Spectrum of the linear water model for a two‐layer liquid with cuspidal geometries at the interface

Abstract

We show that the linear water wave problem in a bounded liquid domain may have continuous spectrum, if the interface of a two‐layer liquid touches the basin walls at zero angle. The reason for this phenomenon is the appearance of cuspidal geometries of the liquid phases. We calculate the exact position of the continuous spectrum. We also discuss the physical background of wave propagation processes, which are enabled by the continuous spectrum. Our approach and methods include constructions of a parametrix for the problem operator and singular Weyl sequences.