The Analyse Of Boundary Integral Equation Methods In Low Frequency Acoustics In 3d

Abstract

In this thesis we consider the direct integral formulations for boundary value problems of the Helmholtz equation. We reduce the boundary value problems to boundary integral equation and analyses the boundary potentials associated with low frequency. From boundary potential we introduce the Calderon projectors and give the basic mapping properties. We discuss unique solvability for the corresponding boundary integral equations (BIE) and its relations to the interior eigenvalue problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the boundary value problems (BVP) when the wave number tends to zero in R3. 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 potentials for the Laplacian and our integral equation procedures developed here are general enough and can be adapted for treating similar low frequency scattering problems.