Infinite Kirchhoff Plate on a Compact Elastic Foundation May Have an Arbitrarily Small Eigenvalue

Abstract

An inhomogeneous Kirchhoff plate composed of a semi-infinite strip waveguide and a compact resonator that is in contact with a Winkler foundation of low variable compliance is considered. It is shown that, for any $$\varepsilon > {\text{0}}$$, a compliance coefficient $$O({{\varepsilon }^{2}})$$ can be found such that the described plate possesses the eigenvalue e4 embedded into the continuous spectrum. This result is quite surprising, because, in an acoustic waveguide (the spectral Neumann problem for the Laplace operator) a small eigenvalue does not exist for any slight perturbation. The cause of this disagreement is explained.