Oblique Derivative Boundary Conditions and the Image Method

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Abstract
The image method of solving boundary value problems is extended to oblique derivative boundary conditions on plane boundaries for a class of equations including the wave equation, the heat equation and Laplace’s equation. This generalizes the recent result of Gilbarg and Trudinger [Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1977] for Laplace’s equation. The usefulness of the method is illustrated by solving with it both the wave equation and the heat equation in a half-space with an initial point source and an oblique derivative boundary condition. In both cases a surface wave is found. For the wave equation there is a region near the boundary where the surface wave is the first arrival. For the heat equation the surface wave is the largest wave in this region for small values of the time.

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