A Comprehensive Review of Boundary Integral Formulations of Acoustic Scattering Problems

This is a review presenting an overview of the developments in boundary integral formulations of the acoustic scattering problems. Generally, the problem is formulated in one of two ways viz. Green’s representation formula, and the Layer-theoretic formulation utilizing either a simple-layer or a double-layer potential. The review presents and expounds the major contributions in this area over the last four decades. The need for a robust and improved formulation of the exterior scattering problem (Neumann or Dirichlet) arose due to the fact that the classical formulation failed to yield a unique solution at (acoustic) wave-numbers which correspond to eigenvalues (eigenfrequencies) of the corresponding interior scattering problem. Moreover, this correlation becomes more pronounced as the wave-numbers become larger i.e. as the (acoustic) frequency increases. The robust integral formulations which are discussed here yield Fredholms integral equations of the second kind which are more amenable to computation than the first kind. However, the integral equation involves a hypersingular kernel which creates ill-conditioning in the final matrix representation. This is circumvented by a regularisation technique. An extensive useful list of references is also presented here for researchers in this area.