Retrospective Comments on the Elastic Wave Scattering Problem

Developments over the past several years have led to renewed study of theoretical methods for treating scattering of ultrasound by defects in elastic solids. Characteristically, many theories of scattering, until a few years ago, dealt with scalar waves and simple obstacles. Within that context two distinct regimes were apparent -the long wave length, and the short wave or imaging regime. The treatment of vector fields in elastic solids is considered the more cumbersome, but we now have made progress --still for idealized configurations. Specifically, the talk will deal with hypotheses on the types of problems which face us in the next generation of situations. As the abstract indicates, the remarks to be given in this talk are retrospective rather than constituting a report of a particular current piece of research. Perhaps the history and the flow of emphasis may be of use in giving us some idea of .what has been accomplished and what might be looked for in the next phase of this very interesting area. Before going further, however, I haye to acknowledge. my colleagues. Gubernatis, Doman.v, Rose, Teitel and seve~al other students at Cornell have contributed significantly to this work. Stimulatin~ interactions with Kahn, Budiansky, Rice, and Kino, among others at the Materials Research Council,have also been extremely useful. In 1973 at the first workshop meeting for this group, the point was made that much of the experimental work that was being done was interpreted using acoustic wave results. Examples of scattering which had been studied were given which showed that, in fact, these scalar wave interpretations gave exactly wrong results for some important directions of scattering. This experience really set the stage for what I am reviewing here. The Cornell group began its effort by attempting to place these results in contex~.with some of the ather approaches that have been carried out in the past. Clearly, elastic wave scattering has been one of the most venerable problems in mathematical physics. A partial inventory of some useful techniques include: partial wave expansions, the collection of groups of these partial waves to form aT-matrix (i.e., transition matrix, using quantum mechanical terminology), integral equation techniques, reciprocity methods, variational methods, and geometric diffraction methods. There are two regimes that must be considered in choosing a method: the short wave regime, which is the imaging regime, and the long wave length regime, in which the differences between the acoustic and the elastic cases are most pronounced. The question is "which method should you use in which regime?" Our feelings are quite definite as regards the purposes at hand. We are concerned with applied physics and engineering applications. There is another closely related enterprise, namely mathematical physics and applied mathematics. In engineering, a useful theoretical framework should be one in which physical .intuition or engineerin~ data can be entered as conveniently as possible. On the other hand, it may well be that the method is approximate and the limits of that approximation *Presently at the National Science Foundation. 317 have to be known. For that reason it is important that one have, as a resource from mathematical physics, some exact solution to the problem. For that reason, we examined this set of possible methods and chose to concentrate, at ·least in the initial phase, on the integral equation method. The partial wave expansion, including work by Ying and Truel 1 , and the matrix work by Pao and Varadan , all furnish an important reference for the theoretician and, eventually, for the experimentalist. However, the point about the integral equation methods and, more recently. the reciprocity methods, to which I will just refer briefly because Kino and Auld will develop these in detail, is that engineering data or physical intuition can be entered relatively easily. Given that motivation as a background, we now write down the key equation and indicate the. strategy which has been formulated, particularly by Gubernatis , in going about solving it at two levels of approximation. At the frequency w the scattering process is described by the inte-