HIGH WAVENUMBER INSTABILITIES OF A STOKES WAVE

A stability analysis for high wavenumber perturbations of a Stokes wave of wave-number k1 and slope ε is presented. Except for a correction term the governing equation is shown to be of Hill's type. The analysis predicts instability at wave-numbers k2 = 1/4(m + 1)2 k1. The two lowest and strongest instabilities are the Benjamin-Feir instability at m = 1, and the quartet resonance at m = 2. Both are incorrectly treated by the present method. For m ≥ 3 the analysis should be asymptotically (ε → 0) correct, yielding instability O(εm) due to m-fold Bragg-scattering. The non-resonant perturbations behave as predicted by WKBJ theory. The instability is too weak for experimental detection; numerical tests should be possible, but are not available at present. © 1979, Cambridge University Press