title: High-frequency homogenization for travelling waves in periodic media.
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 frequencyω1plus a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequencyω2. 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 unlessω1=ω2and [Formula: see text] whereΛ=(λ1λ2…λd) is the periodicity cell of the medium and for any two vectors [Formula: see text] the producta⊙bis defined to be the vector (a1b1,a2b2,…,adbd). 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.
authors: Department of Mathematics , University of Utah , Salt Lake City, UT 84112, USA.
authors: Davit
authors: 0000-0002-8685-2778
authors: D
authors: Harutyunyan
authors: Department of Mathematics , University of Utah , Salt Lake City, UT 84112, USA.
authors: Graeme W
authors: GW
authors: Milton
authors: Department of Mathematics , Imperial College London , London SW7 2AZ, UK.
authors: Richard V
authors: RV
authors: Craster
keywords: Bloch waves; asymptotics; homogenization