Wave Scattering by Time-Dependent Perturbations: An Introduction: 19 (Princeton Series in Applied Mathematics) - Hardcover

Gary Roach

Hardcover

ISBN 10: 0691113408 ISBN 13: 9780691113401

Publisher: Princeton University Press, 2007

Synopsis

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, sonar, diagnostics in engineering and manufacturing, geophysical prospecting, and ultrasonic medicine such as sonograms.


New methods in recent years have been developed to assess the structure and properties of materials and surfaces. When light, sound, or some other wave energy is directed at the material in question, "imperfections" in the resulting echo can reveal a tremendous amount of valuable diagnostic information. The mathematics behind such analysis is sophisticated and complex. However, while problems involving stationary materials are quite well understood, there is still much to learn about those in which the material is moving or changes over time. These so-called non-autonomous problems are the subject of this fascinating book. Roach develops practical strategies, techniques, and solutions for mathematicians and applied scientists working in or seeking entry into the field of modern scattering theory and its applications.



Wave Scattering by Time-Dependent Perturbations is destined to become a classic in this rapidly evolving area of inquiry.

About this title

About the Author

G. F. Roach is professor emeritus of mathematics at the University of Strathclyde. His books include "An Introduction to Linear and Nonlinear Scattering Theory" and "Green's Functions".

From the Back Cover

"Professor Roach is an acknowledged expert in applied analysis. Wave Scattering by Time-Dependent Perturbations is a significant contribution to the mathematical literature--there are no similar books. There is a need to bring some of this analysis to the attention of those people who actually want to know how best to solve time-dependent scattering problems."--Paul Martin, executive editor of The Quarterly Journal of Mechanics and Applied Mathematics

"This book is an excellent rigorous treatise of modern scattering theory. The main new characteristic is that it places the correct emphasis on non-autonomous problems. I strongly believe that it comes at the appropriate time, as it includes the latest developments in the field. This book is an important and highly instructive piece of work, and an invaluable source of information."--George Makrakis, University of Crete

From the Inside Flap

"Professor Roach is an acknowledged expert in applied analysis. Wave Scattering by Time-Dependent Perturbations is a significant contribution to the mathematical literature--there are no similar books. There is a need to bring some of this analysis to the attention of those people who actually want to know how best to solve time-dependent scattering problems."--Paul Martin, executive editor ofThe Quarterly Journal of Mechanics and Applied Mathematics

"This book is an excellent rigorous treatise of modern scattering theory. The main new characteristic is that it places the correct emphasis on non-autonomous problems. I strongly believe that it comes at the appropriate time, as it includes the latest developments in the field. This book is an important and highly instructive piece of work, and an invaluable source of information."--George Makrakis, University of Crete

Excerpt. © Reprinted by permission. All rights reserved.

Wave Scattering by Time-Dependent Perturbations

By G. F. Roach

Princeton University Press

Copyright © 2007 Princeton University Press
All right reserved.
ISBN: 978-0-691-11340-1

Chapter One

Introduction and Outline of Contents

1.1 INTRODUCTION

The use of various types of wave energy as a probe is an increasingly promising nondestructive means of detecting objects and of diagnosing the properties of quite complicated materials.

An analysis of this technique requires a detailed understanding of, first, how waves evolve in the medium of interest in the absence of any inhomogeneities and, second, the nature of the scattered or echo waves generated when the original wave is perturbed by inhomogeneities that might exist in the medium. The overall aim of the analysis is to calculate the relationships between the unperturbed waveform and the echo waveform and to indicate how these relationships can be used to characterise inhomogeneities in the medium.

The central problem with which we shall be concerned in this monograph can be simply stated as follows.

A system consists of a medium containing a transmitter and a receiver. The transmitter emits a signal that is eventually detected at the receiver, possibly after it has been perturbed, that is, scattered, by some inhomogeneity in the medium. We are interested in the manner in which the emitted signal evolves through the medium and the form that it assumes at the receiver. Properties of the scattered or echo signal are then used to estimate the properties of any inhomogeneity in the medium.

