A class of boundary value problems

Abstract

A simple method is presented for the solution of the partial differential equation of diffusion type with constant values of the solution or its normal derivative prescribed on the surfaces of a wedge of arbitrary angle. It is shown that this solution may be transformed in such a manner that it yields the solution to a similar class of boundary value problems for the time-harmonic wave (i.e. Helmholtz’s) equation. Direct solutions are also obtained for the problem of the diffraction of acoustic or electromagnetic plane waves by a perfectly absorbing or a perfectly reflecting wedge. In one solution the difraction problem is solved by modifying the solution of a similar boundary value problem for the diffusion equation. In the second formulation the solution of the diffraction problem is obtained by expressing the total solution as a sum of a diffracted field and geometrical optics terms. The diffracted field is then obtained by imposing the conditions of continuity across the shadow lines of geometrical optics. The solution for the diffraction of an arbitrary plane pulse by a wedge is also obtained; the solution being valid even if the boundary conditions are of the impedance type.