, on membrane, 310, 335 
localized, 198, 361 
uniform, 193, 200, 361 
Allowed frequencies (see Frequencies, 

natural) 
Amplitude, average, of diaphragm, 201 
of harmonic, 87, 91 
of oscillation, 9, 87, 190, 201 
radiation, 301, 320 
velocity, 23, 224 
of wave, 266 
Analytic function, 14 
Analogous circuits, 233, 365 
Analogous impedance (see Acoustic 

impedance) 
Angle, of lag (see Phase angle) 
Angle-distribution function, 321 
Angular distribution, of radiated sound, 
from cylinder, 304 
from dipole, 318 
from piston, 323, 328, 331 
from point on sphere, 321 
of scattered sound, from cylinder, 
349 
from sphere, 354 
Angular momentum, of whirled string, 

119 
Antinode, 84 

Approximate calculation (see Perturba- 
tion calculations) 



459 



460 



VIBRATION AND SOUND 



Attenuation index, for porous material, 

366 
Attenuation, of sound, in ducts, 307, 

371 
Axial waves, 391, 405 



B 



Bar, 151-170 

clamped-clamped, 161, 170 

clamped-free, 157, 170 

free-free, 162, 170 

nonuniform, 164, 171 

plucked and struck, 160 

whirled, 171 
Bel, 225 

Bending, of plate, 209 
Bending moments, in bar, 153 
Bessel functions, 6, 17, 188, 298, 444 

spherical, 316, 446 
Boundary conditions, 75, 117, 175, 187 

for air in duct, 306, 369 

for bar, 157, 162 

for flexible string, 75, 84, 135 

for membrane, 175, 194 

at plane surface, 360, 367, 380, 388 

for plate, 210 

for stiff string, 167 

at wall of room, 388, 423 
Branch point, 13 



Capacitance, of crystal, 41, 69 
Catenoidal horn, 267, 271, 281-283, 292 
Cavity resonance, 228, 258, 359 
Characteristic frequencies, of vowel 

sounds, 233 
Characteristic functions, 107—109 
for bar, 159, 162, 171 
for membrane, 180-183, 189-191, 

195 
for plate, 211 
for sound in room, 389, 402, 415, 423 

in tube, 309, 373 
for string, 107, 112, 118, 143, 149 
Characteristic impedance, of air, 223, 
238, 364, 452 
in tube, 308, 372 



Characteristic impedance, for string, 93, 
126 

of water, 452 
Characteristic values, 107 

for bar, 158, 162 

for Bessel functions, 189, 399 

for membrane, 180, 189, 195 

for plate, 210 

for sound in room, 390-401, 411 

for string, 107, 113, 117, 149 

for wire, 168 
Circular membrane, 183-195, 214, 297, 

306 
Circular waves, 184-187, 297-311 
Clamped bar, 157, 161 
Clamped capacitance, 41 
Clamped impedance, 36, 38 
Clarinet, 248, 285 

Closed tube, 244, 249, 258-261, 291 
Coefficient, absorption, 385, 388, 452 

wall, 403-405, 425 
Coefficients, of Fourie: series, 87 

of power series, 4, 6, 7 

of series of characteristic functions, 
108, 160, 309, 428 

of series for scattered wave, 348, 354 
Coil, driving, electromagnetic, 34, 67, 

277, 289, 292 
Complex exponential, 7 
Complex numbers, 8 
Complex plane, 8 

integrals in, 12-19, 42-44 
Compliance, 21, 29 
Compressibility, of air, 220, 363 
Condenser microphone, 195-208, 211- 

215, 357-360, 379 .. 
Conditions, boundary (see Boundary 
conditions) 

initial (see Initial conditions) 
Conductance, acoustic (see Acoustic 
conductance) 

mechanical, 35, 144 
Configuration, plane of, 58 

of system, 57, 64 
Conformal transformation, 136, 373, 

453-457 
Conical horn, 271-279, 291-292 
Constriction, in tube, 234, 247 
Continuity, equation of, 218, 294 



