ON THE NULL-FIELD EQUATIONS FOR THE EXTERIOR PROBLEMS OF ACOUSTICS

Publisher Website
Google Scholar
Abstract
A familiar method for solving the exterior Neumann problem of acoustics in two dimensions is to derive an integral equation of the second kind over the boundary curve for the unknown potential, u, say. One way of doing this is to represent u as a continuous distribution of simple wave sources over the boundary, leading to an integral equation for the unknown source strength. Another way is to apply Green's theorem to u and a simple wave source (Helmholtz representation); when the field point lies on the boundary, this gives an integral equation for the unknown boundary values of u. It is well-known that both of these methods yield integral equations which have unique solutions, except at the same discrete set of wave numbers (the irregular values), corresponding to the eigenfrequencies of the interior Dirichlet problem. The same methods can be modified to solve the exterior Dirichlet problem, and both yield integral equations of the second kind which have unique solutions except at the eigenfrequencies of the interior Neumann problem. When the field point lies inside the boundary curve, the Helmholtz representation gives an integral relation. Using the known bilinear expansion for the simple wave source (in cylindrical polar coordinates), this integral relation may be reduced to an infinite set of equations, called the ‘null-field’ equations; equations of this type were first derived by Waterman, in 1965. In two dimensions, we show that the null-field equations always have a unique solution—irregular values do not occur. This result is proved here for both the exterior Neumann problem and the exterior Dirichlet problem. Similar results may be obtained in three dimensions.

This publication has 0 references indexed in Scilit: