# Wave Scattering by Time-Dependent Perturbations: An Introduction

G. F. Roach
Pages: 300
https://www.jstor.org/stable/j.ctt7s7hp

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Preface
(pp. ix-xii)
GFR
4. Chapter One Introduction and Outline of Contents
(pp. 1-25)

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...

5. Chapter Two Some Aspects of Waves on Strings
(pp. 26-54)

The main purpose of this chapter is to introduce some of the methods used when investigating two classes of problem we refer to asfree problemsandperturbed problems, respectively. We illustrate these methods by taking as prototype problems those that arise when studying waves on strings. The FP with which we shall be concerned in this chapter deals with wave motions on an infinite homogeneous string. Associated with this FP we consider a number of naturally occurring PPs that arise when such aspects as, for example, variable coefficients or boundary conditions, are introduced into the discussion. Of particular interest are...

6. Chapter Three Mathematical Preliminaries
(pp. 55-112)

The mathematical analysis of scattering phenomena and the development of associated scattering theories are conducted in the framework oflinear spaces. These linear spaces involve generalisations of familiar concepts introduced in elementary courses in vector algebra and in the analysis of functions of a real variable. In these first courses we invariably worked with the value of a function at some point in its domain of definition rather than with the abstract quantity of the function itself. This strategy, as we shall see, will no longer be adequate for our purposes.

Some of the concepts and associated results that we...

7. Chapter Four Spectral Theory and Spectral Decompositions
(pp. 113-145)

Spectral theory provides mechanisms for decomposing quite complicated problems into a number of simpler problems with properties that are more manageable. We shall illustrate how this can be done in the following sections. The account is motivated by considering a typical abstract problem firstly when the underlying space is finite-dimensional and then when the space is infinite-dimensional. In both cases we will work mainly in a Hilbert space setting and assume that the operator that characterises the problem is self-adjoint. When working in an infinite-dimensional setting, we shall also require that the operator be compact on the Hilbert space. The...

8. Chapter Five On Nonautonomous Problems
(pp. 146-173)

In Chapter 1 we saw that in the acoustic case an IBVP could be reduced, formally at least, to an IVP. The IBVP was defined in Rn×R in terms of a partial differential equation that was of second order in time, whilst the IVP was a Cauchy problem for a system of ordinary differential equations that was first order in time and defined on an appropriate energy space. It turns out that for these first-order equations, results concerning existence, uniqueness and stability of solutions can be obtained in an efficient and elegant manner using results from the theory of...

9. Chapter Six On Scattering Theory Strategies
(pp. 174-208)

In this chapter we first recall salient features of scattering theories that have been developed for APs. Some of these have already been indicated in chapter 1, however, here we provide a rather more detailed account. Topics to be covered include propagation aspects, solution decay, scattering states, solutions with finite energy, representations of solutions, expansion theorems and construction of solutions. The comparison of solutions for large time is discussed, as is the evolution operator for a wave equation and the asymptotic equality of solutions. Results are recalled concerning the existence, uniqueness and completeness of wave and scattering operators, and mention...

10. Chapter Seven Echo Analysis
(pp. 209-224)

In the last subsection of chapter 2 we gave an indication of some of the influences of time-dependent perturbations by considering a one-dimensional problem involving a moving bead on a string. This was a simple illustration of a target scattering problem. In this chapter we turn our attention to a more detailed study of this class of problem using the various notations and techniques introduced in chapters 1, 5 and 6.

As mentioned in chapter 1, we consider systems consisting of a medium containing a transmitter and a receiver. The transmitter emits a signal that is eventually detected at the...

11. Chapter Eight Wave Scattering from Time-Periodic Perturbations
(pp. 225-234)

There are many cases of interest in the applied sciences that involve vibrating or pulsating or rotating media. A powerful diagnostic for investigating the properties of such systems is provided by the effect that such systems have on waves that are incident on them. Typical examples include the ultrasonic investigation of the heart and the reflection of radio, television and radar signals from moving objects.

In [61] and [148] many engineering applications of scattering of electromagnetic waves by rotating bodies can be found. However, one of the main difficulties in the analytical study of such problems lies in being unable...

12. Chapter Nine Concerning Inverse Problems
(pp. 235-245)

One of the more intriguing and, indeed, difficult problems in mathematical physics and the applied sciences is the determination of an impurity in an otherwise homogeneous region from the measurements available of a field scattered by the inhomogeneity. This is the standard inverse scattering problem. Such problems frequently arise when analysing, for example, various ultrasonic diagnostic techniques and other nondestructive testing processes. Typical areas include remote sensing problems associated with radar, sonar, geophysics and medical diagnosis.

We remark that in the majority of practical problems measurements of the scattered field can only be made in the far field of the...

13. Chapter Ten Some Remarks on Scattering in Other Wave Systems
(pp. 246-262)

In the previous chapters we have been concerned with acoustic wave scattering problems. In dealing with such problems we adopted theWilcox theory of acoustic scattering introduced in [154]. The main reason for adopting this particular approach was that the Wilcox theory uses quite elementary results from functional analysis, the spectral theory of self-adjoint operators on Hilbert spaces and semigroup theory and leads quite readily to the development of constructive methods based on generalised eigenfunction expansion theorems. Furthermore, unlike the Lax-Phillips theory [68], the Wilcox theory applies to scattering problems in both even and odd space dimensions.

In this monograph we...

14. Chapter Eleven Commentaries and Appendices
(pp. 263-274)

From the outset it has been emphasised that this book is an introductory text intended for the use of those wishing to begin studying the scattering of waves by time-dependent perturbations. For this reason we offer in this chapter some additional remarks on the material that has been presented in previous chapters. The main intentions are, on the one hand, to give some indications of the work that either has been or is being done for more general situations than those considered here and, on the other hand, to suggest further reading directions. Whilst it is recognised that it is...

15. Bibliography
(pp. 275-284)
16. Index
(pp. 285-287)