A comparison of finite difference analogs for hyperbolic equations in inhomogeneous media

P. F. Daley

In an infinite halfspace with constant media, scalar wave equations may be written including one for acoustic wave propagation (pressure wave propagation in fluids) and another for S H - wave potential propagation. Both of these scalar wave equations will be considered here in a radially symmetric medium with the possibility that parameters related to both problems may vary with depth. This is done such that the different effects of discontinuities with depth of the media parameters may be investigated in the context of the boundary conditions required to be introduced. Once these have been determined, finite different analogues for the two cases are constructed. The simplest of these cases is to consider incidence at the halfspace boundary with the upper halfspace assumed to be a vacuum as was considered by Aki and Richards who presented solutions in the form of Sommerfeld integrals. For the problem types being investigated here, stress continuity conditions for a horizontal boundary within the two media types will be addressed. What has often been noticed in the literature is that a scalar wave equation associated with elastodynamic wave propagation in an isotropic homogeneous medium has its parameters modified, without any mathematical justification, to be spatially variable and the resulting equation employed to model elastodynamic (compressional) P - wave propagation using methods such as finite difference modeling. Liberties appear to be taken regarding continuity conditions at discontinuities of media parameters (interfaces). This is not to say that reasonable looking numerical results cannot be obtained, but the nature of these equations is questioned.