The simplest cases: Dirichlet and Neumann boundary conditions

Abstract

After a short derivation of the Helmholtz equation in acoustics this chapter formulates the direct scattering problem and presents results on uniqueness and existence without proofs (but gives complete and easily accessible references). It then introduces the far field patterns, the far field operator F, and formulates the inverse scattering problem. Section 1.4 derives the basic factorization of F and develops two criteria. The inf-criterion states that a given point z is inside the unknown domain D if, and only if, the infimum of a certain non-negative function (which depends solely on the known operator F and the point z) is strictly positive. Although very expensive from the computational point of view, it is nevertheless quite general and leads — for the cases studied in this chapter — to a more elegant characterization by the convergence of a Picard series involving the eigenvalues and eigenfunctions of the normal operator F. The chapter concludes with the investigation of the Neumann problem and some numerical experiments.