NORTH-HOLLAND SERIES IN APPLIED MATHEMATICS AND MECHANICS EDITORS: H A LAUW ERIER Institute of Applied Mathematics University of Amsterdam W T KOITER Laboratory of Applied Mechanics Technical University, Delft VOLUM E 23 NO RTH-HOLLAND PUBLISH IN G COM PANY AMSTERDAM · NEW YORK · OXFORD UNIDIRECTIONAL WAVE MOTIONS H LE V IN E Department of Mathematics , Stanford University 1978 NO RTH -H OLLAND PU BL ISH IN G CO M PANY AM STERDAM · NEW YORK · OXFORD © North-Holland Publishing Company 1978 All rights reserved N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner North-Holland ISB N : 444 85043 Published by: North-Holland Publishing Company Amsterdam · New York · Oxford Sole distributors for the U S.A and Canada: Elsevier North-Holland, Inc 52 Vanderbilt Avenue New York, N Y 10017 Library of Congress Cataloging in Publication Data Levine, Harold, 1922— Unidirectional wave motions (North-Holland series in applied mathematics and mechanics; v 23) Bibliography: p 499 Includes index Wave-motion, Theory of I Title QA927.L44 531Μ13Ό151 77-15565 IS B N 444 85043 Printed in The Netherlands To my wife, Barbara, who inspires in all ways Preface Wave phenomena, with their multi-faceted aspects and diverse physical associations, sustain a considerable and continuing interest It is well nigh impossible to survey, let alone assimilate, the accumulated findings over the centuries; and hence the acquisition of both perspective and procedural facility in this important sphere becomes ever more difficult Students are conventionally introduced to wave theory as it bears on the particular subject matter of courses in mathematics, physics, and engineering, while the benefits of systematization in concept and technique remain unexplored for the most part Books about wave theory in general are not so numerous, though some recent titles which manifest authoritative content and broad scope facilitate progress towards greater appreciation and technical mastery of the subject This entrant to the published ranks aims to describe, in a comprehensive manner, the formulations and their consequent elaborations which have found demonstrable value in wave analysis; the deliberate focus on unidirectional waves permits a relatively simple mathematical development, without leaving significant gaps in methodology and capability Since it is the full resources of mathematics which underlie successful dealings with all manner of wave problems, there are sufficient grounds for a detailed account of methods per se, encompassing both the active or contemporary representatives as well as those previously - and possibly again to be - judged useful Sections of com parable size constitute the organizational framework of the book, marking a departure from the more typical arrangement of subject matter in chapters, and it is hoped that direct access to individual sections may enhance the utility of the work; problems are included, some taken (with permission) from examination papers set in The University of Cambridge, to display the nature of analytical solutions and further the ends of technical competence An author’s indebtedness, in connection with an undertaking such as this, to information and benefit received from many sources cannot be adequately documented; special thanks are due, however, to T T Wu and F G Leppington for stimulating inputs, and to J Schwinger for initially spotlight ing the elegance and versatility of applied mathematics I am also grateful to Priscilla R Feigen for an abundance of practical support and to the Office of Naval Research for prior material assistance H Levine Introduction Wave motions figure prominently, now as heretofore, in numerous and diverse physical phenomena, and their understanding is mirrored in the evolution of the related theories An impressive roster of technological developments pertains to instances of wave propagation in materials or acoustical/electrical settings; and the less common though significant natural disturbances of seismic or tidal origin direct attention to both the practical and theoretical aspects of wave motions on a terrestrial scale Since the ramifications of wave motion have long permeated the physical sciences, a vast corpus of general and special theory, techniques of ap proximation and descriptive patterns exists and is continually being aug mented On the purely mathematical side, investigations have contributed to the fund of knowledge concerning the nature of continuous and discontinuous solutions to the equations of wave motion, and to the conditions for the existence and uniqueness of solutions when various initial and boundary conditions are laid down, as befit the excitation and subsequent interaction of the waves with spatial inhomogeneities The approximations devised by Fresnel (before the differential equations governing the propagation of light in vacuo were known), and later by Kirchhoff, in connection with the theory of optical diffraction at an aperture in an opaque screen, and the rules of geometrical optics, are