2014 • 198 Pages • 1.86 MB • English

Posted April 14, 2020 • Uploaded
by annamarie.wehner

PREVIEW PDF

Page 1

Lecture Notes in Applied and Computational Mechanics 71 Kazumi Watanabe Integral Transform Techniques for Green’s Function

Page 2

Lecture Notes in Applied and Computational Mechanics Volume 71 Series Editors F. Pfeiffer, Garching, Germany P. Wriggers, Hannover, Germany For further volumes: http://www.springer.com/series/4623

Page 3

Kazumi Watanabe Integral Transform Techniques for Green’s Function 123

Page 4

Kazumi Watanabe Department of Mechanical Engineering Yamagata University Yonezawa Japan ISSN 1613-7736 ISSN 1860-0816 (electronic) ISBN 978-3-319-00878-3 ISBN 978-3-319-00879-0 (eBook) DOI 10.1007/978-3-319-00879-0 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013940095 Ó Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Page 5

Dedicated to my teachers, Dr. Akira Atsumi (Late Professor, Tohoku University) and Dr. Kyujiro Kino (Late Professor, Osaka Institute of Technology)

Page 6

Preface When I was a senior student, I found a book on the desk of my advisor Professor and asked him how to get it. His answer was in the negative, saying its content was too hard, even for a senior student. Some weeks later, I found it again in a bookstore, the biggest in Osaka. This was my ﬁrst encounter with ‘‘Fourier Transforms’’ written by the late Professor I. N. Sneddon. Since then, I have learned the 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 crucial information to understand the essential feature of the phenomena. Unfortunately, we cannot solve every differential equation, and almost all phenomena are governed by nonlinear differential equations, of which most are not tractable. The differential equations which can be solved analytically are limited to a very small number. But their solutions give us the essence of the event. The typical partial differential equations which can be solved exactly are the Laplace, the diffusion and the wave equations. These three partial differential equations, which are linearized for simplicity, govern many basic phenomena in physical, chemical and social events. In addition to single differential equations, some coupled linear partial differential equations, which govern somewhat complicated phenomena, are also solvable and their solutions give much information about, for example, the deformation of solid media, propagation of seismic and acoustic waves, and ﬂuid ﬂows. In any case where phenomena are described by linear differential equations, the solutions can be expressed by superposition of basic/fundamental solutions. The integral transform technique is a typical superposition technique. The integral transform technique does not require any previous knowledge for solving differ- ential equations. It simply transforms partial or ordinary differential equations to reduced ordinary differential equations or to simple algebraic equations. However, a substantial difﬁculty is present regarding the inversion process. Many inversion integrals are tabulated in various formula books, but typically, it is not enough. If a suitable integration formula cannot be found, the complex integral must be considered and Cauchy’s integral theorem is applied to the inversion integral. Thus, integral transform techniques are intrinsically connected with the theory of complex integrals. vii

Page 7

viii Preface This book intends to show how to apply integral transforms to partial differential equations and how to invert the transformed solution into the actual space–time domain. Not only the use of integration formula tabulated in books, but also the application of Cauchy’s integral theorem for the inversion integrals are described concisely and in detail. A particular solution for a differential equation with a nonhomogeneous term of a point source is called the ‘‘Green’s function’’. The Green’s functions for coupled differential equations are called ‘‘Green’s dyadic’’. The Green’s function and Green’s dyadic are the basic and fundamental solutions of the differential equation and give the principal features of the event. Furthermore, these Green’s functions and dyadics have many applications for numerical computation techniques such as the Boundary Element Method. However, the Green’s function and Green’s dyadic have been scattered in many branches of applied mechanics and thus, their solution methods are not uniﬁed. The book intends to present and illustrate a uniﬁed solution method, namely the method of integral transform for the Green’s function and Green’s dyadic. Thus, the fundamental Green’s function for the Laplace and wave equations and the Green’s dyadic for the elasticity equations are gathered in this single book so that the reader can have access to a proper Green’s function and understand the mathematical process of its derivation. Chapter 1 describes roughly the deﬁnition of the integral transforms and the distributions to be used throughout the book. Chapter 2 shows how to apply an integral transform for solving a single partial differential equation such as the Laplace and wave equations. The basic technique of the integral transform method is demonstrated. Especially, in the case of the time-harmonic response for the wave 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 so that the reader can easily ﬁnd the difference in the functional form among the Green’s functions. An evaluation technique for a singular inversion integral which arises in a 2D static problem of Laplace equation is developed. The Green’s dyadic for 2D and 3D elastodynamic problems are discussed in Chap. 3. Three basic responses, impulsive, time-harmonic and static responses, are obtained by the integral transform method. The time-harmonic response is derived by 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 viscous ﬂuid. Introducing a small parameter, the nonlinear ﬁeld equations are linearized and reduced to a single partial differential equation for velocity potential or pressure deviation. The Green’s function which gives the acoustic ﬁeld in a uniform ﬂow is derived by the method of integral transform. A conversion tech- nique for the inversion integral is demonstrated. That is, to transform an inversion integral 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 dynamic response produced by a point load on the surface of a beam and a plate is discussed. The impulsive and time-harmonic responses are derived by the integral

