Springer Series on 11 \V~y~ Phenomen~ Edited by L.M. Brekhovskikh Springer Series on ~-;Ei;,,~ Phenomen~ Editors: L.M. Brekhovskikh L.B. Felsen H.A. Haus Managing Editor: H.K.V. Lotsch Volume Mechanics of Continua and Wave Dynamics By L. Brekhovskikh, V. Goncharov Volume 2 Rayleigh-Wave Theory and Application Editors: E.A. Ash, E.G.S. Paige Volume 3 Electromagnetic Surface Excitations Editors: R.F. Wallis, G.I. Stegeman Volume 4 Asymptotic Methods in Short-Wave Diffraction Theory By V.M. Babic, V.S. Buldyrev Volume 5 Acoustics of Layered Media I Plane and Quasi-Plane Waves By L.M. Brekhovskikh, O.A. Godin Volume 6 Geometrical Optics of Inhomogeneous Media By Yu.A. Kravtsov, Yu.I. Orlov Volume 7 Recent Developments in Surface Acoustic Waves Editors: D.F. Parker, G.A. Maugin Volume 8 Fundamentals of Ocean Acoustics 2nd Edition By L.M. Brekhovskikh, Yu. Lysanov Volume 9 Nonlinear Optics in Solids Editor: O. Keller Volume 10 Acoustics of Layered Media II Point Source and Bounded Beams By L.M. Brekhovskikh, O.A. Godin Volume 11 Resonance Acoustic Spectroscopy By N.D. Veksler Volume 12 Scalar Wave Theory Green's Functions and Applications By J. DeSanto Volume 13 Modern Problems in Radar Target Imaging Ed. by W.-M. Boerner, F. Molinet, H. Uberall Volume 14 Random Media and Boundaries By K. Furutsu N.D. Veksler Resonance Acoustic Spectroscopy With 153 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Naum D. Veksler, DSc in Physics Institute of Cybernectics, Estonian Academy of Sciences, Akadeemia tee 21, EE0108 Tallinn, Estonia Translation edited by Professor Herbert Uberall, PhD Department of Physics, Catholic University of America, Hannan Hall, Washington, DC 20064, USA Series Editors: Professor Leonid M. Brekhovskikh, Academician P.P. Shirsov Institute of Oceanology, Russian Academy of Sciences, Krasikowa Street 23, 117218 Moscow, Russia Professor Leopold B. Felsen, Ph.D. Department of Electrical Engineering, Weber Research Institute, Polytechnic University, Farmingdale, NY 11735, USA Professor Hermann A. Haus Department of Electrical Engineering & Computer Science, MIT, Cambridge, MA 02139, USA Managing Editor: Dr.-Ing. Helmut K.V. Lotsch Springer-Verlag, Tiergartenstrasse 17, W-69OO Heidelberg, Fed. Rep. of Germany ISBN -13: 978-3-642-84797-4 e-ISBN -13: 978-3-642-84795-0 DOl: 10.1007/978-3-642-84795-0 Library of Congress Cataloging-in-Publication Data. Veksler, Naum Davidovich. [Akusticheskaia spektro skopiia. English] Resonance acoustic spectroscopy / N.D. Veksler; [translation edited by Herbert Oberall]. p. cm. - (Springer series on wave phenomena; v. 11). Includes bibliographical references and index. ISBN-!3:978-3-642-84797-4 1. Acoustic surface waves. 2. Sound- waves - Scattering. 3. Solids - Acoustic properties. I. Title. n. Series: Springer series on wave phe- nomena; II. QGI76.8.A3V4313 1993. 530.4'12-dc20 92-30272 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Softcoverreprint of the hardcover 1st edition 1993 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset by Macmillan India Ltd, Bangalore-25 54/3140/SPS-54321O - Printed on acid-free paper Preface This monograph is devoted to the analysis of waves generated in an elastic body by a plane harmonic acoustic wave. It concentrates on the "peripheral" (or "circumferential") elastic waves: Rayleigh and whispering gallery waves, which are generated on solid and thick-walled elastic bodies, and the Lamb waves generated in thin-walled bodies. Franz and Stoneley waves are considered to a lesser extent only. Franz waves have been treated in detail for several two dimensional scattering problems, and therefore I decided to touch only lightly upon this subject. Franz waves propagating on helical paths are considered in the case of scattering of an obliquely incident plane acoustic wave by a solid elastic cylinder of infinite extent. The physical phenomena of excitation, propagation, and re-radiation of elastic waves during the scattering of an incident plane wave are investigated in this book. Known methods are applied for solving the traditional problems of scattering by elastic spheres and cylinders. Special emphasis is laid on the interpretation of the solution. I tried to fill the gaps existing between the papers in which new methods in scattering