Resonance regimes of scattering by small bodies with impedance boundary conditions

The paper concerns scattering of plane waves by a bounded obstacle with complex valued impedance boundary conditions. We study the spectrum of the Neumann-to-Dirichlet operator for small wave numbers and long wave asymptotic behavior of the solutions of the scattering problem. The study includes the case when $k=0$ is an eigenvalue or a resonance. The transformation from the impedance to the Dirichlet boundary condition as impedance grows is described. A relation between poles and zeroes of the scattering matrix in the non-self adjoint case is established. The results are applied to a problem of scattering by an obstacle with a springy coating. The paper describes the dependence of the impedance on the properties of the material, that is on forces due to the deviation of the boundary of the obstacle from the equilibrium position.