  In order to investigate corrections to the common KdV approximation to long
waves, we derive modulation equations for the evolution of long wavelength
initial data for a Boussinesq equation. The equations governing the corrections
to the KdV approximation are explicitly solvable and we prove estimates showing
that they do indeed give a significantly better approximation than the KdV
equation alone. We also present the results of numerical experiments which show
that the error estimates we derive are essentially optimal.
