Boundary integral methods in low frequency acoustics

This expository paper is concerned with the direct integral formulations fo r boundary value problems of the Helmholtz equation. We discuss unique solv ability for the corresponding boundary integral equations and its relations to the interior eigenvalue problems of the Laplacian. Based on the integra l representations, we study the asymptotic behaviors of the solutions to th e boundary value problems when the wave number tends to zero. We arrive at the asymptotic expansions for the solutions, and show that in all the cases , the leading terms in the expansions are always the corresponding potentia ls for the Laplacian. Our integral equation procedures developed here are g eneral enough and can be adapted for treating similar low frequency scatter ing problems.