Buried Compressional Line Source in a Halfspace with an Irregular Boundary.

Abstract

By an application of the method of small perturbations an approximate solution of the problem of a compressional line source in a halfspace whose free surface is irregular is obtained in an integral form, on the assumption that the surface slope and the magnitude of the irregularities are small. The resulting integrals for the secondary waves whose amplitudes are proportional to the depth of the surface corrugation are evaluated approximately by the method of multiple steepest descent. Two particular surfaces are considered: a sinusoidal surface and a symmetrical curvilinear surface in the form of a basin. It is found that if the ratio of the wavelength of the incident wave to the halfwidth of the basin is small the secondary waves are reflected in the directions of specular reflection and if the ratio is large, the waves appear to come from the basin. With the sinusoidal surface, if the ratio of surface wavelength to the length of the incident waves is large the secondary waves are reflected in the specular directions and if the ratio is small then the surface acts as a grating, the secondary waves being reflected from the troughs of the surface. The amplitudes of the secondary waves are inversely proportional to the square root of range and the deeper the source the smaller the amplitudes.