INDEX 



461 



Contour integrals, 12-16, 19, 42-46, 

100, 132, 144, 206, 263, 288, 428 
Convention, for complex numbers, 9 
Coordinate systems, 174, 296 
Coordinates, normal, 56, 58, 90-91 
Cosine, 4, 438-439 
Coupled oscillators, 52-66, 68 
Coupling, small, 59 

of source, with standing waves, 415— 
418 
Coupling constant, 54 

electromagnetic, 35 

piezoelectric, 40 
Crystal, piezoelectric {see Piezoelec- 
tric force) 
Cutoff frequency, 235, 280, 282, 308 
Cylinder, force on, 352, 378 

radiation from, 297-305, 376 

scattering from, 347—352 

transmission inside, 305-311, 377 
Cylindrical coordinates, 296 
Cylindrical room, 398-401, 408-409 



D 



Damped vibrations, 23-27 

of air, in room, 386, 401-409, 427-429 
in tube, 262-264 

of bar, 170 

of membrane, 206-208 

of string, 106, 132, 145-146 
Damped waves, 243, 291, 307, 371, 

375, 380 
Damping constant, 24, 133, 243, 386, 

402, 416, 419, 426-429 
Decay, modulus of, 25 
Decay curve, for sound, in room, 406 
Decibel scale, 225 
Decrement, 25 
Degeneracy, 181, 190, 395 
Delta function, 48, 97 
Density of air, 218, 363, 452 

of solids, 152 
Design, of condenser microphone, 357- 
360 

of dynamic speaker, 339-344 

of horn loud-speaker, 274-279 
Dielectric susceptibility, 40 
Diffraction, of sound, 329, 350 



Dipole source, 318 

Dirac, 429 

Dirac delta function (see Delta func- 
tion) 

Directionality, of sound from piston, 329 
(See also Angular distribution) 

Dispersion, of waves, 154, 209, 307 

Displacement, of air in wave, 218 

Distribution, of natural frequencies, 
390-401 
of sound, in room, 383, 401 

Divergence, of vector, 294 

Driver, 27, 59, 62 

Ducts, sound transmission through, 
368-376 

Dynamic loud-speaker, 34, 67-69, 323- 
326, 338-344, 377-379 

Dyne, 2 



E 



Ear, response of, 226-228 
Echo, flutter, 261-264, 291 
Effective length, of tube, 235, 247 
Elastic modulus, of bar, 152 
of crystal, 40 
of plate, 209 
Electrical analogue, 233, 275, 365 
Electromagnetic drive, for bar, 171 
for diaphragm, 341-344, 377 
for horn, 276-279, 289, 292 
for oscillator, 34—38 
for string, 148 
Electromechanical driving force, 34 
Element, of cylinder, radiation from, 
300-303 
of sphere, radiation from, 321-323 
Energy, 2 

kinetic, 2, 89, 163 
in normal coordinates, 59, 91 
potential, 2, 90, 163 
of vibration, 23 
of bar, 162 

of coupled oscillators, 58 
of driven oscillator, 33 
of oscillator, 23 
of sound wave, 223, 224-226, 402, 

414 
of string, 89 



462 



VIBRATION AND SOUND 



Energy density of sound, 223, 240, 383, 

414 
Energy loss, 26, 33, 300, 319, 367, 403 
Energy transfer, 61 
Erg, 2 

Exponential function, 7, 438-439 
Exponential horn, 279-281, 292 
Extended reaction of surface to sound, 

200, 361 



F 



Fermi, Enrico, 422 

Filters, acoustic, 235, 290 , 

Flange, on tube, 246, 258, 326, 336- 

338 
Flexible string (see String) 
Flow resistivity, 363 
Fluctuations of sound in room, 408 
Flute, 248 

Flutter echo, 261-264, 291 
Force, 2 

on cylinder, due to wave, 352 
reaction of membrane to, 176 
on sphere, due to wave, 356 
transient, 16 
Forced vibration, of air, in tube, 242- 
258, 305-311, 368-372 
in horn, 271-287 
of bar, 166 

of coupled oscillators, 62 
of membrane, 195-206 
of oscillator, 27 
of plate, 211-213 
of sound, in room, 413-429 
of string, 91-100, 119-121, 129-132 
138-144 
Fourier series, 85, 97 
Fourier transform, 16, 43, 51, 94, 229, 

288,372 
Fraunhofer diffraction, 350 
Frequencies, complex, 45, 424 
natural, 22, 24, 107, 133 
of bar, 158, 162 
of coupled oscillators, 55, 68 
of kettledrum, 194 
of membrane, 180, 189, 195 
of plate, 210 
of resonating horn, 286 



