Trapped modes in thin and infinite ladder like domains. Part 1 : existence results

TL;DR

This paper investigates the existence of localized wave modes in a thin, ladder-like domain with a localized geometric perturbation, using asymptotic analysis and spectral theory, demonstrating that such perturbations can produce multiple localized eigenmodes.

Contribution

It provides a rigorous asymptotic analysis showing how localized eigenmodes emerge from geometric perturbations in thin ladder-like structures, extending spectral theory to these domains.

Findings

Localized modes exist when the rung width is decreased.

Number of localized eigenvalues can be made arbitrarily large.

Numerical experiments support theoretical predictions.

Abstract

The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter $\epsilon > 0$) whose limit (as $\epsilon$ tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the $\epsilon$ tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that $\epsilon$ is small enough. Numerical experiments illustrate the theoretical results.