examples of efficient and economical procedures for some restricted purposes; such improvisations not rest on firm mathemati cal foundations or provide quantitatively accurate details for all circum stances, but they testify nevertheless to the utility of sound physical insight in dealing with complex situations In this book we propose to follow a middle course between the extremes of a formal mathematics of wave motion on the one hand, and of a collection of physically motivated approximations on the other; the object will be to describe and apply a sufficiently general ma thematical basis which is capable of furnishing both the gross and fine details of varied wave phenomena, and which brings to the subject the benefits of integration and systematization The subject of wave motions may be broadly (though not exclusively) subdivided under the headings: Description, Excitation, and Interaction To the first belongs the particulars of individual wave equations, the geometry or kinematics of wave forms, and general matters bearing on the transport of energy and momentum, as well as dispersion or frequency sensitivity Since an ever growing number of wave problems find their origins rooted in anisotropic or random media (including elastic substances, plasmas, and electrically conducting fluids), geometrical and analytical techniques of con siderable scope are essential to achieving a full account of the manifold and distinctive phenomena possible The second heading embraces aspects of wave generation by localized and extended sources, respectively; and it is noteworthy, in this context, that plane wave forms, attributable to infinitely Xll INTRODUCTION remote sources, shed no light on the (non-dissipative) geometrical attenuation of wave amplitudes which reflects physical reality Point source, or so-called Green’s, functions for all types of wave motions enjoy a prominent status; that is, both differential and integral representations of said functions, and their asymptotic form when the source and observation points are widely separated, find ready use in the formulation and analysis of specific problems Sources with finite extension, such as acoustical oscillators and electrical antennas, have a practical importance, prompting studies of directional characteristics (or radiation patterns) and the influence thereupon of changes in the ratio between the emitted wave length and the source dimension Apart from such actual or primary sources, it is advantageous to obtain represen tations of the wave function (via the Green’s functions) in terms of equivalent sources, whose strengths are related to assigned or unknown values of the former on particular lines or surfaces; these representations are especially useful as regards the third and final heading of our subject matter, wherein the effects of interaction between specified primary waves and medium ir regularities (e.g., obstacles, inclusions, or local variations in the material parameters) Information about these irregularities or scatterers, ranging from microscopic to terrestrial scales, may be gleaned through the attendant phenomena of reflection, refraction, and diffraction, which are fundamental to wave theory There is an impressive roster of exact and approximate results for scattering in different configurations, whose existence rests on utilizing the full resources of mathematical analysis, and the application of complex function theory, integral transforms, asymptotic expansions, and variational formulations in particular; point source functions are recognizable throughout the analysis, being either the very object of calculation in composite (rather than uniform) media, or acting as a kernel for the representation of secondary waves that symbolize an interaction of the primary wave with the in homogeneity Topics allocated to different subdivisions of the subject matter are in terwoven, of course, in particular studies and their exposition need not therefore be rigidly circumscribed by the original ordering Since it is mani festly impossible to detail even a fraction of the innumerable wave problems and their resolutions within the confines of a single volume, we shall focus our attention on the class of problems which involve wave propagation along a fixed direction; this affords a relative simplicity in procedure and ex pression, without incurring any significant loss of perspective as would impede analogous studies of multi-directional/dimensional wave phenomena PART I §1 Flexible String Movements The vibrations of strings have long enjoyed an especial pride of place for introducing (and demonstrating) elementary concepts of wave motion in asmuch as linear mass distributions afford a simple prototype of continuous systems and permit the realization of unidirectional wave motions Despite the restricted scope inherent in uni- rather than multi-directional wave pro pagation, analytical techniques whose capabilities are fully brought out in the latter circumstances can already be utilized, with the least