Skip to main content
×
Home
    • Aa
    • Aa

Parasitic capillary waves: a direct calculation

  • Michael S. Longuet-Higgins (a1)
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.

Copyright
References
Hide All
Chang, J. H., Wagner, R. N. & Yuen, H. C. 1978 Measurement of high frequency capillary waves on steep gravity waves. J. Fluid Mech. 86, 401413.
Chen, B. & Saffman, P. G. 1979 Steady gravity-capillary waves on deep water – I. Weakly nonlinear waves. Stud. Appl. Maths 60, 183210.
Chen, B. & Saffman, P. G. 1980 Steady gravity-capillary waves on deep water – II. Numerical results for finite amplitude. Stud. Appl. Maths 62, 95111.
Cox, C. S. 1958 Measurements of slopes of high-frequency wind waves. J. Mar. Res. 16, 199225.
Crapper, G. D. 1970 Non-linear capillary waves generated by steep gravity waves. J. Fluid Mech. 40, 149159.
Davies, T. V. 1951 The theory of symmetrical gravity waves of finite amplitude. I. Proc. R. Soc. Lond. 208, 475486.
Duncan, J. H., Philomin, V., Qiao, H. & Kimmel, J. 1994 The formation of a spilling breaker. Phys. Fluids 6, S2.
Ebuchi, N., Kawamura, H. & Toba, Y. 1987 Fine structure of laboratory wind-wave surfaces studied using an optical method. Boundary-Layer Met. 39, 133151.
Jäauhne, B., & Riemer, K. 1990 Two-dimensional wave number spectra of small scale water surface waves. J. Geophys. Res. 95, 1153111546.
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, 238159.
Longuet-Higgins, M. S. 1985a Bifurcation in gravity waves. J. Fluid Mech. 151, 457475.
Longuet-Higgins, M. S. 1985b Accelerations in steep gravity waves. J. Phys. Oceanogr. 15, 15701579.
Longuet-Higgins, M. S. 1987 The propagation of short surface waves on longer gravity waves. J. Fluid Mech. 177, 293306.
Longuet-Higgins, M. S. 1992 Theory of weakly damped Stokes waves: a new formulation and its physical interpretation. J. Fluid Mech. 235, 319324.
Longuet-Higgins, M. S. & Cleaver, R. P. 1994 Crest instabilities of gravity waves. Part 1. The almost-highest wave. J. Fluid Mech. 258, 115129.
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, 333344.
Longuet-Higgins, M. S. & Fox, M. J. H. 1977 Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80, 721741.
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, 769786.
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stresses in water waves; a physical discussion, with applications. Deep-Sea Res. 11, 529562.
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, 597620.
Phillips, O. M. 1981 The dispersion of short wavelets in the presence of a dominant long wave. J. Fluid Mech. 107, 465485.
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, 339353.
Ruvinsky, K. D. & Freidman, G. I. 1981 On the generation of capillary-gravity waves by steep gravity waves. Izv. Atmos. Ocean. Phys. 17, 548553.
Schwartz, L. W. & Vanden-Broeck, J.-M. 1979 Numerical solution of the exact equations for capillary-gravity waves. J. Fluid Mech. 95, 111139.
Shyu, J.-H. & Phillips, O. M. 1990 The blockage of gravity and capillary waves by longer waves and currents. J. Fluid Mech. 217, 115141.
Watson, K. M. & Mcbride, J. B. 1993 Excitation of capillary waves by longer waves. J. Fluid Mech. 250, 103119.
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, 561563.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 67 *
Loading metrics...

Abstract views

Total abstract views: 140 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th October 2017. This data will be updated every 24 hours.