THE TWO-DIMENSIONAL DIRECT AND INVERSE SCATTERING PROBLEMS WITH GENERALIZED OBLIQUE DERIVATIVE BOUNDARY CONDITION

Consider the two-dimensional exterior problem for the Helmholtz equation with a generalized oblique derivative boundary condition, which arises in some new scattering problems such as the scattering of tidal waves by islands under suitable assumptions. Different from the well-studied scattering problems with Dirichlet, Neumann, and impedance boundary conditions, the boundary condition in our new model involves a linear combination of the normal and tangential derivatives of the wave field with complex coefficient. Compared with the classical scattering models, the tangential derivative on the obstacle boundary leads to some essential differences such as the symmetric property of the Green function and the reciprocity principle of the scattering data. Both the direct and inverse scattering problems are studied in this paper. First, based on the Lax-Phillips method, we show the unique solvability of the direct scattering problem and the analyticity of the solution on the wave number. Second, we investigate the Green function and the far-field pattern for our scattering model, which play a key role in the corresponding inverse scattering problems. It is found that the symmetry of the Green function and the reciprocity relations need new characterizations in terms of the other scattering problem with conjugate boundary condition in our case. Finally, we study two inverse scattering problems for which the uniqueness results are established using our new results for the direct scattering problem.