A note on the geometrical optics of diffraction by an interface

Extract

1. It is well known that solutions of the Cauchy problem for the wave equation represent disturbances obeying the laws of geometrical optics. Specifically a solution ψ of the wave equation

for which ψ = δψ/δt = 0 initially outside a surface C₀, vanishes at time t in the exterior of a surface C_t parallel to and at a normal distance ct from C₀ (see e.g. (l), page 643). Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary-value conditions of the Cauchy type. Boundary conditions of the type representing reflexion have been treated by Friedlander(2). He showed that as well as the incident and reflected wavefronts, there sometimes exists a ‘shadow’ where diffraction occurs, and that the diffracted wave fronts are normal to the reflecting surface, the corresponding rays travelling along the surface and leaving it tangentially. The purpose of this paper is to extend these results to refraction, where instead of a purely reflecting surface we have an interface between two different homogeneous media.