Hostname: page-component-6b989bf9dc-476zt Total loading time: 0 Render date: 2024-04-14T22:40:57.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Parasitic capillary waves: a direct calculation

Published online by Cambridge University Press:  26 April 2006

Michael S. Longuet-Higgins
Michael S. Longuet-Higgins
Affiliation:
Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA 92093-0402, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

As in a previous theory (Longuet-Higgins 1963) parasitic capillary waves are considered as a perturbation due to the local action of surface tension forces on an otherwise pure progressive gravity wave. Here the 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 influence of gravity on the capillary waves themselves, through the effective gravitational acceleration g* for short waves riding on longer waves.

Nonlinearity in the capillary waves themselves is not included, and certain other approximations are made. Nevertheless, the theory is shown to be in essential agreement with experiments by Cox (1958), Ebuchi, 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 minimum (local) speed of capillary-gravity waves. On subcritical gravity waves, with steepness AK < (AK)c, capillary waves may be generated at all points of the wave surface. On supercritical waves, with AK > (AK)c, capillary waves can only be generated in the wave troughs; they are trapped between two caustics near the crests. Generally, the amplitude of the parasitic capillaries is greatest on gravity waves of near critical (but not maximum) steepness.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 301 , 25 October 1995 , pp. 79 - 107
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chang, J. H., Wagner, R. N. & Yuen, H. C. 1978 Measurement of high frequency capillary waves on steep gravity waves. J. Fluid Mech. 86, 401–413.Google Scholar
Chen, B. & Saffman, P. G. 1979 Steady gravity-capillary waves on deep water – I. Weakly nonlinear waves. Stud. Appl. Maths 60, 183–210.Google Scholar
Chen, B. & Saffman, P. G. 1980 Steady gravity-capillary waves on deep water – II. Numerical results for finite amplitude. Stud. Appl. Maths 62, 95–111.Google Scholar
Cox, C. S. 1958 Measurements of slopes of high-frequency wind waves. J. Mar. Res. 16, 199–225.Google Scholar
Crapper, G. D. 1970 Non-linear capillary waves generated by steep gravity waves. J. Fluid Mech. 40, 149–159.Google Scholar
Davies, T. V. 1951 The theory of symmetrical gravity waves of finite amplitude. I. Proc. R. Soc. Lond. 208, 475–486.Google Scholar
Duncan, J. H., Philomin, V., Qiao, H. & Kimmel, J. 1994 The formation of a spilling breaker. Phys. Fluids 6, S2.Google Scholar
Ebuchi, N., Kawamura, H. & Toba, Y. 1987 Fine structure of laboratory wind-wave surfaces studied using an optical method. Boundary-Layer Met. 39, 133–151.Google Scholar
Jäauhne, B., & Riemer, K. 1990 Two-dimensional wave number spectra of small scale water surface waves. J. Geophys. Res. 95, 11531–11546.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press, 738 pp.
Longuet-Higgins, M. S. 1963 The generation of capillary waves by steep gravity waves. J. Fluid Mech. 16, 238–159.Google Scholar
Longuet-Higgins, M. S. 1985a Bifurcation in gravity waves. J. Fluid Mech. 151, 457–475.Google Scholar
Longuet-Higgins, M. S. 1985b Accelerations in steep gravity waves. J. Phys. Oceanogr. 15, 1570–1579.Google Scholar
Longuet-Higgins, M. S. 1987 The propagation of short surface waves on longer gravity waves. J. Fluid Mech. 177, 293–306.Google Scholar
Longuet-Higgins, M. S. 1992 Theory of weakly damped Stokes waves: a new formulation and its physical interpretation. J. Fluid Mech. 235, 319–324.Google Scholar
Longuet-Higgins, M. S. & Cleaver, R. P. 1994 Crest instabilities of gravity waves. Part 1. The almost-highest wave. J. Fluid Mech. 258, 115–129.Google Scholar
Longuet-Higgins, M. S., Cleaver, R. P. & Fox, M. J. H. 1994 Crest instabilities of gravity waves. Part 2. Matching and asymptotic expansion. J. Fluid Mech. 259, 333–344.Google Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1977 Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80, 721–741.Google Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769–786.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stresses in water waves; a physical discussion, with applications. Deep-Sea Res. 11, 529–562.Google Scholar
Perlin, M., Lin, H. & Ting, C.-L. 1993 On parasitic capillary waves generated by steep gravity waves: an experimental investigation with spatial and temporal measurements. J. Fluid Mech. 255, 597–620.Google Scholar
Phillips, O. M. 1981 The dispersion of short wavelets in the presence of a dominant long wave. J. Fluid Mech. 107, 465–485.Google Scholar
Ruvinsky, K. D., Feldstein, F. I. & Freidman, G. I. 1991 Numerical simulations of the quasistastionery stage of ripple excitation by stee p gravity-capillary waves. J. Fluid Mech. 230, 339–353.Google Scholar
Ruvinsky, K. D. & Freidman, G. I. 1981 On the generation of capillary-gravity waves by steep gravity waves. Izv. Atmos. Ocean. Phys. 17, 548–553.Google Scholar
Schwartz, L. W. & Vanden-Broeck, J.-M. 1979 Numerical solution of the exact equations for capillary-gravity waves. J. Fluid Mech. 95, 111–139.Google Scholar
Shyu, J.-H. & Phillips, O. M. 1990 The blockage of gravity and capillary waves by longer waves and currents. J. Fluid Mech. 217, 115–141.Google Scholar
Watson, K. M. & Mcbride, J. B. 1993 Excitation of capillary waves by longer waves. J. Fluid Mech. 250, 103–119.Google Scholar
Yermakov, S. A., Ruvinsky, K. D. & Salashin, S. G. 1988 Local correlation of the characteristics of ripples on the crest of capillary gravity waves with their curvature. Izv. Atmos. Ocean. Phys. 24, 561–563.Google Scholar