DIFFERENT FORMS FOR NONLINEAR STANDING WAVES IN DEEP-WATER

Multiple forms for standing waves in deep water periodic in both space and time are obtained analytically as solutions of Zakharov's equatio n and its modification, and investigated computationally as irrotation al two-dimensional solutions of the full nonlinear boundary value prob lem. The different forms are based on weak nonlinear interactions betw een the fundamental harmonic and the resonating harmonics of 2, 3, ... times the frequency and 4, 9,... respectively times the wavenumber. T he new forms of standing waves have amplitudes with local maxima at th e resonating harmonics, unlike the classical (Stokes) standing wave wh ich is dominated by the fundamental harmonic. The stability of the new standing waves is investigated for small to moderate wave energies by numerical computation of their evolution, starting from the standing wave solution whose only initial disturbance is the numerical error. T he instability of the Stokes standing wave to sideband disturbances is demonstrated first, by showing the evolution into cyclic recurrence t hat occurs when a set of nine equal Stokes standing waves is perturbed by a standing wave of a length equal to the total length of the nine waves. The cyclic recurrence is similar to that observed in the well-k nown linear instability and sideband modulation of Stokes progressive waves, and is also similar to that resulting from the evolution of the new standing waves in which the first and ninth harmonics are dominan t. The new standing waves are only marginally unstable at small to mod erate wave energies, with harmonics which remain near their initial am plitudes and phases for typically 100-1000 wave periods before evolvin g into slowly modulated oscillations or diverging.