# Asymptotic Wave Theory

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Asymptotic Wave Theory investigates the asymptotic behavior of wave representations and presents some typical results borrowed from hydrodynamics and elasticity theory. It describes techniques such as Fourier-Laplace transforms, operational calculus, special functions, and asymptotic methods. It also discusses applications to the wave equation, the elements of scattering matrix theory, problems related to the wave equation, and diffraction. Organized into eight chapters, this volume begins with an overview of the Fourier-Laplace integral, the Mellin transform, and special functions such as the gamma function and the Bessel functions. It then considers wave propagation, with emphasis on representations of plane, cylindrical or spherical waves. It methodically introduces the reader to the reflexion and refraction of a plane wave at the interface between two homogeneous media, the asymptotic expansion of Hankel's functions in the neighborhood of the point at infinity, and the asymptotic behavior of the Laplace transform. The book also examines the method of steepest descent, the asymptotic representation of Hankel's function of large order, and the scattering matrix theory. The remaining chapters focus on problems of flow in open channels, the propagation of elastic waves within a layered spherical body, and some problems in water wave theory. This book is a valuable resource for mechanics and students of applied mathematics and mechanics.

## Table of Contents

1. The Fourier-Laplace Integral

1.1. The Laplace transform

1.1.1. The direct problem

1.1.2. The inverse problem

1.1.3. Elementary rules

1.2. The Fourier transform in L1

1.3. The Fourier transform in L2

1.4. The Laplace transform (continued)

1.5. The Mellin transform

2. Special Functions

2.1. The gamma function

2.1.1. A summation formula

2.1.2. The Eulerian definition of the function Γ(z)

2.1.3. The Laplace transform of tv

2.1.4. The relation between the function Γ(z) and the group of linear mappings

of the real line into itself

2.1.5. The error function

2.II. The Bessel functions

2.11.1. Definitions

2.11.2. The Kepler equation and Bessel functions

2.11.3. The group of displacements in the plane and Bessel's functions

2.11.4. The Bessel functions of purely imaginary argument

2.11.5. The Hankel functions

2.11.6. Addition formulae for Bessel's functions

3. The Wave Equation

3.1. Introduction

3.2. The reflexion and refraction of a plane wave at the interface between two

homogeneous media

3.3. Spherical waves

3.4. Cylindrical waves

3.5. Group velocity

3.6. Wave guide

3.7. Successive reflexions of plane waves at two parallel rigid planes

3.8. The relation between spherical and plane waves

3.9. The reflexion of a spherical wave at a plane interface

3.10. An alternative approach to Weyl's formula. Poritsky's generalisation

4. Asymptotic Methods

4.1. Asymptotic expansion

4.2. The asymptotic expansion of Hankel's functions in the neighborhood of the

point at infinity

4.3. The Laplace method

4.4. Asymptotic relations and Laplace's transform

4.5. The Laplace method (continued)

4.6. The method of steepest descent

4.7. Waves in linear dispersive media

4.8. The asymptotic representation of the reflected wave in the problem of a spherical

wave impinging on a plane interface. The lateral wave

4.9. The method of steepest descent; an extension to the case when some pole is

located near the saddle

4.10. The asymptotic representation of Hankel's functions of large order

4.11. An asymptotic representation of Legendre's functions of large order

4.12. The asymptotic representation of Hankel's functions of large order (continued)

4.12.1. A discussion of the equation y+2ry=0

4.12.2. Approximate representations of Hv(1)(v), dHv(2)(z)/dz|z=v for large order

4.12.3. The approximate representations of Hv(1) for (z)=v + rv1/3 and v > 0 large

5. Scattering Matrix Theory

5.1. Introduction

5.1. The direct problem

5.1.1. The one-dimensional model of wave propagation

5.1.2. An asymptotic approach

5.1.3. The matrix S(k) (k real)

5.1.4. Evaluating the reflexion coefficient (k real)

5.1.5. The matrix S(k) for k in the complex plane

5.1.6. The Fourier transform of r(/, k), r(k)

5.II. The inverse problem

5.11.1. Preliminary discussion

5.11.2. Some properties of Wt(x, t), W(x, t)

5.11.3. The Gelfand-Levitan integral equation

6. Flow in Open Channel; Asymptotic Solution of some Linear and Nonlinear Wave Equations

6.1. The kinematics and dynamics of flow in open channel

6.1.1. The geometric assumptions concerning a river bed

6.1.2. The dynamic assumptions

6.1.3. The basic equations of flow

6.1.4. An alternative approach to the problem

6.1.5. The pressure term

6.1.6. Long waves

6.1.7. Fourier transform of th(kh0)/kh0

6.1.8. The linearized equations of flow

6.1.9. The method of characteristics

6.1.10. Stability conditions

6.1.11. The analytic solution of the river flow equation

6.II. The asymptotic representation of the solution of the wave equation

6.11.1. An alternative approach

6.11.2. The asymptotic representation of the solution

6.11.3. The case 0 < c2< a < c1

6.11.4. The approximate treatment of the wave equation

6.III. Non-linear wave theory

6.111.1. The method of characteristics

6.111.2. The progressive wave

6.111.3. Non linear waves and the method of averaging

6.111.4. Non linear dispersive waves and two scale expansion procedure

7. Seismic Waves

7.1. Waves in elastic solids

7.2. Plane waves

7.3. Reflexion and refraction of plane elastic waves

7.4. Waves of kind I, II, III

7.5. Analytic representation of P and S waves

7.6. The layered spherical model

7.7. The energy balance

7.8. The reflexion and transmission coefficients

7.9. The wave system in the layered spherical model

7.9.1. The case of an incident P wave

7.9.2. The case of an incident SV wave

7.10. The P, PcP, PcS, PKP waves produced by a point source located outside the core

7.11. Application of the method of steepest descent to P and PcP wave integrals

7.11.1. The P wave

7.11.2. The PcP wave

7.12. The diffracted PcP wave

8. Some Problems in Water Wave Theory

Introduction

8.1. Oscillations in an infinite channel of variable depth

8.1.1. The shape of the channel

8.1.2. The representation of the solution in terms of Fourier integrals

8.1.3. The functional equations (8.25), (8.26)

8.1.4. Zeros and poles of H(u) Uniqueness property. Functional relations between F(u) and G(u)

8.1.5. Determination of the paths C, Γ and relations between the

constants l, w, p, q

8.1.6. Asymptotic behavior of the solution

8.1.7. Discussion of the amplitudes; the reflexion coefficient

8.II. A diffraction problem

8.11.1. The representation of the solution as a Laplace integral

8.11.2. A heuristic way for choosing the C and Γ contours

8.11.3. The boundary conditions

8.11.4. A solution to the functional equation (8.66)

8.11.5. The boundary conditions

8.11.6. Asymptotic behavior of the solution

References

Index

## Product details

- No. of pages: 359
- Language: English
- Copyright: © North Holland 1976
- Published: January 1, 1976
- Imprint: North Holland
- eBook ISBN: 9780444601919