Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-17T12:04:01.430Z Has data issue: false hasContentIssue false
  • English
  • Français

On the subharmonic instabilities of steady three-dimensional deep water waves

Published online by Cambridge University Press:  26 April 2006

Mansour Ioualalen  and
Christian Kharif
Mansour Ioualalen
Affiliation:
Institut de Mécanique Statistique de la Turbulence, 12 avenue de Général Leclerc, 13003 Marseilles, France Present address: ORSTOM BP A5, Nouméa Cedex, New Caledonia, France.
Christian Kharif
Affiliation:
Institut de Mécanique Statistique de la Turbulence, 12 avenue de Général Leclerc, 13003 Marseilles, France
Get access Rights & Permissions [Opens in a new window]

Abstract

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 262 , 3 October 1994 , pp. 265 - 291
Copyright
© 1994 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

Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417–430.Google Scholar
Bryant, P. J. 1985 Doubly periodic progressive permanent waves in deep water. J. Fluid Mech. 161, 27–42.Google Scholar
Chappelear, J. E. 1961 On the description of the short-crested waves. Beach Erosion Board. US Army Corps Engrs Tech. Memo no. 125.
Fuchs, R. A. 1952 On the theory of short-crested oscillatory waves. Gravity waves. US Natl Bur. Stand. Circular 521, 187–200.Google Scholar
Gilewicz, J. 1978 Approximants de Padé. Lecture Notes in Mathematics, vol. 667. Springer.
Hsu, J. R. C., Tsuchiya, Y. & Silvester, R. 1979 Third-order approximation to short-crested waves. J. Fluid Mech. 90, 179–196.Google Scholar
Ioualalen, M. 1990 Etude de la stabilité linéaire d’ondes de gravité progressives et tri-dimensionnelles en profondeur infinie. Thèse no. 207. 90. 76, Univesité Aix-Marseille II.
Ioualalen, M. 1993 Fourth order approximation of short-crested waves. C. R. Acad. Sci. Paris 316 (II), 1193–1200.Google Scholar
Ioualalen, M. & Kharif, C. 1993 Stability of three-dimensional progressive gravity waves on deep water to superharmonic disturbances. Eur. J. Mech. B/ Fluids 12, 401–414.Google Scholar
Kharif, C. 1987 Etude comparative des instabilités tridimensionnelles et bidimensionnelles des vagues de gravité de forte cambrure. J. Mech. Theor. Appl. 3, 535–550.Google Scholar
Kharif, C. & Ramamonjiarisoa, A. 1988 Deep water gravity wave instabilities at large steepness. Phys. Fluids 31, 1286–1288.Google Scholar
Kharif, C. & Ramamonjiarisoa, A. 1990 On the stability of gravity waves on deep water. J. Fluid Mech. 218, 163–170.Google Scholar
Longuet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. Lond. A 360, 471–488.Google Scholar
Longuet-Higgins, M. S. 1978b The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115–125.Google Scholar
McLean, J. W. 1982 Instabilities of finite amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
McLean, J. W., Ma, Y. C., Martin, D. V., Saffman, P. G. & Yuen, H. C. 1981 Three-dimensional instability of finite amplitude water waves. Phys. Rev. Lett. 46, 817–820.Google Scholar
Mercer, G. N. & Roberts, A. J. 1992 Standing waves in deep water. Their stability and extreme form. Phys. Fluids A 4, 259–269.Google Scholar
Okamura, M. 1984 Instabilities of weakly nonlinear standing gravity waves. J. Phys. Soc. Japan 53, 3788–3796.Google Scholar
Okamura, M. 1986 Maximum wave steepness and instabilities of finite-amplitude standing gravity waves. Fluid Dyn. Res. 1, 201–214.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. J. Fluid Mech. 9, 193–217.Google Scholar
Poitevin, J. & Kharif, C. 1991 Subharmonic transition of a nonlinear wave train on deep water. In Mathematical and Numerical Aspects of Wave Propagation Phenomena. Philadelphia: SIAM.
Roberts, A. J. 1983 Highly nonlinear short-crested water waves. J. Fluid Mech. 135, 301–321.Google Scholar
Roberts, A. J. & Peregrine, D. H. 1983 Notes on long-crested water waves. J. Fluid Mech. 135, 323–335.Google Scholar
Roberts, A. J. & Schwartz, L. W. 1983 The calculation of nonlinear short-crested gravity waves. Phys. Fluids 26, 2388–2392.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Z. Angew. Math. Phys. 9, 86–94.Google Scholar
Zhang, J. & Melville, W. K. 1987 Three-dimensional instabilities of nonlinear gravity-capillary waves. J. Fluid Mech. 174, 187–208.Google Scholar