A Note on Embedded Trapped Modes for a Two-layer Fluid over a Rectangular Barrier

Abstract

We prove that the system of equations describing waves in a two-layer fluid over a rectangular barrier of small height possesses embedded trapped modes (eigenvalues submerged in the continuous spectrum) for certain values of the width of the barrier and that these eigenvalues are analytic in the small parameter characterizing the height of the barrier. We do this by means of purely elementary considerations constructing explicit solutions and thus confirm the results of [7] obtained for general perturbations of the depth of the fluid in the particular case of a rectangular barrier.