Classifying inhomogeneities in the medium into identifiable classes by means of their echoes is known as the inverse scattering problem. An associated problem is that of waveform design, which is concerned with the choice of the signal waveform that optimises the echo signal from classes of prescribed inhomogeneities. These problems are of considerable interest and importance in engineering and the applied sciences. However, in order to be able to investigate them, the problem of knowing how to predict the echo signal when the emitted signal and the inhomogeneities are known must be well understood. This is called the direct scattering problem.

When the media involved are either stationary or possess time-independent characteristics-these are called autonomous problems (APs)-the mathematical analysis of the associated scattering effects is now quite well developed and a number of efficient techniques are available for constructing solutions to both the direct and the inverse problems. However, when the media are either moving or have time-dependent characteristics-these are known as nonautonomous problems (NAPs)-the investigations of corresponding scattering phenomena have not reached such a well-developed stage. Nevertheless, there are many significant problems of interest in the applied sciences that are NAPs. For instance, this type of problem can often arise when investigating sonar, radar, nondestructive testing and ultrasonic medical diagnosis methods. Indeed, they occur in any system that is either in motion or has components that either can be switched on or off or can be altered periodically. We shall study some of these systems in later chapters. These NAPs are intriguing both from a theoretical standpoint and from the point of view of developing constructive methods of solution; they certainly present a nontrivial challenge.

In our study here of NAPs we take as a starting point the assumption that all media involved consist of a continuum of interacting infinitesimal elements. Consequently, a disturbance in some small region of a medium induces an associated disturbance in neighbouring regions with the result that some sort of disturbance eventually spreads throughout the medium. We call the progress or evolution of such disturbances propagation. Typical examples of this phenomenon include, for instance, waves on water, where the medium is the layer of water close to the surface, the interaction forces are fluid pressure and gravity and the resulting waveform is periodic. Again, acoustic waves in gases, liquids and solids are supported by an elastic interaction and exhibit a variety of waveforms which can be, for example, sinusoidal, periodic, transient pulse or arbitrary. However, in principle any waveform can be set in motion in a given system provided suitable initial or source conditions are imposed.

The above discussion can be conveniently expressed in symbolic form as follows.

Consider first a system that has no inhomogeneities. Let [f.sub.0](x, s) be a quantity that characterises the state of the system at some initial time t = s and let [u.sub.0](x, t) be a quantity that characterises the state of the system at some later time t > s. We shall be concerned with systems for which states can be related by means of an "evolution rule," denoted by [U.sub.0](t - s), that determines the evolution in time of the system from its initial state [f.sub.0](x, s) to a state u0(x, t) at a later time t > s. This being the case, we write

[u.sub.0](x, t) = [U.sub.0](t - s)[f.sub.0](x, s),

where it is understood that [U.sub.0](0) = I = the identity.

In a similar manner, when inhomogeneities are present in the system, we will assume that we can express the evolution of the system from an initial state [f.sub.1](x, s) to a state u1(x, t) at a later time t > s in the form

[u.sub.1](x, t) = [U.sub.1](t - s)[f.sub.1](x, s), [U.sub.1](0) = I,

where [U.sub.1](t - s) denotes an appropriate evolution rule. Thus we see that we are concerned with two classes of problems. When there are no inhomogeneities present in the system, we shall say that we have a free problem (FP). When inhomogeneities are present in a system, we shall say that we have a perturbed problem (PP). We shall express this situation symbolically in the form

[u.sub.j](x, t) = [U.sub.j](t - s)[f.sub.j](x, s), [U.sub.j](0) = I, j = 0, 1,

where when j = 0 we will assume that we have a FP whilst when j = 1 we have a PP.

The principal aim of this monograph is to make the preceding discussions more precise and, in so doing, indicate means of developing sound, constructive methods of solution from what might be originally thought to be a purely abstract mathematical framework. In this connection we are immediately faced with a number of fundamental questions.

What are the mathematical equations that define (model) the systems of interest?

