Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T06:58:49.482Z Has data issue: false hasContentIssue false
  • English
  • Français

Limiting forms for capillary-gravity waves

Published online by Cambridge University Press:  21 April 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK and Institute of Oceanographie Sciences. Wormley, Surrey, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The form of steep capillary waves is of interest as a possible initial condition for the formation of air bubbles at a free surface. In this paper the limiting forms of pure capillary waves and of quasi-capillary waves are studied analytically. Crapper's finite-amplitude solution is expressed in a simple form, and is shown to be one of several exact elementary solutions to the pure-capillary free-surface condition. Among others are the solution z = w+sinh w, where w is the velocity potential, and also z = w3. The latter solution, though it represents a self-intersecting flow, can be used as the first in a sequence of approximations to the form of the steepest wave. Hence it is shown that the influence of gravity on the shape of the limiting ‘bubble’ is very small. The result is confirmed by an examination of Hogan's numerical calculations of limiting capillary-gravity waves.

In the crest of a limiting wave the particle velocity is almost constant and equal to the phase speed. This property makes it possible to apply a quasi-static approximation so as to determine the form of the crest, and hence to find an expression for the complete profile of a capillary-gravity wave of limiting steepness. It appears that there exists a solitary wave of capillary-gravity type on deep water.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 194 , September 1988 , pp. 351 - 375
Copyright
© 1988 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

Abramowitz, M. & Stegun, I. A. (eds) 1965 Handbook of Mathematical Functions. Dover. 1046 pp.
Byrd, P. F. & Friedman, M. D. 1971 Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn. Springer. 358 pp.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon. 654 pp.
Chen, B. & Saffman, P. G. 1979 Steady gravity-capillary waves on deep water - I. Weakly nonlinear waves. Stud. Appl. Math. 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. Math. 62, 95–111.Google Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532–540.Google Scholar
Hogan, S. J. 1980 Some effects of surface tension on steep water waves. Part 2. J. Fluid Mech. 96, 417–445.Google Scholar
Hogan, S. J. 1981 Some effects of surface tension on steep water waves. Part 3. J. Fluid Mech. 110, 381–410.Google Scholar
Kinnersley, W. 1976 Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77, 229–241.Google Scholar
Schooley, A. H. 1958 Profiles of wind-created water waves in the capillary-gravity transition region. J. Mar. Res. 16, 100–108.Google Scholar
Schwartz, L. W. & Vanden-Broeck, J.-M. 1979 Numerical solution of the exact equations for capillary-gravity waves. J. Fluid Mech. 95, 119–139.Google Scholar
Taylor, G. I. 1959 The dynamics of thin sheets of fluid. II. Waves on fluid sheets. Proc. R. Soc. Lond. A 253, 296–312.Google Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1980 A new family of capillary waves. J. Fluid Mech. 98, 161–169.Google Scholar
Wilton, J. R. 1915 On ripples. Phil. Mag. (6) 29, 688–700.Google Scholar