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.
['Wolf-Patrick Düll', 'Kourosh Sanei Kashani', 'Guido Schneider']
The validity of Whitham's approximation for a Klein-Gordon-Boussinesq model