Traveling Waves for the Whitham Equation

The existence of traveling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surface water waves of finite depth. It is found that there exist small-amplitude periodic traveling waves with sub-critical speeds. As the period of these traveling waves tends to infinity, their velocities approach the limiting long-wave speed c0. It is also shown that there can be no solitary waves with velocities much greater than c0. Finally, numerical approximations of some periodic traveling waves are presented. It is found that there is a periodic wave of greatest height ∼ 0.642h0. Periodic traveling waves with increasing wavelengths appear to converge to a solitary wave.