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Highest waves, phase speeds and particle trajectories of nonlinear capillary waves on sheets of fluid

Published online by Cambridge University Press:  21 April 2006

S. J. Hogan
S. J. Hogan
Affiliation:
Mathematical Institute, University of Oxford, 24/29 St Giles, Oxford OX1 3LB
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Abstract

We derive exact criteria for the highest capillary waves on sheets of fluid and present the profiles for both symmetric and antisymmetric waves. In addition we provide correct definitions of the phase speed of these waves. Previous ‘essentially nonlinear’ solutions to the problem are shown to have been misinterpreted.

Trajectory properties of particles in both types of wave are presented. We also give results for the drift velocity ratios as a function of the mean displacement of a streamline from the free surface. Many of the results are exact, being given in terms of elliptic functions and elliptic integrals. An application to wave motion in thin films on pipe walls is considered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 172 , November 1986 , pp. 547 - 563
Copyright
© 1991 Cambridge University Press

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References

Byrd, P. F. & Friedman M. D.1971 Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn. revised. Springer.
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.1984 Particle trajectories in nonlinear capillary waves. J. Fluid Mech. 143, 243–252.Google Scholar
Hogan S. J.1985a Particle trajectories in nonlinear gravity-capillary waves. J. Fluid Mech. 151, 105–119.Google Scholar
Hogan S. J. 1985b Particle trajectories in nonlinear capillary waves. In The Ocean Surface: Wave Breaking, Turbulent Mixing and Radio Probing (ed. Y. Toba & H. Mitsuyasu), pp. 25–30. Dordrecht: D. Reidel.
Kinnersley W.1976 Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77, 229–241.Google Scholar
Longuet-Higgins M. S.1979 The trajectories of particles in steep, symmetric gravity waves. J. Fluid Mech. 94, 497–517.Google Scholar
Matsuuchi K.1974 Modulational instability of nonlinear capillary waves on thin liquid sheet. J. Phys. Soc. Japan 37, 1680–1687.Google Scholar
Stokes G. G.1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.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