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## Über dieses Buch

This book describes mathematical techniques for integral transforms in a detailed but concise manner. The techniques are subsequently applied to the standard partial differential equations, such as the Laplace equation, the wave equation and elasticity equations. Green’s functions for beams, plates and acoustic media are also shown, along with their mathematical derivations. The Cagniard-de Hoop method for double inversion is described in detail and 2D and 3D elastodynamic problems are treated in full.

This new edition explains in detail how to introduce the branch cut for the multi-valued square root function. Further, an exact closed form Green’s function for torsional waves is presented, as well as an application technique of the complex integral, which includes the square root function and an application technique of the complex integral.

## Inhaltsverzeichnis

### Chapter 1. Definition of Integral Transforms and Distributions

Abstract
This first chapter describes a brief definition of integral transforms, such as Laplace and Fourier transforms, a rough definition of delta and step functions which are frequently used as the source function, and a concise introduction of the branch cut for a multi-valued square root function. The multiple integral transforms and their notations are also explained. The newly added Sect. 1.3 explains closely how to introduce the branch cut for the multi-valued square root function. The branch cut and the argument of the square root function along the branch cut are employed throughout the book. The discussion on the argument of the root function along the branch cut is unique and instructive for the reader, when he/she starts to apply the complex integral to the inverse transform. The last short comment lists some important formula books which are crucial for the inverse transform, i.e. the evaluation of the inversion integral.
Kazumi Watanabe

### Chapter 2. Green’s Functions for Laplace and Wave Equations

Abstract
This chapter shows how to apply the integral transform to the single partial differential equation such as 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 1 and 2D wave equations, the integration path for the inversion integral is discussed in detail with use of the results in Sect. 1.3. 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 of 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 also developed.
Kazumi Watanabe

### Chapter 3. Green’s Dyadic for an Isotropic Elastic Solid

Abstract
The Green's dyadic for 2D and 3D elastodynamic problems are discussed in this chapter. 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 differential equations for the time-harmonic source. In the last section, two exact closed form Green's functions for torsional waves are also presented.
Kazumi Watanabe

### Chapter 4. Acoustic Wave in a Uniform Flow

Abstract
This chapter 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 shows the acoustic field in a uniform flow is derived by the method of integral transform. The convolution integral for the inversion transform is demonstrated to obtain the time-harmonic response. An application technique of the complex integral is also demonstrated in order to transform an infinite integral along the complex line to that along the real axis in the complex plane. It enabled us to apply the tabulated integration formula.
Kazumi Watanabe

### Chapter 5. Green’s Functions for Beam and Plate

Abstract
This chapter presents dynamic Green’s functions for the elastic beam and plate. The dynamic responses produced by a point load on the surface of the beam and plate are discussed. Two dynamic responses, the impulsive and time-harmonic responses, are derived by the integral transform method. In addition to the tabulated integration formulas, an inversion integral is evaluated by the application of the complex integral theory.
Kazumi Watanabe

### Chapter 6. Cagniard-de Hoop Technique

Abstract
This chapter 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 the form of Laplace transform integral and then its Laplace inversion is carried out by inspection without using any integration formula. The Green's function for a single SH-wave and the Green's dyadics for coupled P, SV and SH-waves are obtained exactly.
Kazumi Watanabe

### Chapter 7. Miscellaneous Green’s Functions

Abstract
This last chapter presents five Green’s functions and one application technique of the complex integral. The first and second sections consider the 2D static Green’s dyadic for an orthotropic elastic solid, and for an inhomogeneous elastic solid. The third section discusses the Green’s function for torsional waves in an anisotropic solid. The fourth section discusses wave reflection at a moving boundary. The fifth section is concerned with wave scattering by a rigid inclusion.Both are for SH-wave and their Green's functions are obtained in the exact closed form. The last section shows an excellent application technique of the complex integral. It reduces a semi-infinite integral, which includes the product of two Bessel functions, to a finite integral that is very suitable for numerical evaluation.
Kazumi Watanabe

### Backmatter

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