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.