Some effects of surface tension on steep water waves

Surface tension provides a restoring force which cannot reasonably be ignored for water waves of short crest-to-crest length. Even for large wavelengths its presence precludes any sharp corner developing on the free surface. In this paper we begin an investigation of the effects of surface tension on steep water waves. The work of Longuet-Higgins (1975) is generalized to show how the integral properties of the wave train are affected. In particular it is shown that for pure capillary waves in deep water the mean fluxes of energy, mass and momentum are given by 3Tc, 2T/c and 4T − V respectively, where c is the phase velocity, T the kinetic energy and V the potential energy. Also the exact solution for the wave profile of deep-water pure capillary waves (Crapper 1957) is used to obtain wave profiles, all with the same mean level. This yields the unexpected result that the height of the wave crest above the mean level is not a monotonic function of wave steepness. With subsequent papers this work will form one limiting case of the general problem of deep-water gravity—capillary waves.