Frequencies, natural, of sound, in room, 
390-401, 411 
of string, 84, 107, 113, 117, 149 
of wire, 168 
Frequency, 5 

antiresonance, 203, 212 
cutoff, 235, 280, 282, 308 
fundamental, 84 

natural (see Frequencies, natural) 
resonance, 31, 95, 198, 418-420 
Frequency distribution, of normal 
modes, in room, 390-401 
of sounds, 229 
Frequency space, 391-400 
Frequency spread, of sounds, 229, 419- 

420 
Fresnel diffraction, 350 
Friction, effect of, 23, 27 
on air in pores, 363—366 
on bar, 166 

on membrane, 198-203, 207 
on string, 104-106, 130 
Functions, analytic, 14 

Bessel, 6, 17, 188, 298, 316, 444-446 
characteristic (see Characteristic 

functions) 
delta, 48, 97 

hyperbolic, 18, 136, 438-439 
Legendre, 18, 118, 314, 448 
Neumann, 7, 196, 444-446 
step, 47 

trigonometric, 4, 438—439 
Fundamental frequency, 84 

(See also Frequencies, natural) 

G 

Gradient, 295 
Gyration, radius of, 152 



H 



Half-breadth, of resonance peak, 68 
Half width of pulse, 231 
Hanging string, 149 
Harmonics, 85 
even and odd, 88 

in clarinet, 252 

in closed tube, 245 

in open tube, 248 



INDEX 



403 



Helmholtz resonator (see Resonator) 
Horn, 265-271 

catenoidal, 267, 281-283 

conical, 271-279 

exponential, 267, 279-281 

resonance in, 263—287 

shape of, 269-271 

transients in, 287-288 

transmission coefficient of, 273, 281, 
282 

waves in, 265-269 
Hyperbolic Bessel functions, 210, 445 
Hyperbolic functions, 18, 136, 438-439 
Hyperbolic tangent, 136, 239-243, 255, 
284, 369, 423, 438-443 



Imaginary units i and j, 10 
Impedance, acoustic (see Acoustic 
impedance) 

blocked, of coupled oscillator, 63 

clamped, of coil, 36, 38 

input, of coupled oscillators, 63 

magnetomotive, 37, 38 

mechanical (see Mechanical imped- 
ance) 

motional, 38 
of crystal, 41 

mutual, of coupled oscillator, 63 

radiation (see Radiation load) 

of string support, 133, 144 

of surface, 360 

transfer, of coupled oscillators, 63 
for string, 93, 95, 105, 128 

of wave, in room, 417 

wave, for string, 91, 139 
Impulse excitation, 426 
Impulse function, 47 
Incident and reflected waves, 76, 134, 

238, 366, 388 
Index of refraction, for porous material, 

366 
Initial conditions, 21 

for bar, 160 

for membrane, 183 

for oscillator, 21 

for string, 74, 86 



Integral, contour, 12-16, 19, 42-46, 
100, 132, 144, 206 
infinite, 14^16 
Intensity, 223, 226, 414 

of radiation, from cylinder, 297-305, 
376 
from piston, 328-332, 377-379 
from simple source, 312—314, 386, 

414-^15 
from sphere, 311-324, 377 
of reflected sound, 367-368, 388 
of scattered sound, from cylinder, 
350-351 
from sphere, 354—356 
of sound, in room, 383, 386, 414 
in plane wave, 223, 226 
in standing wave, 240 
Intensity level, 226 

of sound, in room, 386, 415 
Interference of waves, 258, 303, 328, 

349-351, 355 
Interference fluctuations, of sound, 408 
Isothermal expansion, 221, 363 



K 



Kettledrum, 193-195 

L 

Laplace transform, 51, 293, 428 
Laplacian operator, 174, 295 
Legendre functions, 18, 118, 314, 448 
Level, intensity, 226, 386, 414 

pressure, 226, 415 

spectrum, 229 
Lippman color photographs, 423 
Loaded bar, 171 
Loaded membrane, 214 
Loaded plate, 216 
Local reaction of surface to sound, 198, 

