Acoustic scattering and radiation problems, and the null-field method

The best known methods for solving the scattering and radiation problems of acoustics are integral-equation methods. However, it is also known that the simplest of these methods yield equations which are not uniquely solvable at certain discrete sets of frequencies (the irregular frequencies). In this paper, we shall analyse an alternative method (the null-field method, or T-matrix method). We prove that the infinite system of null-field equations always has precisely one solution, i.e. the unphysical irregular frequencies do not occur with this method. Moreover, we also prove that the solution of the original boundary-value problem can always be determined (at any point exterior to the scatterer) from the solution of the null-field equations. We prove these results in two dimensions, for two radiation problems (the exterior Neumann problem and the exterior Dirichlet problem) and two scattering problems (scattering by a sound-hard body and scattering by a sound-soft body); similar results can be proved in three dimensions. We also prove some subsidiary results, concerning the solvability of certain boundary integral equations and the completeness of certain sets of radiating wave-functions, and give a discussion of related numerical techniques.