What is meant by a solution of the defining equations?

Under what conditions do the defining equations have unique solutions?

When solutions of the defining equations exist, can they be expressed in the form

[u.sub.j](x, t) = [U.sub.j](t - s)[f.sub.j](x, s), [U.sub.j](0) = I, j = 0, 1,

where [U.sub.j](t - s), j = 0, 1, is an evolution rule?

How can [U.sub.j](t - s) be determined?

What are the basic properties of [U.sub.j](t - s), j = 0, 1?

If a given problem is regarded as a PP, then an associated FP can be taken to be a problem that is more easily solved than the PP. We then ask, Is it possible to determine an initial state of the system defined by the FP so that the state of this system at some later time t, denoted [u.sub.0](x, t), is equal in some sense to [u.sub.1](x, t), the state at time t of the system defined by the PP? If we can do this, then it is readily seen that we will have taken a large step towards creating a firm basis from which to develop robust, constructive methods for determining the required quantity [u.sub.1](x, t) in terms of a more readily obtainable quantity [u.sub.0](x, t).

In connection with the possible equality of [u.sub.j](x, t), j = 0, 1, we first recognise and make use of the fact that in most experimental procedures, measurements in a system are made far away from any inhomogeneities that might exist in the system. Consequently, we will be mainly concerned here with the nature of the solutions [u.sub.j](x, t), j = 0, 1, and of their differences in the far field of any nonhomogeneity. With this in mind we shall see that it will be sufficient for our purposes to ask, What is the behaviour of [u.sub.j](x, t), j = 0, 1, as t [right arrow] [infinity]? Once this asymptotic behaviour is known, we can clarify what we mean by the equality of [u.sub.j](x, t), j = 0, 1, and turn to determining the conditions that will actually ensure when, in the far field at least, [u.sub.j](x, t), j = 0, 1, can be considered equal.

Although we are particularly interested in addressing these various questions when dealing with NAPs, we would point out that even for APs a detailed mathematical analysis of such questions can be technically very demanding. However, we would emphasise that this monograph is not a book on such topics as functional analysis, mathematical scattering theory, linear operator theory and semigroup theory but rather is meant as a guide through these various areas with the intention of highlighting their uses in practical problems. Examples will be given whenever it is practical to do so, as it is felt that abstract theories are often best appreciated, at first, by means of examples. The presentation in this monograph will frequently be quite formal, and we rely very much on the often held view expressed by Goldberger and Watson that "any formal manipulations which are not obviously wrong are assumed to be correct". Nevertheless, references will always be given in the text to where more precise and often quite general details can be found. Furthermore, we emphasise that in this monograph we are not interested in investigating the evolutionary processes mentioned above in full generality but rather shall confine our attention to those systems involving waves.

1.2 SOME ILLUSTRATIONS

As we have already mentioned, our main interest in this monograph will centre on those physical phenomena whose evolution can be described in terms of propagating waves. In the case of APs a simple example of a propagating wave is provided by a physical quantity y, which is defined by

y(x, t) = f(x - ct), (x, t) [element of] R x R, (1.1)

where c is a real constant. We notice that y has the same value at all those x and t for which (x - ct) has the same values. Thus (1.1) represents a wave that moves with constant velocity c along the x-axis without changing shape. If f is assumed to be sufficiently differentiable, then on differentiating (1.1) twice with respect to x and t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Similarly, we notice that a physical quantity w defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where c is a real constant, represents a wave moving with constant velocity c along the x-axis without changing shape but moving in the opposite direction to the wave y(x, t) defined in (1.1). Furthermore, we see that w(x, t) also satisfies an equation of the form (1.2). Equation (1.2) is referred to as the classical wave equation. The term "classical" will only be used in order to avoid possible confusion with other wave equations that we might consider.

