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 wavelengths. The singularity or small divisor caused by the exactly or nearly 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 utilizes 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.

Automated summary

The study reveals the existence of two independent 'external' and 'internal' modes in a two-layer fluid, where exact harmonic resonance occurs when they share the same phase speed. The singularity or small divisor caused by resonant components is successfully eliminated using the homotopy analysis method, leading to convergent series solutions for steady-state interfacial wave groups with harmonic resonance. As the water depth varies, the energy carried by surface waves mirrors that carried by interface waves, showing a transfer of energy from shorter resonant components to longer primary ones as the upper layer depth increases.