Finite-amplitude steady-state resonant waves in a circular basin

The steady-state second-harmonic resonance between the fundamental and the second-harmonic modes for waves in a circular basin is investigated by solving the water-wave equations as a nonlinear boundary-value problem. The resulting waves are called (1,2)-waves. The geometry of the basin allows for both travelling waves (TW) and standing waves (SW). A solution procedure based on a homotopy analysis method (HAM) approach is used. In the HAM framework, the mathematical obstacle due to the singularity corresponding to the resonant-wave component can be overcome by adding the resonant term in the initial guess of the velocity potential. Approximate homotopy-series solutions can be obtained for both (1,2)-TW and (1,2)-SW. Two branches of (1,2)-TW and two branches of (1,2)-SW are found. They bifurcate from the trivial solution. For (1,2)-TW, the HAM-based approach is combined with a Galerkin numerical-method-based approach to follow the branches of nonlinear solutions further. The approximate homotopy-series solutions are used as initial guesses for the Galerkin method. As the nonlinearity increases, an increasing number of wave components are involved in the solution.

Automated summary

The study investigates the steady-state second-harmonic resonance between fundamental and second-harmonic modes for waves in a circular basin, resulting in (1,2)-waves. Using a homotopy analysis method approach, approximate homotopy-series solutions are obtained for both (1,2)-TW and (1,2)-SW. The nonlinear solutions branch out from trivial solutions as nonlinearity increases.