Surface Waves on Vertically Sheared Flows: Approximate Dispersion Relations

Assuming linear wave theory for [1974] for deep water. We then obtain solutions waves riding on a weak current of O(c) compared to to the general problem to 0(1⁄2) (reproducing the the wave phase speed, an approximate dispersion result of Skop [1987]) and to 0(1⁄22). In section 3 relation is developed to 0(1⁄2 2 ) for arbitrary curwe apply the method to linear, cosine, and 1/7rent U(z) in water of finite depth. The 0(1⁄2 2 ) power current variations and compare results to approximation is shown to be a significant imanalytically or numerically obtained exact provement over the 0(1⁄2) result, in comparison with solutions. numerical and analytic results for the full The results of the analysis show that the soluproblem. Various current profiles in the full tions are valid in the regime (maxlU-•l/•) << 1, range of water depths are considered. Comments on where U is the depth-averaged current, leading to approximate action conservation and application to the conjecture that the expansions are valid for depth-averaged wave models are included. arbitrarily large currents having weak vorticity.