Über die Autorin bzw. den Autor

G. F. Roach

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"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

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"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

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Wave Scattering by Time-Dependent Perturbations

Princeton University Press

Chapter One

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...