Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-18T22:47:17.153Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical calculations of steady gravity-capillary waves using an integro-differential formulation

Published online by Cambridge University Press:  19 April 2006

James W. Rottman  and
D. B. Olfe
James W. Rottman
Affiliation:
Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, California 92093
D. B. Olfe
Affiliation:
Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, California 92093
Get access Rights & Permissions [Opens in a new window]

Abstract

A new integro-differential equation is derived for steady free-surface waves. Numerical solutions of this equation for periodic gravity-capillary waves on a fluid of infinite depth are presented. For the two limiting cases of gravity waves and capillary waves, our results are in excellent agreement with previous calculations. For gravity-capillary waves, detailed calculations are performed near the wave-number at which the classical second-order perturbation solution breaks down. Our calculations yield two solutions in this region, which in the limit of small amplitudes agree with the results obtained by Wilton in 1915; one solution has the small amplitude behaviour of a gravity wave and the other that of a capillary wave, but the numerical results show that at large amplitudes both waves have the characteristics of capillary waves. The calculations also show that the wavenumber range in which two solutions exist increases with increasing wave height.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 94 , Issue 4 , 29 October 1979 , pp. 777 - 793
Copyright
© 1979 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

Block, M. I. G. 1978 Large amplitude surface waves. J. Fluid Mech. 84, 167–179.Google Scholar
Byatt-Smith, J. G. B. 1970 An exact integral equation for steady surface waves. Proc. Roy. Soc. A 315, 405–418.Google Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. Roy. Soc. A 286, 183–230.Google Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532–540.Google Scholar
Fox, L. 1967 Romberg integration for a class of singular integrands. Comp. J. 10, 87–93.Google Scholar
Harrison, W. J. 1909 The influence of viscosity and capillarity on waves of finite amplitude. Proc. Lond. Math. Soc. 7, 107–121.Google Scholar
Kelvin, Lord 1910 Ripples and waves. In Mathematical and Physical Papers, vol. 4, pp. 86–92. Cambridge University Press.
Levi-Civita, T. 1925 Détermination rigoureuse des ondes permanentes d'ampleur finie. Math. Ann. 93, 264–314.Google Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. Roy. Soc. A 342, 157–174.Google Scholar
Muskhelishvili, N. I. 1953 Singular Integral Equations. Noordhoff.
Nekrasov, A. I. 1921 On waves of permanent type I. Izv. Ivanovo-Voznesensk, Politekhn. Inst. 3, 52–65.Google Scholar
Pierson, W. J. & Fife, P. 1961 Some nonlinear properties of long crested periodic waves with length near 2·44 centimeters. J. Geophys. Res. 66, 163–179.Google Scholar
Rottman, J. W. 1978 Numerical calculations of periodic gravity-capillary waves. Ph.D. dissertation, University of California, San Diego.
Sasaki, K. & Murakami, T. 1973 Irrotational progressive, surface gravity waves near the limiting height. J. Ocean. Soc. Japan 29, 94–105.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Stokes, G. G. 1880a Supplement to a paper on the theory of oscillatory waves. In Mathematical and Physical Papers, vol. 1, pp. 314–326. Cambridge University Press.
Stokes, G. G. 1880b Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form. In Mathematical and Physical Papers, vol. 1, pp. 225–228. Cambridge University Press.
Thomas, J. W. 1968 Irrotational gravity waves of finite height: a numerical study. Mathematika 15, 139–148.Google Scholar
Thomas, J. W. 1975 A numerical study of the relationship between the dimensionless parameters in the problem of periodic waves of permanent type in a liquid of finite depth. Quart. Appl. Math. 32, 403–410.Google Scholar
Whittaker, E. T. & Watson, G. N. 1927 A Course of Modern Analysis. Cambridge University Press.
Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688–700.Google Scholar