Water-waves modes trapped in a canal by a body with the rough surface

TL;DR

This paper investigates how small rough surfaces on a body in an infinite water channel can trap water wave modes, showing that multiple eigenvalues indicating trapped modes occur for sufficiently small surface roughness.

Contribution

It establishes the existence of multiple trapped water wave modes caused by a rough surface with small amplitude, under symmetry assumptions, and characterizes their spectral properties.

Findings

Existence of at least N eigenvalues in (0,d) for small roughness parameter

Eigenfunctions decay exponentially and have finite energy

Trapped modes are induced by surface roughness and symmetry

Abstract

The problem about a body in a three dimensional infinite channel is considered in the framework of the theory of linear water-waves. The body has a rough surface characterized by a small parameter $\epsilon>0$ while the distance of the body to the water surface is also of order $\epsilon$. Under a certain symmetry assumption, the accumulation effect for trapped mode frequencies is established, namely, it is proved that, for any given $d>0$ and integer $N>0$, there exists $\epsilon(d,N)>0$ such that the problem has at least $N$ eigenvalues in the interval $(0,d)$ of the continuous spectrum in the case $\epsilon\in(0,\epsilon(d,N)) $. The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes.