High-frequency homogenization for travelling waves in periodic media

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 Schrodinger 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 k and frequency omega(1) plus a modulated Bloch carrier wave having crystal wavevector m and frequency omega(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 omega(1) = omega(2) and (k - m) circle dot Lambda is an element of 2 pi Z(d), where Lambda = (lambda(1)lambda(2) ... lambda(d)) is the periodicity cell of the medium and for any two vectors a = (a(1), a(2), ..., a(d)), b = (b(1), b(2), ..., b(d)) is an element of R-d, the product a circle dot 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.