Mode competition in cross-waves.

Abstract

Cross-waves generated by an oscillating wavemaker in a rectangular wave tank are examined when two or more modes are simultaneously unstable. The partial differential equations governing the evolution of the complex amplitude of inviscid cross-waves are shown to be two coupled nonlinear Schrodinger equations with transverse modulations. Energy dissipation in the system is taken into account by the inclusion of a linear viscous damping term into the amplitude equations. A linearized stability analysis is performed on these equations to determine the critical modes, the growth rates and the stability curves. A center manifold analysis is used to reduce the PDE's to a system of ODE's in the neighborhood of a codimension two point where two adjacent spanwise modes are simultaneously nearly marginal. Four possible steady states of the system are found, one of which is a mixed mode state. A Hopf bifurcation from the mixed mode is predicted for a certain region of the parameter plane, suggesting the possibility of energy interchange between the two modes. The stability of the Hopf bifurcation is determined by studying a fifth order problem, where the quintic contributions come from the higher modes as well as the perturbations of damping and detuning.