Lecture Notes in Applied and ComputationalMechanics

Volume 71

Series Editors

F. Pfeiffer, Garching, GermanyP. Wriggers, Hannover, Germany

For further volumes:http://www.springer.com/series/4623

Kazumi Watanabe

Integral TransformTechniques for Green’sFunction

123

Kazumi WatanabeDepartment of Mechanical EngineeringYamagata UniversityYonezawaJapan

ISSN 1613-7736 ISSN 1860-0816 (electronic)ISBN 978-3-319-00878-3 ISBN 978-3-319-00879-0 (eBook)DOI 10.1007/978-3-319-00879-0Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013940095

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Dedicated to my teachers,Dr. Akira Atsumi(Late Professor, Tohoku University)

and

Dr. Kyujiro Kino(Late Professor, Osaka Institute ofTechnology)

Preface

When I was a senior student, I found a book on the desk of my advisor Professorand asked him how to get it. His answer was in the negative, saying its content wastoo hard, even for a senior student. Some weeks later, I found it again in abookstore, the biggest in Osaka. This was my first encounter with ‘‘FourierTransforms’’ written by the late Professor I. N. Sneddon. Since then, I have learnedthe power of integral transform, i.e. the principle of superposition.

All phenomena, regardless of their fields of event, can be described by differ-ential equations. The solution of the differential equation contains the crucialinformation to understand the essential feature of the phenomena. Unfortunately,we cannot solve every differential equation, and almost all phenomena aregoverned by nonlinear differential equations, of which most are not tractable. Thedifferential equations which can be solved analytically are limited to a very smallnumber. But their solutions give us the essence of the event. The typical partialdifferential equations which can be solved exactly are the Laplace, the diffusionand the wave equations. These three partial differential equations, which arelinearized for simplicity, govern many basic phenomena in physical, chemical andsocial events. In addition to single differential equations, some coupled linearpartial differential equations, which govern somewhat complicated phenomena,are also solvable and their solutions give much information about, for example, thedeformation of solid media, propagation of seismic and acoustic waves, and fluidflows.

In any case where phenomena are described by linear differential equations, thesolutions can be expressed by superposition of basic/fundamental solutions. Theintegral transform technique is a typical superposition technique. The integraltransform technique does not require any previous knowledge for solving differ-ential equations. It simply transforms partial or ordinary differential equations toreduced ordinary differential equations or to simple algebraic equations. However,a substantial difficulty is present regarding the inversion process. Many inversionintegrals are tabulated in various formula books, but typically, it is not enough.If a suitable integration formula cannot be found, the complex integral must beconsidered and Cauchy’s integral theorem is applied to the inversion integral.Thus, integral transform techniques are intrinsically connected with the theory ofcomplex integrals.

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This book intends to show how to apply integral transforms to partialdifferential equations and how to invert the transformed solution into the actualspace–time domain. Not only the use of integration formula tabulated in books,but also the application of Cauchy’s integral theorem for the inversion integralsare described concisely and in detail. A particular solution for a differentialequation with a nonhomogeneous term of a point source is called the ‘‘Green’sfunction’’. The Green’s functions for coupled differential equations are called‘‘Green’s dyadic’’. The Green’s function and Green’s dyadic are the basic andfundamental solutions of the differential equation and give the principal featuresof the event. Furthermore, these Green’s functions and dyadics have manyapplications for numerical computation techniques such as the BoundaryElement Method. However, the Green’s function and Green’s dyadic have beenscattered in many branches of applied mechanics and thus, their solutionmethods are not unified. The book intends to present and illustrate a unifiedsolution method, namely the method of integral transform for the Green’sfunction and Green’s dyadic. Thus, the fundamental Green’s function for theLaplace and wave equations and the Green’s dyadic for the elasticity equationsare gathered in this single book so that the reader can have access to a properGreen’s function and understand the mathematical process of its derivation.

Chapter 1 describes roughly the definition of the integral transforms and thedistributions to be used throughout the book. Chapter 2 shows how to apply anintegral transform for solving a single partial differential equation such as theLaplace and wave equations. The basic technique of the integral transform methodis demonstrated. Especially, in the case of the time-harmonic response for thewave equation, the integration path for the inversion integral is discussed in detail.At the end of the chapter, the obtained Green’s functions are listed in a table sothat the reader can easily find the difference in the functional form among theGreen’s functions. An evaluation technique for a singular inversion integral whicharises in a 2D static problem of Laplace equation is developed.

The Green’s dyadic for 2D and 3D elastodynamic problems are discussed inChap. 3. Three basic responses, impulsive, time-harmonic and static responses, areobtained by the integral transform method. The time-harmonic response is derivedby the convolution integral of the impulsive response without solving the differ-ential equations for the time-harmonic source.