336, 361 
Loop, 84 

Loudness, 226-228 
Loud-speaker, dynamic, 34, 67-69, 

323-326, 338-344, 377-379 
Loud-speaker horn, 265-293 



464 



VIBRATION AND SOUND 



M 



Magnetomotive impedance, 37 
Magnitude, of complex number, 10 
Mass, 2 

effective, for air load, 234, 247, 300, 
319 
for membrane, 202 
for string, 108, 124, 127 

nonuniform, of bar, 164 
of string, 111-114 
Mass-controlled vibrations, 33 
Mechanical admittance, 50 
Mechanical impedance, 29, 38 

of coupled oscillators, 63 

of oscillator, 29, 31, 50 
Mechanical input and transfer imped- 
ance, 63 
Mechanical reactance, 29 
Mechanical resistance, 24, 29 
Membrane, 172-208 

circular, 183-208 

forced motion of, 195-208 

forces on, 173, 176 

plucked, 185 

rectangular, 177-183 

struck, 183, 186 

transient motion of, 206-208 

waves on, 172, 184-187 
Microphone, condenser, 195—208, 211- 
215, 357-360, 379 

ribbon, 353, 378 
Modulus, of decay, 25 

of elasticity, 40, 152, 209 
Moment, bending, 153 
Motional impedance, 35, 38 



N 



Natural frequency {see Frequencies, 

natural) 
Neumann functions, 7, 196, 444-446 
Nodal line, 180-182, 190, 199, 202, 

211 
Nodal point, 84, 140, 241 
Noise, analysis of, 230 
Nonuniform bar, 164 
Nonuniform string, 107-121, 123-130 
Normal coordinates* 56, 58 



Normal modes of oscillation, 84, 107- 
109 
of bar, 156-160, 171 
of coupled oscillators, 55 
of membrane, 179-183, 189-191, 195 
of plate, 210-211 
for room, 390, 402, 415, 423 
for sound in tube, 309, 373 
of strings, 84, 91, 107, 112, 118, 143, 
149 
effective impedance for, 128 
Normalization constant, 108 



O 



Oblique waves, 391, 405 

Oboe, 286 

Open tube, 246-247, 255-258 

small-diameter, 247-253 
Operational calculus, 50, 104 
Organ pipe, 245, 248 
Orthogonality, of characteristic func- 
tions, 108 
Oscillations (see Vibrations) 
Oscillator, coupled, 52-66 

clamped, 23-27 

forced, 27-42 

simple, 20-23 
energy of, 23, 33 
Overtones, 84, 158 



Panel, reaction of, to sound, 361 

Parallel waves, 172 

Particle velocity, in sound wave, 222, 

224 
Period of vibration, 5 
Periodic motion, of string, 78, 86 
Perturbation calculations, 122—133, 

164-165, 402-412, 415, 422 
Phase, constant, surfaces of, 266-269 
Phase angle, 10 

for driven oscillator, 31 
Phase shift, scattering, for cylinder, 
301, 449 

for sphere, 320, 450 
Piano string, 102-104 
Piezoelectric constant, 40 
Piezoelectric force, 38 



INDEX 



465 



Piston, in duct, 309-311 
in sphere, 323-326 

in plane wall, 326-336, 344-346, 
447, 451 
Pitch, 226 

Plate, vibrations of, 208-213 
Point source, 313 
Poisson's ratio, 209 
Polar coordinates, 174, 187, 398, 409 
Polarization, electrical, 39 
Pole, of complex function, 13, 45 
Porosity, 363 
Porous surface, 363-366 
Power, absorbed by wall, 367-368, 385, 
388 
lost to friction, 33, 36 
radiated, from cylinder, 299, 302, 305 
from horn, 272, 278, 280, 282 
from opening, 235 
from piston, in plane, 338, 344 
from simple source, 313 
from sphere, 313, 318, 321, 322 
from various sources, 228 
scattered, by cylinder, 351 
by sphere, 355 
Power series, 4 
Pressure, of air, 218, 452 
mean square of, 414 
in tube, 240 

maxima and minima of, 241 
Pressure level, 226, 415 
Principal wave, in duct, 308, 371 
Pulse, of sound, frequency distribution 
of, 230 
in room, 126 
in tube, 261-264, 291 



Q 



Q of system, 25, 403 
Quadrupole source, 377 
Quality, tone, of sound, from bar, 158 
dependence on intensity, 228 
from string, 86 

