Three methods for the description of the temporal response to a SH plane impulsive seismic wave in a soft elastic layer overlying a hard elastic substratum

Abstract

We treat the case of a flat stress-free surface (i.e., the ground in seismological applications) separating air from a homogeneous, isotropic, solid substratum overlain by a homogeneous, isotropic, solid layer (in contact with the ground) solicited by a SH plane body wave incident in the substratum. The analysis is first carried out in the frequency domain and subsequently in the time domain. The frequency domain response is {\it normal} in that no resonances are excited (a resonance is here understood to be a situation in which the response is infinite in the absence of dissipation). The translation of this in the time domain is that the scattered pulse is of relatively-short duration. The duration of the pulse is shown to be largely governed by radiation damping which shows up in the imaginary parts of the complex eigenfrequencies of the configuration. Three methods are elaborated for the computation of the time history and give rise to the same numerical solutions for a large variety of configurations of interest in the geophysical setting under the hypothesis of non-dissipative, dispersionless media. The method appealing to the complex eigenfrequency representation is shown to be the simplest and most physically-explicit way of obtaining the time history (under the same hypothesis). Moreover, it is particularly suited for the case in which modes can be excited as occurs when the incident wave is not plane or the boundary condition is not of the stress-free variety for all transverse coordinates on the ground plane.