Green’s Functions for Axially Symmetric Elastic Waves in Unbounded Inhomogeneous Media Having Constant Velocity Gradients

Abstract

This paper treats the propagation of elastic waves in one class of inhomogeneous media. The properties of the media are proportional to powers of the Cartesian co-ordinate z in such a way that Poisson’s ratio remains constant and the velocities of propagation of P and S waves are proportional to z. Exact expressions are obtained for the P, SV, and SH displacements generated by impulsive point sources buried in unbounded media of this class. The sources are taken to be symmetric about the z axis. Separation of the vector-wave equation is achieved by use of a potential representation that is a generalization of the familiar Stokes-Helmholtz representation; the P, SV, and SH displacement vectors are expressed in terms of scalar potentials that satisfy independent second-order wave equations. The SH displacement is solenoidal, but it is found that the products of the P and SV displacement vectors with appropriate weighting functions, rather than the displacement vectors themselves, are irrotational and solenoidal, respectively. The media are found to be dispersive, with the result that decaying tails follow the advancing wave fronts.