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 ε⁴ 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.