Effects of Rayleigh waves on the essential spectrum in perturbed doubly periodic elliptic problems

TL;DR

This paper demonstrates how mirror reflection modifications in doubly periodic elliptic operators induce Rayleigh waves and create additional spectral bands in the essential spectrum, using asymptotic analysis and high contrast coefficients.

Contribution

It introduces a novel method of generating Rayleigh waves and spectral band modifications through coefficient reflection in doubly periodic elliptic operators.

Findings

Mirror reflection induces Rayleigh type waves near the line.

Modification causes an additional spectral band in the essential spectrum.

Results are proven using asymptotic analysis with high contrast coefficients.

Abstract

We give an example of a scalar second order differential operator in the plane with double periodic coefficients and describe its modification, which causes an additional spectral band in the essential spectrum. The modified operator is obtained by applying to the coefficients a mirror reflection with respect to a vertical or horizontal line. This change gives rise to Rayleigh type waves localized near the line. The results are proven using asymptotic analysis, and they are based on high contrast of the coefficient functions.