R 

Radiation amplitudes, for cylinder, 301, 
449 
for sphere, 320, 450 



Radiation of sound, from cylinder, 
298-305 
from dipole, 318 
from piston, in plane, 326-336 
in sphere, 323-326 
in tube, 309-311 
from distributed source, 313 
from dynamic speaker, 338—344 
from open tube, 246-247, 336-338 
from simple source, 313 
from sphere, 311—326 
from vibrating wire, 299-300 
Radiation load, due to medium, 38, 104 
on diaphragm, 193, 198, 200 
on dipole source, 319 
on piston in sphere, 324 

in wall, 332, 451 
on wire, 300 

(See also Acoustic impedance) 
Radius of gyration, 152 
Reactance, acoustic (see Acoustic react- 
ance) 
mechanical, 29 
Reaction, of surface, to sound, 310, 
333 
extended and local, 361 
Reciprocity, principle of, 63, 352 
Rectangular membrane, 177—183, 199 
Reed instruments, 248—253 
Reed motion, 249 

Reflection, of waves, from absorbing 
surface, 366 
from obstacles, 347, 355 
on string, 76, 134 
Residue of function at pole, 13 
Resistance, acoustic (see Acoustic resist- 
ance) 
mechanical, 24, 29 
Resistance constant, 24 
Resistance-controlled vibrations, 33 
Resistivity, flow, 363 
Resonance, 31 

cavity, 228, 258-261 

of coupled oscillators, 64 

of horn, 283, 286-287 

of membrane, 202 

of oscillator, 31 

of plate, 212 

of room, 381, 418-420 



466 



VIBRATION AND SOUND 



Resonance, of string, 95, 99 

of tank, 235 

of tube, 245, 248, 252, 253, 256 
Resonance peak, half-breadth of, 68, 

419 
Resonator, Helmholtz, 235 
Response, of microphone, 205, 212, 358 

of room, 395, 419, 421 

transient (see Transient vibrations) 
Reverberation, 382, 386, 401 
Reverberation time, 387 
Ribbon microphone, 353, 378 
Rigidity, of panel, 361 
Room, cubical, 396 

cylindrical, 398-401, 409 

nonrectangular, 397-401 

rectangular, 389-394, 402-408, 418- 
429 

response of, 395, 419, 421 

reverberation in, 386-388 



S 



Salmon, Vincent, 271 

Scattered and reflected sound, 347, 350 

Scattering, of sound, 346-347 

from absorbing patches, 410, 412 
from -cylinder, 347-352 
from sphere, 354-357 
Scattering shift, of phase, for cylinder, 
348, 449 
for sphere, 354, 450 
Separation of variables, 179 
Series, of characteristic functions, 108, 
183, 415 
Fourier, 85, 97 
power, 4 
Series coefficients, 87, 108 
Shadow, acoustic, 350 
Shearing force, in bar, 153 
Simple harmonic vibrations, 23, 80, 

156 
Simple source (see Source) 
Sine, 4, 438-439 

Sound power of various sources, 228 
Source, dipole, 318-319 

simple, 312-313, 415, 418-421 
on sphere, 321-322 
Source function, 313 



Specific acoustic impedance (see Acous- 
tic impedance) 
Specific heats of gas, 220, 452 
Specific impedance of surface, 306 
Spectrum level, 229 
Sphere, radiation from, 311-326 

scattering from 354—357 
Spherical Bessel functions, 316; 446 
Spherical coordinates, 296, 311 
Spherical room, 401 
Spring supports, for string, 146 
Spring, 3 

mass on, 3, 20-33 
Standing waves, in air in tube, 240 

in room, 381, 389 

on string, 83, 140 
Standing wave ratio, 142, 456 
Statistical analysis, of sound in room, 

382-385 
Steady state, 29, 31 
Step function, 47 
Stiff string, 166-170 
Stiffness constant, 21 
Stiffness-controlled vibrations, 33 
Strain in crystal, 39 
Strength of simple source, 313 
Stress, in crystal, 39 
String, damped, 104-106 

energy of, 89-90 

forced motion of, 91-104, 129-132, 
138-141 

free vibration of, 84-90 

hanging, 149 

impedance of, 91, 95, 126-129 
characteristic, 93 

nonuniform, 111-113 

plucked and struck, 80, 87 

stiff, 166 

waves in, 72-80, 134-136 

weights on, 68 

whirling, 114-121 
String support, effect of, 76, 134 

impedance of, 133 
Surface, acoustic impedance of, 362 
porous, impedance of, 363-366 

sound absorption at, 360-368, 388 
Susceptance, acoustic (see Acoustic 
susceptance) 

mechanical, of oscillator, 35 



INDEX 



467 



Susceptaace, mechanical, of string sup- 
port, 144 
Susceptibility, dielectric, 40 
Symmetry of room, effect of, 395 



