BOUNDARY-DISTRIBUTION SOLUTION OF THE HELMHOLTZ EQUATION FOR A REGION WITH CORNERS

A technique is described for the solution of the Helmholtz equation together with associated boundary conditions based on a generalization of a method used for the solution of the Dirichlet problem of potential theory, in which a dipole distribution is introduced on the boundary of a region to generate the potential inside. In order that the boundary conditions be satisfied, the distribution must be found as the solution of an integral equation. If the boundary is smooth, the equation is of Fredholm type, but if it has a corner the equation is singular. The problem of a sharp corner is analyzed, and properties of the solution are developed using the theory of singular integral equations. Direct use of the technique can be made impossible in some cases by the presence of partner problem" eigenvalues. A simple method for avoiding this difficulty is presented. © 1979."