Justification of geometrical optics for non-convex obstacles

Abstract

Using new results about the propagation of singularities, we shall show that it suffices to approximate the exact solution of the Helmholtz equation locally for justification of geometrical optics. The methods which we describe allow us to show that the geometrical-optics approximation converges to the exact solution of the Helmholtz equation for a large class of obstacles. Apart from some technical assumptions concerning the behaviour of the reflected rays, we only need that solutions of the wave equation decay exponentially outside of the obstacle. This is true, if rays are not trapped, cf. [S]. In this paper, however, we shall not treat the most general case, but approximate the solution of the Helmholtz equation only at points X, satisfying the additional conditions (A2) and (A3) given below. A more general case will be considered in a sequel to this paper. Let B be a bounded open set with C” boundary in R3, and Q = R3 B. The solution of Au + k2u = 0, (1.1)