Tangential waves, 391, 405 

Tank, 235 

Temperature change due to sound, 222, 

288 
Tension, on membrane, 173 
on string, 72 
variable, 116 
Thermodynamic relations for gas, 220- 

221 
Threshold, of hearing, 227 

of pain, 227 
Transducer, electromagnetic, 34-38 

electromechanical, 38-42 
Transfer admittance and impedance, 
of coupled oscillator, 63 
for string, 93, 95, 105 
"for walls of room, 411 
Transform, Fourier, 16, 43, 93 

Laplace, 51, 293, 428 
Transformation, conformal, 136 
Transient force, 16, 42, 93, 110 
Transient vibrations, 42-52 
of air, in horn, 287-288, 293 
in room, 386, 426-129 
in tube, 261-264, 372 
of coupled oscillators, 64 
of diaphragm, 206-208 
of oscillator, 29, 44-52 
of radiation, from piston, 344-346 
of string, 93, 100-106, 110, 132, 145, 
147-148 
Transmission, of sound, through ducts, 
368-376 
inside cylinders, 305-311 
inside horns, 265-283 
through porous material, 365-366 
in room, 418-425 

inside tubes, 224, 237-240, 254- 
258 
Transmission coefficient, 273 
of horn, 273, 281, 282 
of piston, 338 



Transmission line, analogous, 254, 366 
Trigonometric functions, 4, 438 
Tube, analogy of, with transmission line, 
254 

cavity resonance in, 258—261 

closed, 244-245 

constriction in, 234 

effective length of, 234 

flaring (see Horn) 

open, 246-253, 255-258 

waves in, 222-225, 238-244, 305-311, 
368-376 
Tuning fork, 158 



U 



Unit impulse function, (see Delta 

function) 
Unit step function, (see Step function) 
Units, physical, 2 



Velocity, particle (see Particle velocity) 
wave (see Wave velocity) 

Velocity amplitude, 23 

Vibrations, damped (3ee Damped vibra- 
tions) 
free, 20-23 

of air, in room, 390-401 
of bar, 156-160, 168 
of coupled oscillators, 54-59 
energy of, 23, 58, 89, 214, 223, 296 
of membrane, 180-183, 187-191 
of plate, 208 
simple harmonic, 20-23 
of string, 84-91, 112-118 
forced (see Forced vibrations) 

Vibrator (see Oscillator) 

Vowel sounds, 232 



W 



Wall coefficient (see Coefficient) 
Warble tone, 379 
Water, acoustic constants of, 452 
Wave, 72-74 
axial, tangential and oblique, 391- 
400, 405, 409 



468 



VIBRATION AND SOUND 



Wave, in bar, 155-156 

circular, 184, 191 

damped, 243, 307, 371 

longitudinal and transverse, 217 

in membrane, 172, 184-186, 191 

plane, 217, 266-268 

in porous material, 365-366 

principal, in duct, 308, 371 

standing, 83, 140, 240, 381, 389 

in string, 71-80, 92-93 

in wire, 167 
Wave admittance, 91 
Wave equation, 81 

for air, 221, 294 

for horn, 269 

for membrane, 174 

for porous material, 364 

separation of, 179, 268 

for string, 81 
Wave impedance, of principal wave, in 
duct, 372 

for string, 91, 139 

(See also Acoustic impedance) 
Wave number, 225 
Wave velocity, 72 



Wave velocity, in air, 222, 452 

in bar, 154 

in membrane, 177, 187 

in* string, 72 

in tube, 307, 371-374 

in water, 452 
Wavelength, 83, 134, 225 
Weighted string, 68, 71 
Width, of pulse, 231 

of resonance peak, 68 
Whirled bar, 171 
Whirled string, 114 
Wind instruments, 253, 285 
Wire, radiation from, 299 

vibrations of, 166-169 
Work, 2 



Young's modulus of elasticity, 152 

Z 

Zero response of diaphragm, 202-203, 
212-213 



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Ll 



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