Current-modified evolution equation for a broader bandwidth capillary-gravity wave packet

Abstract

We derive a higher order nonlinear evolution equation for a broader bandwidth three-dimensional capillary–gravity wave packet, in the presence of a surface current produced by an internal wave. Instead of a set of coupled equations, a single nonlinear evolution equation is obtained by eliminating the velocity potential for the wave-induced slow motion. Finally, the equation is expressed in an integro-differential equation form, similar to Zakharov’s integral equation. Using the evolution equation derived here, we show that the two sidebands of a surface capillary–gravity wave get excited as a result of resonance with an internal wave, all propagating in the same direction. It is also shown that surface waves can grow exponentially with time at the expense of the energy of the internal wave. doi:10.1017/S1446181116000225