Nonlinear dynamics of two-mode interactions in parametric excitation of surface waves

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Abstract
Parametric excitation of surface waves in a container under vertical forcing is investigated in detail, by an averaged Lagrangian method due to John Miles, and a system of evolution equations of third-order nonlinearity is presented for the case that the forcing frequency is chosen to be near twice the frequencies of two nearly degenerate free modes. The system of first-order differential equations in four variables which are derived from an averaged Hamiltonian is considered in a unified fashion, and the analytical results are compared with three experimental observations. It is found with the help of numerical integration that this dynamical system yields not only excitation of a single-mode state, but also interaction between two modes in which each mode oscillates either periodically or chaotically. These results are in good agreement with the observations, except for one case in which nonlinearity is considered to be too strong. As a fourth case, homoclinic chaos in the Hamiltonian system of two-degrees of freedom without damping is studied numerically. It is suggested that the chaotic mode competition observed in the experiments is different from the homoclinic chaos.