theory are applied to model (test) problems. The material is presented systematically, including the formulation of the problem, method of solution, algorithm, computation, and analysis. A large number of new computational results concerning the solution of the scattering problem are given as form functions, modal resonances, dispersion curves, and acoustic spectrograms. Each numerical example is carefully constructed to elucidate one or the other aspect of the scattering process. Although each chapter can be read independently, they are all closely connected and are mutually complementary. The limited number of geometrical shapes of scatterers considered here is occasioned by the aim to analyze the solution in a rather broad frequency band. Analytical methods (including the asymptotic ones) permit one to obtain the solution of very difficult problems, for example, three-dimensional scattering problems by elastic bodies of smooth shape. Sometimes the asymptotics trace back the physics of the scattering process. However, as a rule, they break down in the resonance frequency range, to which the main attention is paid in this book. In spite of the fact that new numerical methods and fast computers now allow the three-dimensional scattering problem for elastic bodies to be solved in principle, only the solution of scattering by a spheroid (at an arbitrary angle of incidence) is actually obtained in the low-frequency range. VI Preface In the case of scattering by elastic spheres and cylinders, the availability of the exact solution in series form allows one to obtain the solution in a very broad frequency range, and the specially elaborated procedures permit it to be analyzed qualitatively. The problem is considered as a steady-state one, assum ing the loading to be in the form of a plane harmonic wave. Such an approach is common in acoustics. Once the solution of the steady-state problem has been found, one can obtain the solution for a loading that arbitrarily changes in time by using the convolution theorem. The solutions of two classical problems are very useful for the analysis of the elastic peripheral waves generated in spheres and cylinders, namely, first, the Rayleigh wave on an elastic half-space, and second, the Lamb waves in a plane "dry" layer (without any ambient liquid). The elastic scatterers considered in this book vary from solid to thick- or thin-walled bodies. The equations of linear elasticity theory are generally used to describe the motion of the elastic body. In one case only, are additionally the equations of Timoshenko-type and of the membrane theory of thin shells used for a thin-walled elastic body. The models of acoustically rigid and soft bodies commonly employed in acoustics are very useful and sometimes indispensable. Such a variety of models is in no degree connected with the wish to enlarge the proportion of numerical results. On the contrary, the models arise naturally and are used only for qualitative understanding of the scattering process; a new model is considered only when it helps the main thought further. The treatment in this book is descriptive rather than based on rigorous proof. We assume that some of the phenomena considered here will later be rigorously justified on the level of a theorem. We use the physical level of rigor typical of the original papers. From the point of view of content, the issues presented in the book are closely connected with the well-studied topics of acoustics and mechanics of a solid deformable body: the free vibrations of "dry" elastic spheres and cylinders - -solid, thick-and thin-walled; the vibrations of plates and shells; and the dispersion relations. The contact with the liquid surrounding the scatterer and the presence of an incident plane wave are the main differences of the scattering problem considered here from that corresponding to eigenvibrations of the body. Certainly, realistic. problems are more complicated than those which are treated in the book. This is connected both with the geometry of the scatterer (variable thickness, totality of different geometrical shapes bounding a common volume, presence of bulkheads, framing, and reinforcements) and with non uniformity, for example, the lamination of the liquid surrounding the