Dense Sets and Far Field Patterns in Acoustic Wave Propagation

We consider the Dirichlet, Neumann, and transmission boundary value problems corresponding to the scattering of an entire, time harmonic acoustic wave by a bounded obstacle in the plane. We first construct sets of solutions to these problems such that the restrictions of these solutions to the boundary $\partial \Omega $ of the scattering obstacle are dense in $L^2 (\partial \Omega )$. These results are then used to determine when the class of far field patterns corresponding to each of these scattering problems is dense or not dense in $L^2 [0,2\pi ]$.