Layer potentials and boundary value problems for the helmholtz equation in the complement of a thin obstacle

Abstract

Our purpose is to show in a precise manner the mathematical approach of the problem of the acoustic diffraction by an infinitely thin screen. The classical equations of acoustics are transformed into integral equations. The sound field diffracted by the obstacle is described by a double layer potential, the density of which is equal to the step of the potential across the screen. To avoid the mathematical difficulties, the infinitely thin screen will be considered as the limit of a sequence of obstacles with finite thickness. On such obstacles, the operators can be defined, thanks to the theory of Pseudo‐differential operators and Pseudo‐Poisson kernels. Then a limiting process is used and gives existence and unicity of the desired solutions in Sobolev spaces.