Variational formulations for scattering in a three-dimensional acoustic waveguide

Variational formulations for direct time-harmonic scattering problems in a three-dimensional waveguide are formulated and analyzed. We prove that the operators defined by the corresponding forms satisfy a Garding inequality in adequately chosen spaces of test and trial functions and depend analytically on the wavenumber except at the modal numbers of the waveguide. It is also shown that these operators are strictly coercive if the wavenumber is small enough. It follows that these scattering problems are uniquely solvable except possibly for an infinite series of exceptional values of the wavenumber with no finite accumulation point. Furthermore, two geometric conditions for an obstacle are given, under which uniqueness of solution always holds in the case of a Dirichlet problem.