Chapter 4 presents the governing equations for acoustic waves in a viscousfluid. Introducing a small parameter, the nonlinear field equations are linearizedand reduced to a single partial differential equation for velocity potential orpressure deviation. The Green’s function which gives the acoustic field in auniform flow is derived by the method of integral transform. A conversion tech-nique for the inversion integral is demonstrated. That is, to transform an inversionintegral along the complex line to that along the real axis in the complex plane.It enabled us to apply the tabulated integration formula.

Chapter 5 presents Green’s functions for beams and plates. The dynamicresponse produced by a point load on the surface of a beam and a plate isdiscussed. The impulsive and time-harmonic responses are derived by the integral

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transform method. In addition to the tabulated integration formulas, the inversionintegrals are evaluated by application of complex integral theory.

Chapter 6 presents a powerful inversion technique for transient problems ofelastodynamics, namely the Cagniard-de Hoop method. Transient response of anelastic half-space to a point impulsive load is discussed by the integral transformmethod. Applying Cauchy’s complex integral theorem, the Fourier inversionintegral is converted to an integral of the Laplace transform and then its Laplaceinversion is carried out by inspection without using any integration formula. TheGreen’s function for an SH-wave and Green’s dyadics for P, SV and SH-waves areobtained.

The last Chap. 7 presents three special Green’s functions/dyadics. The 2D staticGreen’s dyadic for an orthotropic elastic solid and that for an inhomogeneous solidare derived. In the last section, moving boundary problems is discussed. Twodifferent Laplace transforms are applied for a single problem, and a conversionformula between two Laplace transforms is developed with use of Cauchy’stheorem. This conversion enables us to apply the integral transform technique to amoving boundary problem.

The integral transform technique has been used for many years. The inversionprocess inevitably requires a working knowledge of the theory of complex func-tions. The author finds the challenge of a complex integral amusing, especially thechallenge of choosing the right contour for the inversion integral. He hopes thatyoung researchers will join the fun and carry on with the inversion techniques.In this respect it must be mentioned that he feels a lack of mathematical skill in therecent research activities, since some researchers tend to use numerical techniqueswithout considering the possibility of an analytical solution. The increasedmathematical techniques expand wider the horizon of the differential equations,and one can extract more firm knowledge from nature which is described by thedifferential equations. The author hopes that the book will give one more techniqueto the younger researchers.

Finally, the author wishes to express his sincere thanks to Dr. MikaelA. Langthjem, Associate Professor of Yamagata University, for his advice andnice comments.

Yonezawa, Japan, January 2013 Kazumi Watanabe

Preface ix

Contents

1 Definition of Integral Transforms and Distributions . . . . . . . . . . . 11.1 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Distributions and Their Integration Formulas . . . . . . . . . . . . . . 41.3 Comments on Inversion Techniques

and Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Green’s Functions for Laplace and Wave Equations . . . . . . . . . . . 112.1 1D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 1D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 2D Static Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 2D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 2D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 3D Static Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 3D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8 3D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 39References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Green’s Dyadic for an Isotropic Elastic Solid . . . . . . . . . . . . . . . . 433.1 2D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 2D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 2D Static Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4 3D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.5 3D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.6 3D Static Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Acoustic Wave in a Uniform Flow . . . . . . . . . . . . . . . . . . . . . . . . 774.1 Compressive Viscous Fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Linearization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3 Viscous Acoustic Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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4.4 Wave Radiation in a Uniform Flow . . . . . . . . . . . . . . . . . . . . . 854.5 Time-Harmonic Wave in a Uniform Flow . . . . . . . . . . . . . . . . 91References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Green’s Functions for Beam and Plate . . . . . . . . . . . . . . . . . . . . . 935.1 An Impulsive Load on a Beam . . . . . . . . . . . . . . . . . . . . . . . . 935.2 A Moving Time-Harmonic Load on a Beam. . . . . . . . . . . . . . . 955.3 An Impulsive Load on a Plate. . . . . . . . . . . . . . . . . . . . . . . . . 995.4 A Time-Harmonic Load on a Plate . . . . . . . . . . . . . . . . . . . . . 101References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Cagniard-de Hoop Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.1 2D Anti-Plane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.2 2D In-Plane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.3 3D Dynamic Lamb’s Problem. . . . . . . . . . . . . . . . . . . . . . . . . 130References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7 Miscellaneous Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1577.1 2D Static Green’s Dyadic for an Orthotropic Elastic Solid . . . . . 1577.2 2D static Green’s Dyadic for an Inhomogeneous

Elastic Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.3 Reflection of a Transient SH-Wave at a Moving Boundary . . . . 173Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

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