Waves Trapped by Semi-Infinite Kirchhoff Plate at Ultra-Low Frequencies

Abstract

In this paper, a semi-infinite Kirchhoff plate with a traction-free edge, which rests partially on a heterogeneous Winkler foundation (the Neumann problem for the biharmonic operator perturbed by a small free term with a compact support), is considered. It is shown that, for arbitrary small e > 0, a variable foundation compliance coefficient (defined nonuniquely) of order e can be constructed, such that the plate obtains the eigenvalue e4 that is embedded into a continuous spectrum, and the corresponding eigenfunction decays exponentially at infinity. It is verified that no more than one small eigenvalue can exist. It is noteworthy that a small perturbation cannot prompt an emergence of an eigenvalue near the cutoff point of the continuous spectrum in an acoustic waveguide (the Neumann problem for the Laplace operator).