Trapped modes and resonances for thin horizontal cylinders in a two-layer fluid

TL;DR

This paper derives exact series solutions for water waves over a thin submerged horizontal cylinder in a two-layer fluid, identifying trapped modes and resonances near cut-off frequencies, and conditions for their transition.

Contribution

It introduces a new analytical approach to model trapped modes and resonances for thin cylinders in layered fluids, revealing conditions for embedded trapped modes.

Findings

Series solutions expressed via exterior Neumann problem

Identification of trapped modes near cut-off frequencies

Conditions for transition of resonances into trapped modes

Abstract

Exact solutions of the linear water-wave problem describing oblique waves over a submerged horizontal cylinder of small (but otherwise fairly arbitrary) cross-section in a two-layer fluid are constructed in the form of convergent series in powers of the small parameter characterizing the "thinness" of the cylinder. The terms of these series are expressed through the solution of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the cylinder. The solutions obtained describe trapped modes corresponding to discrete eigenvalues of the problem (lying close to the cut-off frequency of the continuous spectrum) and resonances lying close to the embedded cut-off. We present certain conditions for the submergence of the cylinder in the upper layer when these resonances convert into previously unobserved embedded trapped modes.