Spectral gaps in a double-periodic perforated Neumann waveguide

TL;DR

This paper analyzes the spectral band-gap structure of a Neumann Laplacian in a perforated waveguide with small, dense holes, deriving asymptotic formulas for band edges and demonstrating the existence of spectral gaps.

Contribution

It provides new asymptotic formulas for spectral band edges in a perforated waveguide, accounting for complex boundary layers and Floquet parameter effects, advancing homogenization techniques.

Findings

Some spectral gaps are proven to be open for small perforations.

The position and size of gaps depend on geometric and perforation parameters.

Asymptotic behavior near band edges involves boundary layers and a fast Floquet variable.

Abstract

We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter $\varepsilon>0$. The periodicity cell itself contains a string of holes at a distance $O(\varepsilon)$ between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as $\varepsilon \to 0$. We demonstrate that, for $\varepsilon$ small enough, some spectral gaps are open. The position and size of the opened gaps depend on the strip width, the perforation period, and certain integral characteristics of the holes. The asymptotic behavior of the dispersion curves near the band edges is described by means of a `fast Floquet variable' and involves boundary layers in the vicinity of the perforation string of holes. The dependence on the Floquet parameter of the model problem in the periodicity cell requires a serious modification of the standard justification scheme in homogenization of spectral problems. Some open questions and possible generalizations are listed.