  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
Schr{\"o}dinger 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 wave vector $\Bk$ and frequency $\omega_1$ plus a modulated Bloch
carrier wave having crystal wave vector $\Bm$ 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 $(\Bk-\Bm)\odot\Lambda \in 2\pi
\mathbb Z^d,$ where $\Lambda=(\lambda_1\lambda_2\dots\lambda_d)$ is the
periodicity cell of the medium and for any two vectors $a=(a_1,a_2,\dots,a_d),
b=(b_1,b_2,\dots,b_d)\in\mathbb R^d,$ the product $a\odot b$ is defined to be
the vector $(a_1b_1,a_2b_2,\dots,a_db_d).$ 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 rigor as that of Allaire and coworkers 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.
