  The viability of the Whitham equation as a nonlocal model for
capillary-gravity waves at the surface of an inviscid incompressible fluid is
under study. A nonlocal Hamiltonian system of model equations is derived using
the Hamiltonian structure of the free surface water wave problem and the
Dirichlet-Neumann operator. The system features gravitational and capillary
effects, and when restricted to one-way propagation, the system reduces to the
capillary Whitham equation. It is shown numerically that in various scaling
regimes the Whitham equation gives a more accurate approximation of the
free-surface problem for the Euler system than other models like the KdV, and
Kawahara equation. In the case of relatively strong capillarity considered
here, the KdV and Kawahara equations outperform the Whitham equation with
surface tension only for very long waves with negative polarity.
