II - The scalar Helmholtz equation

Abstract

This chapter is devoted to presenting the foundations of obstacle scattering problems for time-harmonic acoustic waves. The chapter provides brief discussion of the physical background of the scattering problem, and then formulates the boundary-value problems for the Helmholtz equation. It will synthetically recall the basic concepts as they were presented by Colton and Kress. However, some of the details are left in the analysis. In this context, the technical proof is not repeated for the jump relations and the regularity properties for single- and double-layer potentials with continuous densities. However, Lax's theorem will be presented that enables to extend the jump relations from the case of continuous densities to square integrable densities. It then establishes some properties of surface potentials vanishing in sets of R3. These results play a significant role in the completeness analysis. Discussing the Green representation theorems will help in deriving some estimates of the solutions. The chapter concludes by analyzing the general null-field equation for the exterior Dirichlet and Neumann problems. In particular, it will establish the existence and uniqueness of the solutions and will prove the equivalence of the null-field equations with some boundary integral equations.