Modulation of the amplitude of steep wind waves

Lake & Yuen (1978) have suggested that in very steep wind waves the modulation-frequency of the wave amplitude may correspond to the frequency of the fastest-growing subharmonic instability of a uniform train of waves whose amplitude equals the mean wave amplitude ā . The approximate theory of Benjamin & Feir (1967) gives this frequency as (āk)fd , where κ is the wavenumber and fd the frequency of the unperturbed waves. This expression applies strictly only to very small values of the wave steepness āk . More recently (Longuet-Higgins 1978) the present author calculated accurately all the normal-mode instabilities of steep gravity waves on deep water. In this note these calculations are used to determine the frequency of the fastest-growing sub-harmonic instabilities precisely. When compared with the experimental data of Lake & Huen, these frequencies show even closer agreement.