A CAUCHY-TYPE PROBLEM FOR THE WAVE-EQUATION

This paper discusses the solution of a wave equation in a region above the x(3) = 0 plane, with given Cauchy-tppe boundary conditions in the x(3)-direction. Two problems are treated, one, where the given Dirich let and Neumann data on the plane x(3) = 0 are independent, and the ot her, where they are linearly related in terms of the Neumann operator (corresponding to a down-going wave). Fourier transforms are used to r educe the three-dimensional wave equation to a one-dimensional Klein-G ordon equation. By Riemann's method, the solution of the Klein-Gordon equation is constructed with given Dirichlet and Neumann boundary cond itions in the x(3)-direction. A regularization procedure is developed for transforming the boundary data into a form for which the problem i s well-posed. A finite difference formula is obtained which can be use d in the numerical implementation. These results an applicable to the layer-stripping approach to the inverse problem.