scatterer, presence of'the bottom and the surface, and other circumstances. Therefore the problems considered can be understood as model ones. Without comprehending the nature of the scattering process in the model problem it is difficult to formulate, and to solve, real problems. The material presented has appeared as original papers during the last ten years and is treated here in a English-language monograph for the first time. Preface VII There are some difficulties with terminology in this field because the material is essentially new and not all terms are settled yet. However, I have endeavored to use a consistent terminology throughout the whole book. Technical applications and problems of identification are not treated here. I consider this to be a book about the physical phenomena which can be discovered in the course of a mathematical experiment. The restricted size of the book forced me to select the material carefully and to omit all extreme ramifications. Recently published monographs and surveys have made the problem of selection substantially easier. Preference was given to the subjects of broad interest that are reasonably well studied and comprehended, and which are important in terms of new methods or new concepts. Much of the material was first published in Russian in my book Acoustic Spectroscopy. Tallinn, September 1991 N.D. Veksler Acknowledgements The author is thankful to Dr. Victor Korsunskii for help with computer codes, to Mrs. Valentina Furik and Mrs. Monika Perk~nn for speedily and accu rately typing the text, and to Mr. Ants Kivilo, who prepared all the drawings with great care. Prof. Herbert Uberall, translation editor of this book, has improved the text very carefully and has given it a readable form. The author is very grateful to him for this. Contents 1 Scattering of an Obliquely Incident Plane Acoustic Wave by a Circular-Cylindrical SheD of Infinite Extent . . . . . . 1 1.1 Formulation of the Problem and Its Solution in Series Form 1 1.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Resonance Scattering by Elastic Bodies of Cylindrical Shape. . . .. 16 2.1 Scattering by a Solid Elastic Cylinder . . . . . . . . . . . . .. 16 2.2 Numerical Results for Scattering by a Solid Elastic Cylinder. 24 2.3 Scattering by a Circular-Cylindrical Shell. . . . . . . . . . .. 28 2.4 Numerical Results for the Problem of a Plane Acoustic Pressure Wave Scattered by a Circular-Cylindrical Shell. . . . . . . '. . 31 3 Application of the Resonance Scattering Theory to Problems of Acoustic Wave Scattering by Elastic Spheres. . . . . . . . . 46 3.1 Scattering by a Solid Elastic Sphere. . . . . . . . . . . . . 47 3.2 Numerical Results for Scattering by a Solid Elastic Sphere.. 50 3.3 Scattering by an Elastic Spherical Shell . . . . . . . . . . . .. 64 3.4 Numerical Results for Scattering by Thick-Walled Spherical Shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67 4 Synthesis of the Backscattering Form Function for a Solid Elastic Sphere Using the Sommerfeld-Watson Transformation. . . . . . .. 73 4.1 Approximate Formula for the Description of the Form Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Features of the Rayleigh Wave . . . . . . . . . . . . . . . . . . 76 4.3 Synthesis of the Form Function. . . . . . . . . . . . . . . . . . 80 4.4 Comparison with Experiment. . . . . . . . . . . . . . . . . . . 81 5 Resonance ~solation and Identification Method. . . . . . . . . . . . . 83 5.1 The Essence of the Method. . . . . . . . . . . . . . . . . . . .. 83 5.2 Results................................. 87 5.2.1 Solid Elastic Cylinder. . . . . . . . . . . . . . . . . . . . 87 5.2.2 Thick-Walled Cylindrical Shell . . . . . . . . . . . . . 87 5.3 Acoustic Spectrogram. . . . . . . . . . . . . . . . . . . . . . . 92 5.4 Resonances. of Guided Waves. . . . . . . . . . . . . . . . . . 93 X Contents 6 Pulsed Resonance Identification Method. . . . . . . . . . . . . . . .. 98 6.1 The Essence of the Method. . . . . . . . . . . . . . . . . . . .. 98 6.2 Results.................................. 104 6.2.1 Solid Elastic Cylinder. . . . . . . . . . . . . . . . . . .. 104 6.2.2 Cylindrical Shells. . . . . . . . . . . . . . . . . . . . . .. 106 6.2.3 Spherical Shells. . . . . . . . . . . . . . . . . . . . . . .. 