Point Source in a Layer Overlying a Semi-infinite Solid with Special Reference to Diffracted Waves

Abstract

Summary The problem of a point source of dilatational waves in a homogeneous, isotropic, perfectly elastic layer overlying a homogeneous, isotropic, perfectly elastic halfspace has been discussed. The terms corresponding to various reflected waves are obtained from the representation of the point source as a summation of plane waves by making use of reflection coefficients. These are then integrated along the modified Sommerfeld contour and various minimum-time-path and diffracted waves are obtained. Numerical values of the elastic constants are inserted and the dependence of the amplitudes of these waves on frequency and sourcedepth observed. The amplitudes of the diffracted waves are compared with those of the minimum-time-path waves and it has been found that the contribution due to the former is always small as compared to that due to the latter. In fact the former are appreciable only when the frequency is small. Chopra (1957) discussed the disturbance due to a point source of dilatational waves situated inside a stratum bounded on either side by a halfspace. The application of the Bromwich method of expansion followed by evaluation along the Sommerfeld loops, resolved the disturbance into various reflected, refracted and diffracted waves. Chopra showed that the successive terms in the Bromwich expansion of displacement potentials in one of two halfspaces can be obtained from the representation of the point source as a summation of plane waves by multiplying the integrand in this representation with appropriate reflection and refraction coefficients and also changing the exponent suitably. This method of obtaining the successive terms in the Bromwich expansion of displacement potentials can be extended to problems involving line or point sources in multilayered elastic media. 2. Statement of the problem We have taken the problem of a point source of dilatational waves in a homogeneous, isotropic, perfectly elastic layer overlying a homogeneous, isotropic, perfectly elastic halfspace. The layer is assumed to be of uniform thickness. The waves starting from the point source as dilatational waves are multiply reflected . from the free surface and from the interface variously as dilatational and distortional