Current-depth refraction of water waves

Surface gravity waves propagating on large-scale currents over a gently sloping sea bed are considered. Assuming irrotational flow a general proof is given of the existence of a mean energy level, leading directly to the current wave set-down. The complete set of conservation equations describing combined current-depth refraction is presented and solved in two special ssituations. Also a case with a rotational current is considered. Dissipation is neglected.