Homogenization of the spectral equation in one-dimension

TL;DR

This paper investigates the high-frequency spectral behavior of a one-dimensional periodic system using Bloch wave homogenization, resulting in a combined microscopic-macroscopic spectral problem supported by numerical validation.

Contribution

It introduces a novel homogenization approach for high-frequency spectral analysis in one dimension, integrating microscopic and macroscopic eigenmodes.

Findings

Derivation of a spectral problem incorporating microscopic and macroscopic modes

Numerical simulations confirming the theoretical predictions

Enhanced understanding of high-frequency behavior in periodic structures

Abstract

The asymptotic behavior of a one-dimensional spectral problem with periodic coefficient is addressed for high frequency modes by a method of Bloch wave homogenization. The analysis leads to a spectral problem including both microscopic and macroscopic eigenmodes. Numerical simulation results are provided to corroborate the theory.