The low-frequency theory of elastic wave scattering

Abstract

Abstract. An incident longitudinal, or transverse, plane wave is scattered by a bounded region immersed in an infinite isotropic and homogeneous elastic medium. The region could be either a rigid scatterer or a cavity. Integral representations for the total displacement field, as well as for the introduced spherical scattering amplitudes are given explicitely in a compact form. Representations for the scattering cross-section whenever the incident wave is a longitudinal or a transverse wave are also provided. Using Papkovich potentials and low-frequency techniques the scattering problems are reduced to an iterative sequence of potential problems which can be solved successively in terms of expansions in appropriate harmonic functions. In each one of the four cases (longitudinal and transverse incidence on rigid scatterer and cavity) the corresponding exterior boundary value problems that specify the approximations as well as the analytic expressions for the scattering amplitudes and the scattering cross-section are given explicitly. The leading low-frequency term of the scattering cross-section for a rigid scatterer is independent of the wave number while for the case of a cavity it is proportional to the fourth power of the wave number. The low-frequency limit of the displacement field which corresponds to the static problem when the scatterer is a cavity, does not depend on the geometrical characteristics of the scatterer and it is always a constant.

Low Frequency Scattering by a Planar Crack

“…In this paper, a method to obtain the low frequency asymptotic solution to the problem of scattering of elastic waves by a planar crack of arbitrary shape is described in the spirit of the low frequency scattering theory [7].…”

The ellipsoidal cavity in the presence of a low-frequency elastic wave

“…It is also proved in [7] that the normalized spherical scattering amplitudes for the case of a cavity are given by gf(?,k) = k2pHp:…”

“…Introduction. In [7] we gave a systematic analysis of the elastic scattering problem at low frequencies. We studied the four basic problems corresponding to either a longitudinal or a transverse incident wave which is scattered by a rigid body or a cavity consisting of a smooth, convex, and bounded three-dimensional set.…”

“…In [7] we indicated that the total field ^ which satisfies the prescribed scattering problem has the following integral representation: *(r) = <£(r) + J *(r') • 7rf (r, r') ^(r'), r e V.…”

“…If we denote by ap the scattering cross section corresponding to P-incidence and by a1 the one corresponding to S-incidence, then, as shown in [7], we obtain°p…”