On the resonant interaction between a surface wave and a weak surface current

An internal wave motion, below a layer of uniform fluid, induces a weak current on the free surface in the form of a long wave with phase velocity cI. A uniform progressive train of surface waves, whose wave-length is much shorter than that of the current is incident on it from infinity and undergoes modification. In particular, when the group velocity cg of the progressive wave is equal to cI, the resonance takes place and then, even though the amplitude of the current is small, the interaction builds up near a number of its wavelengths until the train of surface waves is significantly modified. The equations governing the modifications are derived, using the method of multiple scales, and the roles of the Döppler shift and the radiation stress in resonant situations are elucidated. Three-dimensional interactions are discussed and an analogy is drawn between the fundamental equation describing the interactions and Schrödinger's equation.