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

Abstract

Steady-state interfacial waves under a one-dimensional (1-D) class-IV exact triad resonance are investigated in a two-layer fluid with a free surface upper boundary. Four groups (G1–G4) of convergent series solutions are obtained by the homotopy analysis method. Though the harmonic resonance conditions are contained in the class-IV resonance criteria, the influences of 1:2 harmonic resonances on the energy spectrum could be neglected. Unlike former steady-state resonant interfacial wave spectrum where all the components joining the resonance are significant, the energy of one primary component can be ignored for two groups (G2 and G4) of wave solutions obtained in this paper. It is found that a little energy induced from the external environment might greatly change the energy spectrum for G1. However, the energy introduced from the outside cannot vary the wave energy distribution for G2. The reason for the extremely high crests on the instantaneous profiles of free surfaces of G2 and G4 is that the peaks of the class-IV exactly resonant and some trivial components momentarily overlap at some special horizontal positions. The class-IV exact triad resonance curve could be divided into four pieces containing the existence ranges of G3 and G4 and two regions with no solution found. One of the regions without convergent solutions results from an infinite number of singularities or small divisors caused by infinite exact or near resonances. Our results indicate that steady-state interfacial waves with class-IV triad resonance interactions among one surface and two internal wave modes could exist.