Steady-state harmonic resonance of periodic interfacial waves with free surface boundary conditions based on the homotopy analysis method

We investigate the steady-state harmonic resonance of periodic interfacial gravity waves in a two-layer fluid with free surface. Two independent ’external’ and ’internal’ modes with separate linear dispersion relationships exist for this two-layer fluid. Exact harmonic resonance occurs when an external mode and an internal mode share the same phase speed and have an integer ratio of wave lengths. The singularity or small divisor caused by the exact or near resonant component is successfully removed by the homotopy analysis method (HAM). Convergent series solutions are obtained of steady-state interfacial wave groups with harmonic resonance. It is found that steady-state resonant waves form a continuum in parameter space. For finite amplitude interfacial waves, the energy carried by surface waves mirrors that carried by interface waves as the water depth varies. As the upper layer depth increases, energy carried by both surface and interface waves transfers from the shorter resonant component to the longer primary one. The paper utilises a HAM-based analytical approach to obtain a steady-state, periodic, interfacial wave system with exact- and near-resonant interactions between internal and external modes.