PARASITIC CAPILLARY WAVES - A DIRECT CALCULATION

As in a previous theory (Longuet-Higgins 1963) parasitic capillary wav es are considered as a perturbation due to the local action of surface tension forces on an otherwise pure progressive gravity wave. Here th e theory is improved by: (i) making use of our more accurate knowledge of the profile of a steep Stokes wave; (ii) taking account of the inf luence of gravity on the capillary waves themselves, through the effec tive gravitational acceleration g for short waves riding on longer wa ves. Nonlinearity in the capillary waves themselves is not included, a nd certain other approximations are made. Nevertheless, the theory is shown to be in essential agreement with experiments by Cox (1958), Ebu chi, Kawamura & Toba (1987) and Perlin, Lin & Ting (1993). A principal result is that for gravity waves of a given length L > 5 cm there is a critical steepness parameter (AK)(c) at which the surface velocity ( in a frame of reference moving with the phase-speed) equals the minimu m (local) speed of capillary-gravity waves. On subcritical gravity wav es, with steepness AK < (AK)(c), capillary waves may be generated at a ll points of the wave surface. On supercritical waves, with AK > (AK)( c), capillary waves can only be generated in the wave troughs; they ar e trapped between two caustics near the crests. Generally, the amplitu de of the parasitic capillaries is greatest on gravity waves of near c ritical (but not maximum) steepness.