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 the water wave and Boussinesq equations. The
equations governing the corrections to the KdV approximation are identical
for both systems and are explicitly solvable.  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 for the correction to the
Boussinesq equation are essentially optimal.