An inequality for the reduced wave operator and the justification of geometrical optics

We present an inequality for the reduced wave operator in the exterior of a star‐shaped surface in n‐space, with a Dirichlet boundary condition on the surface and a radiation condition at infinity. This inequality is used to demonstrate the continuous dependence (in a suitable norm) of the solution of a scattering problem upon the boundary data and inhomogeneous term in the differential equation. This basic result is then used together with the results of D. Ludwig [7] to prove that the formal solution of the scattering problem for a convex body, which is given by geometrical optics, is asymptotic to the exact solution. Similar results have been given in two dimensions by V. S. Buslaev [1] and R. Grimshaw [2], using different methods, who also consider the Neumann problem. Unfortunately the methods used here are inapplicable in that case.