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The Foundations of Acoustics Basic Mathematics and Basic Acoustics Eugen Skudrzyk 1971 Springer-Verlag New York Wien EUGEN SKUDRZYK Professor of Physics, Ordnance Research Laboratory and Physics Department, The Pennsylvania State University, University Park, Pa., USA With 197 Figures This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re·use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks © 1971 by Springer-VerlagjWien Softcover reprint of the hardcover 1st edition 1971 Library of Congress Catalog Card Number 76-161480 ISBN- 13:978-3-7091-8257 -4 e-ISBN -13:978-3-7091-8255-0 DOl: 10.1007/978-3-7091-8255-0 To Liselotte Skudrzyk Preface Research and scientific progress are based upqn intuition coordinated with a wide theoretical knowledge, experimental skill, and a realistic sense of the limitations of technology. Only a deep insight into physical phenomena will supply the necessary skills to handle the problems that arise in acoustics. The acoustician today needs to be well acquainted with mathematics, dynamics, hydrodynamics, and physics; he also needs a good knowledge of statistics, signal processing, electrical theory, and of many other specialized subjects. Acquiring this background is a laborious task and would require the study of many different books. It is the goal of this volume to present this background in as thorough and readable a manner as possible so that the reader may turn to specialized publications or chapters of other books for further information without having to start at the preliminaries. In trying to accomplish this goal, mathematics serves only as a tool; the better our understanding of a physical phenomenon, the less mathematics is needed and the shorter and more concise are our computa tions. A word about the choice of subjects for this volume will be helpful to the reader. Even scientists of high standing are frequently not acquainted with the fundamentals needed in the field of acoustics. Chapters I to IX are devoted to these fundamentals. After studying Chapter I, which dis cusses the units and their relationships, the reader should have no difficulty converting from one system of units to any other. Years of experience in the teaching of acoustics show that students invariably make mistakes in applying complex notation to problems of acoustics. It is the purpose of Chapter II to thoroughly familiarize the reader with complex notation and with the symbolic method of solving linear differential equations. The chapter ends with brief discussions of the acoustic loss factor and the exact treatment of internal dissipation for harmonic time variations through the introduction of complex elastic constants or a complex sound velocity. Chapter III summarizes the results of the theory of complex functions. This chapter also lays the groundwork for the Sommerfeld method of dealing with diffraction problems with the aid of Riemann spaces. Series are summed by transforming them into contour integrals, and complicated integrals are solved by contour integration. The selection of the path of integration in contour integrals and of the branch cuts in branch-cut integrals is treated in detail. The saddle point and the stationary phase methods are derived since both these methods are needed for computing the farfield radiation of complex sound generators. The chapter ends with VIII Preface a basic discussion about the singular points of differential equations and their effect on the solutions. Chapter IV deals with the Fourier series and the Fourier integral. The theory of the warble tone is treated as an appli cation of the Fourier method. In Chapter V (Advanced Fourier Analysis), theorems are derived that are helpful in evaluating Fourier integrals. Convergence is enforced in special cases by assuming infinitely small damping. Chapter VI relates the Laplace transform to the Fourier transform with damping. The basic rules and formulae are summarized in tables. Chapter VII gives a brief discussion of the various other transforms such as the Hankel and Mellin transform and Chapter VIII deals with correlation analysis and with the basic relations and definitions that apply to power spectra, correlation functions, cross spectral densities and cross correlation functions. Chapter IX introduces a variation of the Fourier analysis that has been derived by WIENER. Convergence is enforced by working with the frequency or wave number integrals of the spectral amplitude, because these integrals exist for statistically varying functions of zero mean. In Chapter X, transients generated as a consequence of a frequency dependent complex transmission factor are studied. Transients are of utmost importance in acoustics. Because the transients of electrical and mechanical systems generate hissing sounds at high frequencies and time delays at low frequencies, they reduce the acoustic quality of a musical reproduction. Transients represent the distinguishing marks for musical instruments and sOl,Ind sources in general; they are responsible for the sensation of distance anU, in closed rooms, for the sensation of direction. The search tone method of spectral analysis has been extensively used in the past. The theory of this analysis is discussed in detail. It is shown that it leads to erroneous results for unsteady sounds and impulses even if they are repeated a few times every second. Chapter XI covers the basics of probability theory and statistics. Many fields of modern acoustics depend on statistics, e. g., signal processing, flow noise, sound scattering from the surface of the sea, and sound