EVOLUTION OF A SHORT SURFACE-WAVE ON A VERY LONG SURFACE-WAVE OF FINITE-AMPLITUDE

To facilitate the theoretical prediction of the evolution of a short gravity wave on a long wave of finite amplitude, we consider a model where the long wave is represented by Gerstner's exact but rotational solution in Lagrangian coordinates. Analytical formulae for the modulation of an infinitesimal irrotational short wave are shown to be qualitatively accurate in comparison with the numerical results by Longuet-Higgins (1987) and with the analytical results by Henyey et al. (1988) for irrotational long waves. Discrepancies are generally of second order in the long-wave steepness, consistent with the vorticity in Gerstner's solution. Weakly nonlinear short waves are shown to be parametrically excited by the long wave over a long time. In particular, multiple bands of modulational instability appear in the parameter space. Numerical calculations of the nonlinear evolution equation show the onset of chaos for sufficiently large parameter alpha = epsilon(kABAR)2/(2-OMEGA/sigma), where epsilon-kABAR is the short-wave steepness and (epsilon-OMEGA, sigma) the frequency of the (long, short) wave. Furthermore, if the short-wave amplitude A is approximated by a two-mode truncated Fourier series, the evolution equation reduces to a non-autonomous Hamiltonian system. The numerical solutions confirm that the onset of chaos is an inherent feature of the parametrically excited nonlinear system.