An introduction to echo analysis : scattering theory and wave propagation / G. F. Roach.

Format:

Book

Author/Creator:

Roach, G. F. (Gary Francis)

Series:

Springer monographs in mathematics

Language:

English

Subjects (All):

Scattering (Physics)--Mathematics.

Scattering (Physics).

Scattering (Mathematics).

Waves--Mathematics.

Waves.

Physical Description:

x, 319 pages : illustrations ; 24 cm.

Place of Publication:

London : Springer, [2008]

Summary:

The use of various types of wave energy is an increasingly promising, non-destructive means of detecting objects and of diagnosing the properties of quite complicated materials. An analysis of this technique requires an understanding of how waves evolve in the medium of interest and how they are scattered by inhomogeneities in the medium. These scattering phenomena can be thought of as arising from some perturbation of a given, known system and they are analysed by developing an associated scattering theory.

This monograph provides an introductory account of scattering phenomena and a guide to the technical requirements for investigating wave scattering problems. It gathers together the principal mathematical topics which are required when dealing with wave propagation and scattering problems, and indicates how to use the material to develop the required solutions.

Both potential and target scattering phenomena are investigated and extensions of the theory to the electromagnetic and elastic fields are provided. Throughout, the emphasis is on concepts and results rather than on the fine detail of proof; a bibliography at the end of each chapter points the interested reader to more detailed proofs of the theorems and suggests directions for further reading.

Aimed at graduate and postgraduate students and researchers in mathematics and the applied sciences, this book aims to provide the newcomer to the field with a unified, and reasonably self-contained introduction to an exciting research area and, for the more experienced reader, a source of information and techniques.

Contents:

1 Introduction and Outline of Contents 1

2 Some One-Dimensional Examples 17

2.2 Free Problems 17

2.3 Solutions of the Wave Equation 19

2.4 Solutions of Initial Value Problems 22

2.5 Integral Transform Methods 27

2.6 On the Reduction to a First Order System 31

2.7 Waves on Sectionally Homogeneous Strings 33

2.7.1 A Two-Part String 33

2.7.2 A Three-Part String 37

2.8 Duhamel's Principle 39

2.9 On the Far Field Behaviour of Solutions 41

2.9.1 Jost Solutions 43

2.9.2 Some Scattering Aspects 46

3 Preliminary Mathematical Material 51

3.2 Notations 51

3.3 Vector Spaces 52

3.4 Distributions 58

3.5 Fourier Transforms and Distributions 69

4 Hilbert Spaces 77

4.2 Orthogonality, Bases and Expansions 83

4.3 Linear Functionals and Operators on Hilbert Spaces 91

4.4 Some Frequently Occurring Operators 97

4.5 Unbounded Linear Operators on Hilbert Spaces 105

5 Two Important Techniques 115

5.2 Spectral Decomposition Methods 115

5.2.2 Spectral Decompositions 121

5.2.3 Spectral Decompositions on Finite Dimensional Spaces 122

5.2.4 Reducing Subspaces 127

5.2.5 Spectral Decompositions on Infinite Dimensional Spaces 132

5.2.6 Functions of an Operator 138

5.2.7 Spectral Decompositions of Hilbert Spaces 140

5.3 Semigroup Methods 145

5.3.1 Well-posedness of Problems 150

5.3.2 Generators of Semigroups 151

6 A Scattering Theory Strategy 157

6.2 Propagation Aspects 158

6.3 Solutions with Finite Energy and Scattering States 163

6.4 Construction of Solutions 166

6.4.1 Wave Operators and Their Construction 169

6.5 Asymptotic Conditions 175

6.6 A Remark about Spectral Families 181

6.7 Some Comparisons of the Two Approaches 182

7 An Approach to Echo Analysis 187

7.2 A Typical Mathematical Model 187

7.3 Scattering Aspects and Echo Analysis 191

7.4 Construction of the Echo Field 193

8 Scattering Processes in Stratified Media 201

8.2 Hilbert Space Formulation 203

8.3 Scattering in Plane Stratified Media 209

8.3.1 The Eigenfunctions of A 213

8.3.2 The Wave Eigenfunctions of A 216

8.3.3 Generalised Eigenfunction Expansions 219

8.3.4 Some Remarks about Asymptotic Wave Functions 222

9 Scattering in Spatially Periodic Media 225

9.2 The Mathematical Model 225

9.3 Elements of Floquet Theory 229

9.3.1 Hill's Equation 232

9.4 Solutions of the Mathematical Model 236

10 Inverse Scattering Problems 245

10.2 Some Asymptotic Formulae for the Plasma Wave Equation 247

10.3 The Scattering Matrix 249

10.3.1 Decomposable Operators 250

10.3.2 Some Algebraic Properties of W[subscript plus or minus] and S 252

10.4 The Inverse Scattering Problem 254

10.5 A High Energy Limit Method 255

10.5.1 The solution of an Integral Equation 256

Appendix A10.1 Proof of Theorem 10.2 259

11 Scattering in Other Wave Systems 263

11.2 Scattering of Electromagnetic Waves 263

11.3 Overview of Acoustic Wave Scattering Analysis 271

11.4 More about Electromagnetic Wave Scattering 274

11.5 Potential Scattering in Chiral Media 277

11.5.1 Formulation of the Problem 277

11.6 Scattering of Elastic Waves 279

11.6.1 An Approach to Elastic Wave Scattering 280

12.2 Remarks on Previous Chapters 285

A12.1 Limits and Continuity 289

A12.2 Differentiability 291

A12.3 The Function Classes C[superscript m](B) and C[superscript m](B) 291

A12.4 Sobolev Spaces 292

A12.5 Retarded Potentials 293

A12.6 An Illustration of the Use of Stone's Formula 296.

Notes:

Includes bibliographical references and index.

ISBN:

9781846288517

1846288517

OCLC:

232658367