The high wavenumber instabilities of a Stokes wave

A stability analysis for high wavenumber perturbations of a Stokes wave of wavenumber 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 wavenumbers k2 = ¼(m + 1)2k1. 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.