Diffraction of elastic waves from a surface source in a heterogenous medium

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Abstract
A theory is developed for wave propagation of a given frequency emerging from a seismic surface source in a medium in which the velocity is a continuous function of one coördinate only. It is assumed that the relative change of the elastic parameters is very small over a wave length. The wave equations are then solved in cylindrical coördinates under suitable boundary conditions and integral representations are obtained for the displacements, which are generally valid. These integrals are then evaluated for a special case with an almost linear velocity gradient and the surface displacements are obtained for long ranges. It is found that the amplitude of the body waves (both P and S) inside the shadow zone decays exponentially with the distance from the source at a rate proportional to one-third power of the frequency and two-thirds power of the velocity gradient.