Spectral gaps for the linear surface wave model in periodic channels

TL;DR

This paper analyzes the spectral band-gap structure of linear water-wave problems in periodic channels with small apertures, providing asymptotic formulas for gap positions and conditions for non-degeneracy of spectral bands.

Contribution

It offers new asymptotic formulas for spectral gaps in periodic channels with small apertures and conditions preventing band degeneration.

Findings

Existence of many spectral gaps for small apertures

Asymptotic formulas for gap positions as aperture size approaches zero

Conditions ensuring spectral bands do not degenerate into eigenvalues

Abstract

We consider the linear water-wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width $\epsilon$. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the continuous spectrum. We show that the for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as $\epsilon \to 0$: the endpoints are determined within corrections of order $\epsilon^{3/2}$. The width of the first bands is shown to be $O(\epsilon)$. Finally, we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity.