  We are interested in the Korteweg-de Vries (KdV), the Burgers, and the
Whitham limit for a spatially periodic Boussinesq model with non-small
contrast. We prove estimates between the KdV, the Burgers, and the Whitham
approximation and true solutions of the original system which guarantee that
these amplitude equations make correct predictions about the dynamics of the
spatially periodic Boussinesq model over the natural time scales of the
amplitude equations. The proof is based on Bloch wave analysis and energy
estimates. The result is the first justification result of the KdV, the
Burgers, and the Whitham approximation for a dispersive PDE posed in a
spatially periodic medium of non-small contrast.
