Spanwise modal competition in cross-waves

An asymptotic theory for cross-waves generated by an oscillating wavemaker in a semi-infinite rectangular wave tank is derived for the limit of large mode number. The possibility of multiple mode excitation is included by introducing a spanwise modulation. The partial differential equations governing the evolution of inviscid cross-waves are shown to be two coupled nonlinear Schrödinger equations. Energy dissipation in the system is taken into account including a linear damping term. A center manifold analysis is used to reduce the PDEs to a system of coupled Landau equations in the neighborhood of a codimension-two point where two adjacent spanwise modes are marginally stable. Four possible steady states of the system are found, one of which is a mixed-mode superposition of two spanwise modes. A Hopf bifurcation from the mixed mode is predicted for some parameters; extending the system to higher order allows the stability of this bifurcation to be determined in terms of perturbations to a Hamiltonian system. Both subcritical and supercritical bifurcation are possible. An experimental study in the neighborhood of a codimension-two point shows good agreement with the theoretical predictions including the discovery of a mixed-mode state.