Steady-state interfacial gravity waves with one-dimensional class-III triad resonance

Abstract

Steady-state interfacial waves exhibiting class-III triad resonance are investigated in a two-layer liquid with a free surface. Two independent linear dispersion relationships related to surface and internal modes exist in the idealized model. One-dimensional class-III triad resonance requires the presence of two short surface wave modes and one long internal wave mode in a copropagating wave system. Convergent series solutions are achieved by the homotopy analysis method (HAM) for steady-state interfacial wave groups involving resonant triads and quartets. Unlike conventional progressive interfacial waves that have small surface amplitude and large interface amplitude, the free surface wave height is far larger than that of the internal interface waves because the resonance interactions comprise multiple surface wave modes and a single internal wave mode. As the upper layer thickness increases, energy from the interface is transported to the free surface, and energy in the whole wave system shifts from shorter to longer resonant triads and quartets. Our results indicate that steady-state interfacial waves with class-III exact and near-resonance interactions among surface and internal wave modes could occur in cases representative of the real ocean.