Analytical Cubic Solution to Weakly Nonlinear Interactions Between Surface and Interfacial Waves

Resonant interaction between one surface wave and two oblique interfacial waves is analyzed in a three dimensional system of a finite-depth, two-layer fluid. A third order perturbation analysis is carried out to obtain the evolution equations of the waves amplitudes. Taking the waves amplitudes as the perturbation small parameter, the evolution equations of the waves are solved simultaneously to obtain the short and long term behavior of the interfacial waves. In contrast to the second order analysis, the current analysis shows that after an initial exponential growth period, the interfacial waves stop growing and stabilize. Furthermore, the influences of surface wave frequency, density ratio and viscosity in the resonant interaction are investigated. Finally, the analytical results are compared with previous experimental observations.