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

Resonance regimes of scattering by small bodies with impedance boundary conditions