  In this paper we prove the validity of a long wave Whitham approximation for
a system consisting of a Boussinesq equation coupled with a Klein-Gordon
equation. The proof is based on an infinite series of normal form
transformations and an energy estimate. We expect that the concepts of this
paper will be a part of a general approximation theory for Whitham's equations
which are especially used in the description of slow modulations in time and
space of periodic wave trains in general dispersive wave systems.
