Hypersingular integral equations not needed in the impedance problem in scattering theory

AbstractWe propose a new approach to the analysis of the impedance problem for the Helmholtz equation in the exterior of a body (obstacle) in two and three dimensions. This approach can be called ‘method of interior boundaries’, because an additional boundary is introduced inside the scattering body. An appropriate boundary condition is specified on the additional boundary. The solution of the problem is obtained in the form of a single-layer potential on the whole boundary. The density in the potential satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact, our method holds for any positive wave numbers. The Neumann problem is a particular case of our model. The method of hypersingular integral equations widely used in the scattering theory is more complicated and has several shortcomings in comparison with our approach