Abstract
This paper discusses the solution of a wave equation in a region above the x3=0 plane, with given Cauchy-type boundary conditions in the x3-direction. Two problems are treated, one, where the given Dirichlet and Neumann data on the plane x3=0 are independent, and the other, where they are linearly related in terms of the Neumann operator (corresponding to a down-going wave). Fourier transforms are used to reduce the three-dimensional wave equation to a one-dimensional Klein-Gordon equation. By Riemann's method, the solution of the Klein-Gordon equation is constructed with given Dirichlet and Neumann boundary conditions in the x3-direction. A regularization procedure is developed for transforming the boundary data into a form for which the problem is well-posed. A finite difference formula is obtained which can be used in the numerical implementation. These results are applicable to the layer-stripping approach to the inverse problem.
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