Near-Limiting Gravity Waves in Water of Finite Depth

Progressive, irrotational gravity waves of constant form, with all crests in a wave train identical, exist as a two-parameter family. The first parameter, the ratio of mean depth to wavelength, varies from zero (the solitary wave) to infinity (the deep-water wave). The second parameter, the wave height or amplitude, varies from zero (the infinitesimal wave) to a limiting value dependent on the first parameter. Solutions of limiting waves, with angled crests, have been presented in a previous paper; this paper considers near-limiting waves having rounded crests with a very small radius of curvature, in some cases as little as 0.0001 of the water depth.The computing method is a modification of the integral equation technique used for limiting waves. Two leading terms are again used to give a close approximation to the flow near the crest and hence minimize the number of subsequent terms needed; the form of these leading terms is suggested by earlier work of G. G. Stokes (Mathematical and physical papers, vol. 1, pp. 225-228. Cambridge University Press (1880)), M. A. Grant (J. Fluid Mech. 59, 257-262 (1973)) and L. W. Schwartz (J. Fluid Mech. 62, 553-578 (1974)). To achieve satisfactory accuracy, however, it is now necessary to add a set of dipoles above the crest in the complex potential plane, as previously used by M. S. Longuet-Higgins & M. J. H. Fox (J. Fluid Mech. 80, 721-741 (1977)). The results include the first fully detailed calculations of non-breaking waves having local surface slopes exceeding 30 degrees. The local profile at the crest, despite its very small scale, is shown to tend with increasing wave height to the asymptotic self-similar form previously computed by Longuet-Higgins & Fox. Their predictions of an ultimate maximum slope of 30.37 degrees and a vertical crest acceleration of 0.388g are supported. The results agree well with earlier calculations for steep waves at the two extremes of solitary and deep-water waves. In particular, it is confirmed that in the approach to limiting height the phase velocity and certain integral quantities possess not only the well-known maximum but also a subsequent minimum, the first in the infinite series of extrema predicted theoretically by M. S. Longuet-Higgins & M. J. H. Fox (J. Fluid Mech. 85, 769-786 (1978)). Briefly considered also are the level of action of near-limiting deep-water waves, the decay of surface drift velocity from the limiting value and the method established for computing waves of all lesser heights.