Scattering of monochromatic acoustic and electromagnetic plane waves by two quarter spaces

Abstract

The plane wave scattering in a two‐dimensional medium consisting of two quarter spaces below a half‐space, also known as the vertical fault model, is the simplest two‐dimensional model applicable to geophysical exploration. The wave field obeys the two‐dimensional Helmholtz equation. Its formulation is equivalent to the electromagnetic case when the fundamental unknown is the horizontally polarized electric field vector and to the acoustic case when the unknown is the pressure. In the electromagnetic case the medium parameters are the electrical conductivity and the magnetic permeability, and in the acoustic case they are the compressibility and the density. The formulation of the scattered field in all space is defined by an integral representation composed of two one‐sided Fourier transforms in each medium. It represents an extension of Weaver's formulation. The spectral constituents of the Fourier integrals are determined by means of the Neumann's series, as a solution to the set of integral equations obtained by application of the boundary conditions. The numerical results show that the series converge and that the field values have a precision always better than 5%. The same formulation can be extended to solve the problem of scattering in more complex two‐dimensional or in three‐dimensional medium constituted of perpendicular blocks. Fields and spectral constituents may be used in the inversion of geophysical data.