scattering in metals. Some of the modern theories of vibrations and sound radiation are based on statistical computations. Chapter XII is an attempt to acquaint the reader with the theories and methods that are of importance in signal processing. Very few acousticians are aware of the importance of this field in acoustic communication and in ·sonar. Signal processing makes it possible to interpret signals that are far below the general noise level. Signal processing makes it possible to trade transmission power for analysis time. High-quality signal processing, for instance, made it possible to receive signals from as far away as the planet Mars with a 0.5 watt transmitter. Acoustics starts with Chapter XIII. Chapters XIII to XVIII deal in the conventional way with the derivation of the wave equation, with reflection and refraction of plane waves and with wave propagation in nonabsorbent channels. Chapters XVIII to XXI are devoted to propagation phenomena and to sound scattering in spherical, cylindrical, and spheroidal coordinates. Spheroidal coordinates are used to compute the sound radi- Preface IX ation and sound scattering of ellipsoidal bodies, piston membranes that are not in a baffle, and needle shaped bodies. The spheroid seems to give a more realistic approximation to the sound radiation of a ship than a finite cylinder. With the exception of the sound radiation for its rigid body and its breathing modes, the sound radiation is extremely small at low frequencies. But it increases with a high power of the frequency and is already appreciable below the coincidence frequency; it then increases slowly to that of a similar cylinder as the frequency is increased above the coincidence frequency. The wave equation in spheroidal coordinates is of particular interest. The eigenvalues are no longer constants but depend on the frequency. The mathematical situation is considerably more complex than it is for waves described by cylindrical and spherical coordi nates. Chapter XXIII presents Green's theorem and the theory of the Helm holtz-Huygens diffraction integral. Chapters XXIV and XXV are devoted to the theories of diffraction. The Rubinowicz-Kirchhoff theory that decomposes the refracted field into a geometrical optical field, and a field that originates at the boundaries of the diffracting object is derived in detail. The theoretical results are compared with results of exact computa tions. The Sommerfeld theory of the straight edge is discussed in detail, because this theory is becoming very important for analyzing sound propa gation around edges and other discontinuous structures. The Sommerfeld theory also gives the background for very good asymptotic approximations to the diffraction problem of bodies of any shape, and it gives exact infor mation about the diffraction caused by perfectly absorbent surfaces. For instance, it would not be possible to make a structure acoustically ain_ visible" by coating it with perfectly absorbent material. Some of the in cident intensity will be reflected just because of the discontinuity of the wave field at the acoustic shadow boundary. Chapter XXVI deals with the fundamentals of arrays of transducers and with the radiation charac teristics of membranes. The properties of the Green's function of the wave equation and of its basic forms are summarized in Chapter XXVII. The last Chapter (XXVIII) is reserved for a discussion of radiation impedance and mutual impedance. The effect of resonance, the use of acoustic impedance methods, room acoustics, vibrations of simple and complex structures, vibration statistics, asymptotic relations, the sound radiation of finite plates and shells, and other subjects will be treated later in special publications. The list of references at the end of the book contains the most important publications dealing with the subject matter and related material. Referen ces are given by name and date, i. e., "H. STENZEL 1946" refers to the paper published by H. STENZEL in 1946. The references given should make it possible for the reader to pursue further a particular subject. Most of the referenced papers and books themselves contain lists of references, so that a complete list for a particular field is easily collected. The supplementary volumes published by JASA give an almost complete survey of the publica tions in acoustics from 1949. x Preface The reader who is not acquainted with acoustics is advised to start this book by studying Chapters I, II, IV, and Chapters XIII through XVIII. Students frequently like problems to test and deepen their knowl edge; however, this author believes that such problems do more harm than good. They make the student waste valuable time that he could use more efficiently in studying the theory and trying his skills by repeating the derivations on his own. Problems are of value only if they also contain detailed discussions of how the answers should be worked out, or the student will derive his results by poor and impractical methods. At a later date, a list of such problems with full instructions on how to arrive at the answers will be published. The advanced student who wants to test his knowledge is advised to look at the references at the end of the book, and to sketch on paper how he would deal with some of the subjects. He can then compare his computations with those in the original paper. Some may feel that the material has been selected in an arbitrary manner. However, there has been little or no freedom of choice. The material presented in this book is needed for further studies of acoustics; the material is basic for later publications that will