Steady-state multiple near resonances of periodic interfacial waves with rigid boundary

Abstract

Steady-state resonant interfacial waves in a two-layer fluid within a frictionless duct are investigated theoretically. A combination of the homotopy analysis method (HAM) and Galerkin’s method is used to search for accurate steady-state resonant solutions with multiple near resonances. In the HAM, a piecewise parameter in the auxiliary linear operators is introduced to remove the small divisors caused by nearly resonant components. Convergent series solutions are then provided to the Galerkin iterations to accelerate the convergence rate. It is found that weakly nonlinear steady-state resonant waves form a continuum in the parameter space. As nonlinearity (wave steepness) increases, energy appears to be progressively shifted to sideband frequency components, effectively broadening the spectrum. The corresponding interfacial wave profile exhibits an almost fixed spatial pattern of repeated relatively high frequency, high-amplitude bursts followed by low-amplitude, longer waves. On examining the influence of density ratio, though changing slightly, the upper layer enlarges the amplitude of components near primary ones, which reduces the amplitude of higher frequency components, enlarges the wave steepness, and reduces the horizontal velocity in the wave field. Our results indicate that steady-state systems with resonant interactions among periodic interfacial wave components could occur naturally in the ocean. All these should enhance our understanding of periodic resonant interfacial waves.