Nonlinear gravity waves on steady non-uniform currents

The interaction of nonlinear gravity waves and steady non-uniform currents is studied using the averaged Lagrangian method due to Whitham (1965a,b). The results are compared with the essentially linear theory of Longuet-Higgins & Stewart (1961, 1964) for three specific problems: waves on a stream (U(x), 0) with variations in the stream balanced by upwelling from below or inflow from the sides, and waves on a shear flow (0, V(x)). It appears that rates of growth of large waves are less than those predicted by linear theory and that the energy density can sometimes decrease when the wave height and steepness are still increasing. The final section discusses the form of the energy equation in terms of the Lagrangian.