Nonlinear Periodic and Solitary Water Waves on Currents in Shallow Water

A variable-coefficient Korteweg-de Vries equation is used to model the deformation of nonlinear periodic and solitary water waves propagating on a unidirectional background current, which is either flowing in the same direction as the waves, or is opposing them. As well as the usual form of the Korteweg-de Vries equation, an additional term is needed when the background current has vertical shear. This term, which has hitherto been often neglected in the literature, is linear in the wave amplitude and represents possible nonconservation of wave action. An additional feature is that horizontal shear in the background current is inevitably accompanied by a change in total fluid depth, to conserve mass, and this change in depth is a major factor in the deformation of the waves. Using a combination of asymptotic analyses and numerical simulations, it is found that waves grow on both advancing and opposing currents, but the growth is greater when the current is opposing.