The Symbols
Historical Introduction
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Research and scientific progress are based upon 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 necesary 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 acomplish 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 computations.
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 discusses 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 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 application 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 sound sourced in general; they are responsible for the sensation of distance and, 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 radiation and sound scattering 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 coordinates. Chapter XXIII presents Green's theorem and the theory of the Helmholtz—Huygens diffraction integral. Chapters XXIV and XXV are devoted to the theories of diffraction. The Rubinowicz—Kirchoff 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 computations. The Sommerfeld theory of the straight edge is discussed in detail, because this theory is becoming very important for analyzing sound propagation 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 information about the diffraction caused by perfectly absorbent surfaces. For instance, it would not be possible to make a structure acoustically "invisible" by coating it with perfectly absorbent material. Some of the incident 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 characteristics 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. References 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 publications in acoustics from 1949. 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 knowledge; 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.