Propagation and trapping of obliquely incident wave groups over a trench with currents

The analytical solutions of secord-order long waves generated by the diffraction of short wave groups by a combination of depth and current velocity discontinuities are presented. The incident wave groups consist of two co-linear short waves with slightly different frequencies. These incident short wave groups are reflected partially by the abrupt changes in depth and current velocity. The incident, reflected, and transmitted wave groups are accompanied by the second-order locked long waves, which propagate together with the wave envelopes of the short waves at their group velocity. Because of the discontinuity in the second-order free surface displacement of the locked long waves, free long waves are generated, which propagate at the shallow-water phase speed and in a direction different from those of short waves and wave envelopes. With certain combinations of angle of wave incidence, depth ratios and current velocity, the free long waves can be trapped in the vicinities of depth and current discontinuities. Several examples are given to illustrate the resonance of trapped free long waves.