109 7 Peripheral Waves in the Scattering by Elastic Cylinders and Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11 0 7.1 Scattering from an Elastic Cylinder. . . . . . . . . . . . . . .. 110 7.2 Scattering from a Solid Elastic Sphere . . . . . . . . . . . . .. 128 8 Analysis of Peripheral Waves via Backscattering by Circular-Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . .. 131 8.1 Introductory Remarks. . . . . . . . . . . . . . . . . . . . . . .. 131 8.2 Model Problem: Lamb-Type Waves in a Plane "Dry" Layer. 136 8.3 Application of the Resonance Scattering Theory . . . . . . .. 138 8.4 Form Function Analysis . . . . . . . . . . . . . . . . . . . . .. 147 8.5 Phase Velocities . . . . . . . . . . . . . . . . . . . . . . . . . .. 149 8.6 Asymptotic Formula for Resonance Frequencies.. . . . . . .. 152 9 Analysis and Synthesis of the Backscattered Acoustic Pressure from a Circular-Cylindrical Shell . . . . . . . . . . . . . . . . . . .. 157 9.1 Analysis of the Acoustic Pressure Scattered by the Shell . .. 160 9.2 Synthesis of the Approximate Formula for the Backscattering Form Function. . . . . . . . . . . . . . . . . . . . . . . . . . .. 168 9.2.1 The Speculady Reflected Wave and the Waves Refracted in a Thin Layer. . . . . . . . . . . . . . . .. 168 9.2.2 The So Wave. . . . . . . . . . . . . . . . . . . . . . . .. 173 9.2.3 The A Wave. . . . . . . . . . . . . . . . . . . . . . . .. 179 9.3 Discussion............................... 183 10 Peripheral Waves in the Scattering by Spherical Shells. . . . . . .. 186 10.1 Form Function and Isolated Modal Resonances. . . . . .. 186 10.1.1 The A Wave. . . . . . . . . . . . . . . . . . . . . . .. 191 10.1.2 The So Wave. . . . . . . . . . . . . . . . . . . . . . .. 191 10.1.3 The Al Wave . . . . . . . . . . . . . . . . . . . . . .. 196 10.1.4 The Sl Wave. . . . . . . . . . . . . . . . . . . . . . .. 197 10.2 Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . . .. 199 10.3 Approximation of Resonance Positions. . . . . . . . . . . .. 200 10.4 Hidden Resonances. . . . . . . . . . . . . . . . . . . . . . . .. 205 11 Peripheral Waves in the Scattering of a Plane Acoustic Wave Obliquely Incident on a Solid Elastic Cylinder . . . . . . . . . . . 207 11.1 Form Function . . . . . . . . . . . . . . . . . . . . . . . . . 207 Contents XI 11.2 Modal Resonances. . . . . . . . . . . . . . . . . . . . . . . .. 210 11.3 Dispersion Curves of the Phase Velocities. . . . . . . . . .. 213 12 Analysis of Peripheral Waves in the Scattering of a Plane Acoustic Wave Obliquely Incident on a Circular-Cylindrical Shell. . . . . .. 219 12.1 Form Function Analysis. . . . . . . . . . . . . . . . . . . . .. 219 12.2 Modal Resonances. . . . . . . . . . . . . . . . . . . . . . . .. 223 12.3 Phase Velocities of Peripheral Waves. . . . . . . . . . . . . 227 12.4 Estimation of the Resonance Frequency Positions . . . . . 232 13 On the Causes of Possible Errors in the Solution of Scattering Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 236 13.1 Application of a Model Outside its Region of Validity. . .. 236 13.2 Application of the Procedure Outside its Region of Validity 239 13.3 Boundedness of the Frequency Domain. . . . . . . . . . . .. 241 13.4 Extremely Large Computation Steps . . . . . . . . . . . . .. 242 13.5 Series Truncation at a Small Number of Terms. . . . . . .. 243 13.6 Ordinary Precision. . . . . . . . . . . . . . . . . . . . . . . .. 251 13.7 Errors in Interpretation . . . . . . . . . . . . . . . . . . . . 252 13.7.1 Indirect Methods of Finding the Resonance Frequencies. . . . . . . . . . . . . . . . . . . . . . . 252 13.7.2 Inadequate Choice of the Background. . . . . . . .. 253 13.7.3 Families of Resonances with Equal Cutoff Frequencies. . . . . . . . . . . . . . . . . . . . . . . .. 254 13.7.4 Superposition of Resonances with Low Q Factor.. 255 14 Estimation of the Shell Thickness from the Form Function . . 256 14.1 Form Function Structure in the Domain of the First Resonances of the So Wave. . . . . . . . . . . . . . . . . 256 14.2 Estimation of the Shell Thickness. . . . . . . . . . . . . . .. 263 14.3 Commentary............................. 264 References. . .' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 266 Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 277