Traveling waves for the Whitham equation

Abstract

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 $c_0$. It is also shown that there can be no solitary waves with velocities much greater than $c_0$. Finally, numerical approximations of some periodic traveling waves are presented. It is found that there is a periodic wave of greatest height $\sim 0.642 h_0$. Periodic traveling waves with increasing wavelengths appear to converge to a solitary wave.