Wave Scattering by Time-Dependent Perturbations An Introduction

This book offers the first comprehensive introduction to wave scattering in nonstationary materials. G. F. Roach's aim is to provide an accessible, self-contained resource for newcomers to this important field of research that has applications across a broad range of areas, including radar, son...

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Bibliographic Details
Main Author	Roach, G. F
Format	eBook Book
Language	English
Published	Princeton, New Jersey : Woodstock, Oxfordshire Princeton University Press 2009
Edition	1
Series	Princeton Series in Applied Mathematics
Subjects	Acoustic wave Angular frequency Approximation Asymptotic analysis Asymptotic expansion Banach space Boundary value problem C0-semigroup Characteristic function (probability theory) Continuous function Continuous function (set theory) Continuous spectrum Differentiable function Differential equation Dimension Dimension (vector space) Dimensional analysis Dirac delta function Dirichlet problem Eigenfunction Eigenvalues and eigenvectors Equation Exponential function Floquet theory Fourier inversion theorem Fourier series Fourier transform Fredholm integral equation Frequency domain General Topics for Engineers Helmholtz equation Hilbert space Initial value problem Integral equation Integral transform Inverse problem Inverse scattering problem Linear differential equation Linear map Locally integrable function Mathematical analysis Mathematics Metric space Ordinary differential equation Orthonormal basis Partial differential equation Perturbation (Mathematics) Physics Plane wave Projection (linear algebra) Propagator Quantity Quantum mechanics Reflection coefficient Riesz representation theorem Scalar (physics) Scattering Scattering (Physics) Scattering (Physics) > Mathematics Scattering theory SCIENCE SCIENCE / Waves & Wave Mechanics SCIENCE / Physics / General SCIENCE / Physics / Mathematical & Computational Self-adjoint Self-adjoint operator Sine wave Spectral theorem Spectral theory Theorem Theory Time domain Time evolution Unbounded operator Unitarity (physics) Vector space Volterra integral equation Wave function Wave packet Wave propagation Waves Waves > Mathematics Characteristic function (probability theory) Linear differential equation Eigenfunction Time domain Partial differential equation Projection (linear algebra) Dirac delta function Self-adjoint operator Spectral theorem Wave propagation Integral transform Vector space Mathematical analysis Eigenvalues and eigenvectors Propagator Unbounded operator Hilbert space Integral equation Angular frequency Frequency domain Approximation C0-semigroup Fredholm integral equation Scattering Initial value problem Helmholtz equation Time evolution Continuous function Riesz representation theorem Scattering theory Dimension (vector space) Boundary value problem Asymptotic expansion Wave packet Continuous spectrum Plane wave Locally integrable function Sine wave Continuous function (set theory) Metric space Theory Differential equation Acoustic wave Floquet theory Fourier transform Inverse scattering problem Exponential function Scalar (physics) Linear map Dimensional analysis Dirichlet problem Theorem Volterra integral equation Asymptotic analysis Differentiable function Wave function Equation Quantity Dimension Ordinary differential equation Fourier series Banach space Unitarity (physics) Fourier inversion theorem Inverse problem Orthonormal basis Spectral theory Self-adjoint Quantum mechanics Reflection coefficient
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ISBN	9780691113401 0691113408 9781400828166 1400828163
DOI	10.1515/9781400828166

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Table of Contents:

