High-frequency homogenization for travelling waves in periodic media
File(s)HFH6.pdf (440.14 KB)
Accepted version
Author(s)
Harutyunyan, D
Milton, GW
Craster, RV
Type
Journal Article
Abstract
We consider high-frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schrödinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 1 plus a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 2. We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless ω 1=ω 2 and [Formula: see text] where Λ=(λ1λ2…λ d ) is the periodicity cell of the medium and for any two vectors [Formula: see text] the product a⊙b is defined to be the vector (a 1 b 1,a 2 b 2,…,a d b d ). This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigour as that of Allaire and co-workers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue.
Date Issued
2016-07-13
Online Publication Date
2017-07-13T06:00:23Z
Date Acceptance
2016-06-03
ISSN
1364-5021
Publisher
The Royal Society
Journal / Book Title
Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences
Volume
472
Issue
2191
Copyright Statement
© 2016 The Author(s). Published by the Royal Society. All rights reserved.
Source Database
pubmed
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/L024926/1
Subjects
Bloch waves
asymptotics
homogenization
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Publication Status
Published
Article Number
20160066