Wave-Number-Explicit Bounds in Time-Harmonic Scattering

Abstract

In this paper we consider the problem of scattering of time-harmonic acoustic waves by a bounded, sound soft obstacle in two and three dimensions, studying dependence on the wave number in two classical formulations of this problem. The first is the standard weak formulation in the part of the exterior domain contained in a large sphere, with an exact Dirichlet-to-Neumann map applied on the boundary. The second formulation is as a second kind boundary integral equation in which the solution is sought as a combined single- and double-layer potential. For the variational formulation we obtain, in the case when the obstacle is starlike, explicit upper and lower bounds which show that the inf-sup constant decreases like $k^{-1}$ as the wave number k increases. We also give an example where the obstacle is not starlike and the inf-sup constant decreases at least as fast as $k^{-2}$. For the boundary integral equation formulation, if the boundary is also Lipschitz and piecewise smooth, we show that the norm of the inverse boundary integral operator is bounded independently of k if the coupling parameter is chosen correctly. The methods we use also lead to explicit bounds on the solution of the scattering problem in the energy norm when the obstacle is starlike. The dependence of these bounds on the wave number and on the geometry is made explicit.