Wave scattering by time-dependent perturbations: An introduction (princeton series in applied mathematics) -- Contents -- Preface -- Chapter 1: Introduction and Outline of Contents -- Chapter 2: Some Aspects of Waves on Strings -- Chapter 3: Mathematical Preliminaries -- Chapter 4: Spectral Theory and Spectral Decompositions -- Chapter 5: On Nonautonomous Problems -- Chapter 6: On Scattering Theory Strategies -- Chapter 7: Echo Analysis -- Chapter 8: Wave Scattering from Time-Periodic Perturbations -- Chapter 9: Concerning Inverse Problems -- Chapter 10: Some Remarks on Scattering in Other Wave Systems -- Chapter 11: Commentaries and Appendices -- Bibliography -- Index
Front Matter Table of Contents Preface Chapter One: Introduction and Outline of Contents Chapter Two: Some Aspects of Waves on Strings Chapter Three: Mathematical Preliminaries Chapter Four: Spectral Theory and Spectral Decompositions Chapter Five: On Nonautonomous Problems Chapter Six: On Scattering Theory Strategies Chapter Seven: Echo Analysis Chapter Eight: Wave Scattering from Time-Periodic Perturbations Chapter Nine: Concerning Inverse Problems Chapter Ten: Some Remarks on Scattering in Other Wave Systems Chapter Eleven: Commentaries and Appendices Bibliography Index
Cover -- Title -- Copyright -- Contents -- Preface -- 1 Introduction and Outline of Contents -- 1.1 Introduction -- 1.2 Some Illustrations -- 1.3 Towards Generalisations -- 1.4 Chapter Summaries -- 2 Some Aspects of Waves on Strings -- 2.1 Introduction -- 2.2 A Free Problem -- 2.3 On Solutions of the Wave Equation -- 2.4 On Solutions of Initial Value Problems for the Wave Equation -- 2.5 Integral Transform Methods -- 2.6 Reduction to a First-Order System -- 2.7 Some Perturbed Problems for Waves on Strings -- 2.7.1 Waves on a Nonuniform String -- 2.7.2 Waves on a Semi-infinite String with a Fixed End -- 2.7.3 Waves on an Elastically Braced String -- 2.7.4 Waves on a String of Varying Length -- 2.7.5 A Scattering Problem on a Semi-infinite String -- 3 Mathematical Preliminaries -- 3.1 Introduction -- 3.2 Notation -- 3.3 Vector Spaces -- 3.4 Distributions -- 3.5 Fourier Transforms and Distributions -- 3.6 Hilbert Spaces -- 3.6.1 Orthogonality, Bases and Expansions -- 3.6.2 Linear Functionals and Operators on Hilbert Spaces -- 3.6.3 Some Frequently Occurring Operators -- 3.6.4 Unbounded Linear Operators on Hilbert Spaces -- 3.6.5 Some Remarks Concerning Unbounded Operators -- 4 Spectral Theory and Spectral Decompositions -- 4.1 Introduction -- 4.2 Basic Concepts -- 4.3 Concerning Spectral Decompositions -- 4.3.1 Spectral Decompositions on Finite-Dimensional Spaces -- 4.3.2 Reducing Subspaces -- 4.3.3 Spectral Decompositions on Infinite-Dimensional Spaces -- 4.4 Some Properties of Spectral Families -- 4.5 Concerning the Determination of Spectral Families -- 4.6 On Functions of an Operator -- 4.7 Spectral Decompositions of Hilbert Spaces -- 4.7.1 An Illustration -- 4.7.2 A Little More About Riemann-Stieltjes Integrals -- 4.7.3 Spectral Measure -- 4.7.4 On Spectral Subspaces of a Hilbert Space -- 5 On Nonautonomous Problems -- 5.1 Introduction
9.4.1 Some Asymptotic Formulae for the Plasma Wave Equation -- 9.4.2 On the Autonomous Inverse Scattering Problem -- 9.4.3 Extension to Nonautonomous Inverse Scattering Problems -- 10 Some Remarks on Scattering in Other Wave Systems -- 10.1 Introduction -- 10.2 Scattering of Electromagnetic Waves -- 10.3 Strategy for Autonomous Acoustics Problems in R[sup(3)] -- 10.4 Strategies for Electromagnetic Scattering Problems -- 10.4.1 Concerning Autonomous Problems -- 10.4.2 Concerning Nonautonomous Problems -- 10.5 Scattering of Elastic Waves -- 10.5.1 Strategy for Autonomous Elastic Wave Scattering Problems -- 11 Commentaries and Appendices -- 11.1 Remarks on Previous Chapters -- 11.2 Appendices -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- V -- W
5.2 Concerning Semigroup Methods -- 5.2.1 On the Well-posedness of Problems -- 5.2.2 On Generators of Semigroups -- 5.3 The Propagator and Its Properties -- 5.4 On the Solution of a Nonautonomous Wave Problem -- 5.4.1 A Mathematical Model -- 5.4.2 Energy Space Setting and Solution Concepts -- 5.4.3 Reduction to a First-Order System -- 5.4.4 On the Construction of the Propagator and the Solution -- 5.5 Some Results from the Theory of Integral Equations -- 6 On Scattering Theory Strategies -- 6.1 Introduction -- 6.2 On Scattering Processes in Autonomous Problems -- 6.2.1 Propagation Aspects -- 6.2.2 Solutions with Finite Energy and Scattering States -- 6.2.3 On the Construction of Solutions -- 6.2.4 Wave Operators and Their Construction -- 6.2.5 More About Asymptotic Conditions -- 6.2.6 A Remark About Spectral Families -- 6.2.7 Some Comparisons of the Two Approaches -- 6.2.8 Summary -- 6.3 On Scattering Processes in Nonautonomous Problems -- 6.3.1 Propagation Aspects -- 6.3.2 Scattering Aspects -- 6.3.3 On the Construction of Propagators and Solutions -- 7 Echo Analysis -- 7.1 Introduction -- 7.2 Concerning the Mathematical Model -- 7.3 Scattering Aspects and Echo Analysis -- 7.4 On the Construction of the Echo Field -- 7.4.1 Zero Approximation for the Echo Field -- 7.4.2 Concerning Higher-Order Approximations -- 7.5 A Remark About Energy in the System -- 8 Wave Scattering from Time-Periodic Perturbations -- 8.1 Introduction -- 8.2 Concerning the Mathematical Model -- 8.3 Basic Assumptions, Definitions and Results -- 8.4 Some Remarks on Estimates for Propagators -- 8.5 Scattering Aspects -- 8.5.1 Some Results for Potential Scattering -- 9 Concerning Inverse Problems -- 9.1 Introduction -- 9.2 Preliminaries -- 9.3 Reduction of the Plasma Wave Equation to a First-Order System -- 9.4 A High-Energy Method
Chapter Four. Spectral Theory and Spectral Decompositions
Chapter Ten. Some Remarks on Scattering in Other Wave Systems
Chapter Six. On Scattering Theory Strategies
Chapter Eight. Wave Scattering from Time-Periodic Perturbations
Index
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Chapter One. Introduction and Outline of Contents
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Chapter Eleven. Commentaries and Appendices
Contents
Chapter Five. On Nonautonomous Problems
Frontmatter --
Chapter Two. Some Aspects of Waves on Strings
Chapter Three. Mathematical Preliminaries
Preface
Chapter Nine Concerning Inverse Problems
Bibliography
Chapter Seven. Echo Analysis