Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-20T04:54:33.039Z Has data issue: false hasContentIssue false
  • English
  • Français

On two-dimensional packets of capillary-gravity waves

Published online by Cambridge University Press:  11 April 2006

V. D. Djordjevic  and
L. G. Redekopp
V. D. Djordjevic
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles Permanent address: Department of Mechanical Engineering, University of Belgrade, Yugoslavia.
L. G. Redekopp
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles
Get access Rights & Permissions [Opens in a new window]

Abstract

The motion of a two-dimensional packet of capillary–gravity waves on water of finite depth is studied. The evolution of a packet is described by two partial differential equations: the nonlinear Schrödinger equation with a forcing term and a linear equation, which is of either elliptic or hyperbolic type depending on whether the group velocity of the capillary–gravity wave is less than or greater than the velocity of long gravity waves. These equations are used to examine the stability of the Stokes capillary–gravity wave train. The analysis reveals the existence of a resonant interaction between a capillary–gravity wave and a long gravity wave. The interaction requires that the liquid depth be small in comparison with the wavelength of the (long) gravity waves and the evolution equations describing the dynamics of this interaction are derived.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 79 , Issue 4 , 23 March 1977 , pp. 703 - 714
Copyright
© 1977 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

Ablowitz, M. S., Kaup, D. J., Newell, A. C. & Segur, H. 1974 The inverse scattering transform - Fourier analysis for nonlinear problems. Studies in Appl. Math. 53, 249.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wavetrains on deep water. Part 1. J. Fluid Mech. 27, 417.Google Scholar
Benney, D. J. & Newell, A. C. 1967 The propagation of nonlinear wave envelopes. J. Math. Phys. 46, 133.Google Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532.Google Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. Roy. Soc. A 388, 191.Google Scholar
Grimshaw, R. H. J. 1975 The modulation and stability of an internal gravity wave. Res. Rep. School Math. Sci., Univ. Melbourne, no. 32–1975.Google Scholar
Harrison, W. J. 1909 The influence of viscosity and capillarity on waves of finite amplitude. Proc. Lond. Soc. Math. 7 (2), 107.Google Scholar
Hasimoto, H. & Ono, H. 1972 Nonlinear modulation of gravity waves. J. Phys. Soc. Japan, 33, 805.Google Scholar
Hayes, W. D. 1973 Group velocity and nonlinear dispersive wave propagation. Proc. Roy. Soc. A 332, 199.Google Scholar
McGoldrick, L. F. 1970a An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40, 251.Google Scholar
McGoldrick, L. F. 1970b On Wilton's ripples: a special case of resonant interactions. J. Fluid Mech. 42, 193.Google Scholar
McGoldrick, L. F. 1972 On the rippling of small waves: a harmonic nonlinear nearly resonant interaction. J. Fluid Mech. 52, 725.Google Scholar
Newell, A. C. 1977 Long waves – short waves, a solvable model. To appear in S.I.A.M.J.
Whitham, G. B. 1967 Nonlinear dispersion of water waves. J. Fluid Mech. 27, 399.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Yuen, H. C. & Lake, B. M. 1975 Nonlinear deep water waves: theory and experiment. Phys. Fluids, 18, 956.Google Scholar
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. J. Exp. Theor. Phys. 34, 62.Google Scholar