Band-gap structure of the spectrum of the water-wave problem in a shallow canal with a periodic family of deep pools

TL;DR

This paper analyzes the spectral properties of water waves in a shallow periodic channel with deep pools, demonstrating the existence of spectral gaps for small depths using asymptotic analysis.

Contribution

It introduces a novel asymptotic approach to establish spectral gaps in water-wave problems with complex geometries involving deep potholes.

Findings

Spectral gaps exist for small channel depths.

Asymptotic analysis effectively handles boundary layer complexities.

The study advances understanding of wave propagation in structured shallow waters.

Abstract

We consider the linear water-wave problem in a periodic channel $\Pi^h \subset \mathbb{R}^3$, which is shallow except for a periodic array of deep potholes in it. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which includes the spectral Steklov boundary condition posed on the free water surface. We apply methods of asymptotic analysis, where the most involved step is the construction and analysis of an appropriate boundary layer in a neighborhood of the joint of the potholes with the thin part of the channel. Consequently, the existence of a spectral gap for small enough $h$ is proven.