concentrate more on practical applications of acoustics. Acknowledgements I am indebted and very grateful to the authors of those books that were of such great help to me in writing this volume. Special acknowledgement goes to P. M. MORSE and H. FESHBACH for their outstanding book "Methods of Mathematical Physics", McGraw-Hili Book Company, New York, 1953. This book contains a wealth of original information that cannot be found elsewhere. MORSE and INGARD, "Acoustics", McGraw-Hill Book Company, New York, 1968, has also been helpful. H. STENZEL'S book, "Leitfaden zur Berechnung von Schallvorgangen", Springer, Berlin, 1939, was of basic use for the study of spherical propagation phenomena and of arrays of trans ducers. The theory of diffraction is discussed in A. RUBINOWICZ, "Die Beugungswelle in der Kirchhoffschen Theorie der Beugung", Springer, Berlin-Heidelberg-New York, 1966. The author of this text devoted his life to the study and development of the theory of diffraction. This book is original and of outstanding quality. Sections of "Basic Mathematics and Acoustics" are based on early papers by A. RUBINOWICZ and chapters of his book. A book that also needs mentioning is: A. SOMMERFELD'S, "Partial Differential Equations in Physics", Academic Press, New York, 1964). This text is very helpful in the study of contour and branch line integrals and of the Hankel and Bessel functions and their integral representations. The quality of scientific research frequently parallels that of one's mathematical tables. Much time is frequently wasted in the derivation of formulae or the solving of integrals that have been derived or solved before. Very good tables have been published by ABRAMOWITZ and STEGUN, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", Dover Publications, Inc., New York, 1964; and the excellent tables of formulae and integrals by GRADSHTEYN and RYZHIK, Preface XI "Table of Integrals Series and Products", Academic Press, New York, 1965, are the most complete tables that have ever been assembled. Some of the tables in this book have been supplemented with material 'computed by these authors. Not as complete but also very good are the "Integral Tables" by W. GROBNER and N. HOFREITER, Springer, Wien, 1957. I am deeply obligated to the Ordnance Research Laboratory and its director, Dr. JOHN C. JOHNSON, for providing every possible assistance in preparing this volume, and also to Prof. D. H. RANK and Prof. W. WEBB of the Physics Department for their valuable support. Without this help, the book could not have been written. I am very grateful to the Office of Naval Research (Dr. N. PERONE and Dr. N. BASDEKAS, Task NR 064-475) for their stimulus and support. I thank my esteemed teacher, Prof. E. MEYER, and my former colleagues, G. BUCHMANN, W. KEIDEL, W. KUHL, H. MEINL, H. OBERST, A. SCHOCH, and K. TAMM, who are responsible for the solution of a great number of problems discussed in this book. Thanks also go to the many of my present friends who helped improve the manuscript, particularly to Prof. R. O. ROWLANDS, and Prof. J. L. BROWN, JR., and to Dr. W. THOMPSON, JR., for critically reading and correcting the manuscript. I am also very grateful to Prof. F. G. BRICKWEDDE for· his help in writing Chapter I, and to Mr. J. LEFRANCOIS, editor at the Ordnance Research Laboratory, for constant assistance in styling and improving the manuscript. Special thanks are directed to Mrs. P AULETTA LEIDY for her help in the preparation and typing of the manuscript, and to the librarians, Mrs. LUCILLE J. STRAl;TSS, Mrs. VIRGINIA FRANK, and Mrs. WILMA HEISER for their generous assistance. The Pennsylvania State University July 1971 Eugen Skudrzyk Table of Contents The Symbols ..... . . XXV Historical Introduction 1 I. Equations and Units . 6 1.1. Dimensional and Numerical Equations 6 1.2. The kg.m.sec·amp System of Units . . 6 1.3. The Definition of the Unit of Electric Current, the Ampere [A] 7 1.4. Derived Electrical Units ................ . 8 1.5. The Practical (Physical) Units for Electrical Quantities 9 1.6. The Fundamental Electrical Laws 9 1. 7. Transformation of Units .... 14 II. Complex Notation and Symbolic Methods 17 2.1. Complex Notation and Rotating Vectors 17 2.2. Computations with Complex Vectors 19 2.2.1. Definitions 19 2.2.2. Addition 20 2.2.3. Subtraction . . 20 2.2.4. Multiplication . 21 2.2.5. Division .... 21 2.2.6. Logarithm of a Complex Number. 22 2.2.7. Raising a Complex Number to a Given Power 22 2.2.8. Differentiation and Integration ........ . 23 2.3. Conjugate Complex Vectors and Their Applications . 24 2.4. Addition of Harmonic Functions of the Same Frequency 24 2.5. Symbolic Method for Solving Linear Differential Equations 25 2.6. Complex Solution and Boundary Conditions 27 2.7. Computation of Power ................ . 28 2.8. Basic Theory of Internal Friction .......... . 29 m. Analytic Functions: Their Integration and the Delta Function 33 3.1. Analytic Functions .................. . 33 3.2. Representation of an Analytic Function by a Power Series 34 3.3. Cauchy's Formula ...... . 35 3.4. The Cauchy Integral Formula 36 3.5. Residues . . . . . . . . . . . . 37 3.6. Examples .......... . 38 2" 3.6.1. Evaluation of Integrals of the type JR(cosO, sinO)dO 38 o 3.6.2. Summation of a Series by Contour Integration 39 j 3.7. Evaluation of Integrals of the Form Q (x) dx •.. 39 -co 3.7.1. Integrals Involving Sines and Cosines .... 41 3.8. Contour Integrals for Hankel and Bessel Functions 41 3.9. Jordan's Lemma .................. . 45 3.10. Integrals Through Poles, Principal Value of Integrals 46 3.11. Multivalued Functions ................ . 48