Because the wave equation is linear, the compound wave

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

where f, g are arbitrary functions, is also a solution of the wave equation (1.2). This is the celebrated d'Alembert solution of the wave equation. In specific problems the functions f, g are determined in terms of the imposed initial conditions that solutions of (1.2) are required to satisfy [71, 105]. Indeed, if u(x, t) is required to satisfy the initial conditions

u(x, 0) = [phi](x), [u.sub.t](x, 0) = [PSI](x),

then (1.4) can be expressed in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

(see chapter 2 and [71, 105]). Consequently, if a system is defined by the wave equation (1.2), then (1.5) indicates how the initial state of the system, characterised by the functions [phi] and [PSI], evolves into a state defined by u(x, t), t > 0.

As a simple illustration of such an evolutionary process, consider a system consisting of an infinite string in the particular case when [PSI] [equivalent to] 0. The initial state of this system is then completely characterised by the function [PSI]. For simplicity, assume that the initial displacement of the string is of finite amplitude and is only nonzero over a finite length of the string. The function [phi] defines this profile. The result (1.5) indicates that this initial state of the system evolves into a state consisting of two propagating waves travelling in opposite directions along the string, each having the same profile as the initial state but with only half the initial amplitude.

As an illustration of a scattering process, again consider waves on a string. We assume, as before, that [PSI] [equivalent to] 0, but now we impose the extra requirement that the displacement of the string should be zero at x = 0 for all time. Essentially, this means that we are considering waves on a semi-infinite string. Arguing as before, we see that the initial state of the system, which again is completely characterised by the function [phi], evolves into a state consisting of two propagating waves travelling in opposite directions and having the same profile as the initial state but with only half the initial amplitude. However, this situation does not persist. After a certain time, say T, which is dependent on the velocity of the waves, one of the propagating waves will strike the "barrier" at x = 0. This wave will bounce off the barrier; that is, it will be reflected or scattered by the barrier. Consequently, from time T onwards the initial state of the system evolves into a state consisting of three components. Specifically, it will consist of the two propagating waves mentioned earlier and a reflected or scattered wave. Thus the evolved state can become quite difficult to describe analytically. Quite how difficult matters can become will be demonstrated in chapter 2. Further complications arise when we have to work either in more than one dimension or with defining equations that are more difficult to analyse than the wave equation (1.2). It was largely with these prospects in mind that scattering theory was developed.

We would emphasise at this stage that not all solutions of the wave equation yield propagating waves. For example, if the wave equation is solved using a separation of variables technique, then stationary wave solutions can be obtained. They are called stationary waves because they have certain features, such as nodes and antinodes, that retain their positions permanently with respect to time. Such solutions can be related to the bound states appearing in quantum mechanics and to the trapped wave phenomenon of classical wave theory.

Since the classical wave equation occurs in so many areas of mathematical physics, we shall adopt it as a prototype for much of our present study. We shall, of course, discuss other types of wave equations after we have settled the main features of wave scattering associated with this particular equation.

1.3 TOWARDS GENERALISATIONS

Scattering means different things to different people. Broadly speaking, it can be thought of as the interaction of an evolutionary process with a nonhomogeneous and possibly nonlinear medium. Certainly, the study of scattering phenomena has played a central role in mathematical physics over the years, with perhaps the earliest investigation of them being attributed to Leonardo da Vinci who studied the scattering of light into the shadow of an opaque body. Subsequently, other scattering effects have been discovered and investigated in such diverse fields as acoustics, quantum mechanics, medical diagnosis and many other nondestructive testing procedures.

Scattering phenomena arise as a consequence of some perturbation of a given known system, and they are analysed by developing an associated scattering theory. These scattering theories are concerned, broadly speaking, with the comparison of two systems, one being regarded as a perturbation of the other, and with the study of the manner in which the two systems might eventually become equal in some sense. Since a NAP can often be regarded as a perturbation of an AP, the development of scattering theories involving NAPs and APs seems to offer good prospects for providing a sound basis from which to develop robust approximation methods for obtaining solutions to a NAP in terms of the more readily obtainable solutions to an associated AP.

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2007
• Language: English
• ISBN 10: 0691113408
• ISBN 13: 9780691113401
• Binding: Hardcover
• Number of pages: 300