4 - Green's Tensor and Integral Representations

Summary

Synopsis

In Chapter 4 we discuss the formulation of integral representations of solutions to rather general problems in elastic-wave propagation. Two constructions are used: the reciprocity identity and the Green's tensor for a full space. For these representations for an infinite domain to be derived, the principle of limiting absorption is introduced. This is needed for time-harmonic problems because the disturbance, in a sense, has been going on forever, resulting in no initial wavefront being present. Moreover, we establish a uniqueness result, indicating as we do so both the role of the principle of limiting absorption and that of specifying an edge condition. The chapter closes with an example that uses these ideas to develop an integral representation for the scattering of an acoustic wave by an elastic inclusion.

Introduction

In Chapter 2 we moved away from discussing plane waves to an introduction of plane-wave spectral representations in Section 2.3. This allowed us to discuss more general wavefields and to understand their propagation characteristics in terms of those of plane waves. We continue with this general theme, but construct, in this chapter, both far more general representations and ones in physical space rather than in wavenumber space. Though we make limited use of it in the chapters that follow, this material is very important because it is the basis for formulating elastic-wave problems in a form suitable to be analyzed numerically.