Page 8

Preface ix transform method. In addition to the tabulated integration formulas, the inversion integrals are evaluated by application of complex integral theory. Chapter 6 presents a powerful inversion technique for transient problems of elastodynamics, namely the Cagniard-de Hoop method. Transient response of an elastic half-space to a point impulsive load is discussed by the integral transform method. Applying Cauchy’s complex integral theorem, the Fourier inversion integral is converted to an integral of the Laplace transform and then its Laplace inversion is carried out by inspection without using any integration formula. The Green’s function for an SH-wave and Green’s dyadics for P, SV and SH-waves are obtained. The last Chap. 7 presents three special Green’s functions/dyadics. The 2D static Green’s dyadic for an orthotropic elastic solid and that for an inhomogeneous solid are derived. In the last section, moving boundary problems is discussed. Two different Laplace transforms are applied for a single problem, and a conversion formula between two Laplace transforms is developed with use of Cauchy’s theorem. This conversion enables us to apply the integral transform technique to a moving boundary problem. The integral transform technique has been used for many years. The inversion process inevitably requires a working knowledge of the theory of complex func- tions. The author ﬁnds the challenge of a complex integral amusing, especially the challenge of choosing the right contour for the inversion integral. He hopes that young 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 the recent research activities, since some researchers tend to use numerical techniques without considering the possibility of an analytical solution. The increased mathematical techniques expand wider the horizon of the differential equations, and one can extract more ﬁrm knowledge from nature which is described by the differential equations. The author hopes that the book will give one more technique to the younger researchers. Finally, the author wishes to express his sincere thanks to Dr. Mikael A. Langthjem, Associate Professor of Yamagata University, for his advice and nice comments. Yonezawa, Japan, January 2013 Kazumi Watanabe

Page 9

Contents 1 Deﬁnition of Integral Transforms and Distributions . . . . . . . . . . . 1 1.1 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Distributions and Their Integration Formulas . . . . . . . . . . . . . . 4 1.3 Comments on Inversion Techniques and Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Green’s Functions for Laplace and Wave Equations . . . . . . . . . . . 11 2.1 1D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 1D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 2D Static Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 2D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 2D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6 3D Static Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7 3D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.8 3D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 Green’s Dyadic for an Isotropic Elastic Solid . . . . . . . . . . . . . . . . 43 3.1 2D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 2D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 2D Static Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 3D Impulsive Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5 3D Time-Harmonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6 3D Static Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Acoustic Wave in a Uniform Flow . . . . . . . . . . . . . . . . . . . . . . . . 77 4.1 Compressive Viscous Fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Linearization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 Viscous Acoustic Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 xi

Page 10

xii Contents 4.4 Wave Radiation in a Uniform Flow . . . . . . . . . . . . . . . . . . . . . 85 4.5 Time-Harmonic Wave in a Uniform Flow . . . . . . . . . . . . . . . . 91 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5 Green’s Functions for Beam and Plate . . . . . . . . . . . . . . . . . . . . . 93 5.1 An Impulsive Load on a Beam . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2 A Moving Time-Harmonic Load on a Beam. . . . . . . . . . . . . . . 95 5.3 An Impulsive Load on a Plate. . . . . . . . . . . . . . . . . . . . . . . . . 99 5.4 A Time-Harmonic Load on a Plate . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 Cagniard-de Hoop Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.1 2D Anti-Plane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2 2D In-Plane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.3 3D Dynamic Lamb’s Problem. . . . . . . . . . . . . . . . . . . . . . . . . 130 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7 Miscellaneous Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.1 2D Static Green’s Dyadic for an Orthotropic Elastic Solid . . . . . 157 7.2 2D static Green’s Dyadic for an Inhomogeneous Elastic Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.3 Reflection of a Transient SH-Wave at a Moving Boundary . . . . 173 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Integral Transform Techniques for Green's Function

2015 • 274 Pages • 4.02 MB

Integral Transform Techniques for Green's Function

2014 • 198 Pages • 4.34 MB

Integral Transforms and Their Applications

1978 • 427 Pages • 8.35 MB

Laplace transform techniques for nonlinear systems.

1970 • 173 Pages • 3.97 MB

The Convolution Transform Isidore Isaac Hirschman

1955 • 275 Pages • 14.51 MB

The Hilbert-Hankel Transform and its Application to Shallow Water Ocean Acoustics

2012 • 492 Pages • 23.3 MB

Integral Transforms in Science and Engineering.

2016 • 493 Pages • 19.5 MB

Pricing and Hedging Asian Options Using Monte Carlo and Integral Transform Techniques Trust ...

2010 • 102 Pages • 1023 KB

Implementation Techniques for the Truncated Fourier Transform

2017 • 86 Pages • 1.88 MB

Wavelet transform based techniques for ultrasonic signal processing

2017 • 82 Pages • 2.49 MB

Transform Techniques in Chemistry

1978 • 394 Pages • 17.69 MB

Novel Hilbert Huang Transform Techniques for Bearing Fault Detection

2013 • 83 Pages • 851 KB

Wavelet Transform-Based Multi-Resolution Techniques For Tomographic Reconstruction And ...

2012 • 293 Pages • 18.64 MB