technical com plications, if a lone position coordinate or specific direction of travel is involved Thus, we may prepare the way for an extensive use of source or Green’s functions in wave theory by describing their application to string vibration problems, and we may also introduce general procedures relating to the estimation of Fourier integrals and the asymptotic solution of differential equations; it is likewise appropriate to illustrate, in a one-dimensional setting, the effective manner whereby scattering matrix and variational formulations contribute to wave analysis The simplest attributes of an ideal string include perfect uniformity, or distribution of mass along its length, and flexibility, which signifies that the tension or resultant force acting at any (small) normal section is directed along the tangent at the corresponding point of the string Two types of vibrations, namely longitudinal and transverse, may be distinguished; in the former, the string retains a straight profile and the displacements of its material elements are collinear; in the latter, the elements move within a plane that is perpendicular to the line of the string If the amplitude of such motions be sufficiently small, these types have an independent character and theoreti cal description Large scale motions of the ideal string are governed by non-linear partial differential equations (which involve both longitudinal and transverse displacements), and their analysis is formidable At the outset, therefore, we consider the features of a linear theory suitable for restricted (infinitesimal) vibrations; and withhold attention from the relationship be tween linear and non-linear formulations as well as the consequences of non-uniformity and imperfect flexibility of the string Let us refer to a string which is located on the x-axis in its equilibrium configuration and envisage planar vibrations specified by time varying dis placements, y(x, t), therefrom We overlook for the present all save the tensile force and suppose that the latter has an invariable magnitude, P The dynamical (i.e., Newtonian) equation for transverse motion of a differential element, obtained after resolving and combining the tensions at the respective endpoints, indicates a proportionality between local acceleration and cur vature of the string; and if the square of the slope, (dyldx)2, be neglected in comparison with unity, as befits small and smooth displacements, their connection is given by the so-called linear wave equation UNIDIRECTIONAL W AVE MOTIONS d 2y d 2y = dX ( 1) C dt Here the coefficient factor c has the dimensions of a velocity and a constant magnitude c = V(Plp) (1.2) if the linear density of the string, p, is uniform The representative ( 1) of second order partial differential equations is atypical insofar as its general solution can be exhibited, namely (1.3) y ( x , t) = f ( x - ct) + g ( x + ct) where f, g designate any twice differentiable functions of one variable; this conclusion is directlyforeshadowed by the form which thedifferential equa tion assumes onintroducing the independent variables ξ = x - ct, η = x + ct, viz., s 2y Οξδη = (1.4) The functional character of the term / ( x —ct) in (1.3) is such as to imply a progressive motion or wave form with invariable aspect, inasmuch as its envelope (or numerical spectrum for all values of x) at the instant t is a replica of that at an earlier time, say t - , in the sense of equality between the values of this term at the pairs of locations x, x + c t ; evidently the constant c defines the speed with which the wave form advances The opposite signs of dfldx and dfldt manifest by the relation * dX C dt - 0.5) are in keeping with a sense of progression towards the right (that is, in creasing values of x ) and the accompanying sketch (see Fig 1) of two success ive profiles makes clear the anti-correlation in sign between the local slope and transverse velocity Likewise, we may verify that the second term in (1.3), g ( x + ct), represents a progressive motion which travels towards the left (or negative x -direction) with the same rate of advance c and satisfies the first order differential equation dX C dt ( 1.6) A fixed value for the magnitude of these oppositely directed motions obtains when the locus of coordinate and time variables is a straight line, viz.: x T ct = constant, ( 1.7) respectively; the distinct families of parallel lines generated by assigning 488 U N ID IR E C T IO N A L W A V E M O TIO N S and νω ± [c2ω2—(v2—ε 2)μ 2]112 Re k = , lm/c = 0, v —c ( v 2\ | ω| > ( — - ΐ ) μ restrict the possibility of wave amplification to the frequency band |ω |< ((v2lc 2) - 1)1/2μ· The inverse function determinations Re ω = vk, Im ω = ± ( μ 2—c2k2)112, Re ω = vk ± (c2k2- μ 2)1/2, Im ω = 0, \k\ > μ/c, \k\ > μ /c confirm the existence of curves (namely, C+, C_ in Fig 50), bridging the cited frequency band, which are correlated with displacement of k along the real line; and the permissible arrangement of integrals, f" Γ00 άω I ®(ω) exp{i[fc(