Abstract
A third-order nonlinear envelope equation is derived for surface waves in finite-depth water by assuming small wave steepness, narrow-band spectrum, and small depth as compared to the modulation length. A generalized Dysthe equation is derived for waves in relatively deep water. In the shallow-water limit, one of the nonlinear dispersive terms vanishes. This limit case is compared with the envelope equation for waves described by the Korteweg-de Vries equation. The critical regime of vanishing nonlinearity in the classical nonlinear Schrödinger equation for water waves (when kh ≈ 1.363) is analyzed. It is shown that the modulational instability threshold shifts toward the shallow-water (long-wavelength) limit with increasing wave intensity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Talanov, Pis’ma Zh. Éksp. Teor. Fiz. 2, 218 (1965) [JETP Lett. 2, 138 (1965)].
P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
M. I. Rabinovich and D. I. Trubetskov, Introduction to the Oscillation and Wave Theory (Nauka, Moscow, 1984) [in Russian].
V. E. Zakharov, Prikl. Mekh. Tekh. Fiz. 2, 86 (1968).
H. Hashimoto and J. Ono, J. Phys. Soc. Jpn. 33, 805 (1972).
A. Davey, J. Fluid Mech. 53, 769 (1972).
T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967).
D. J. Benney and G. J. Roskes, Stud. Appl. Math. 48, 377 (1969).
A. Davey and K. Stewartson, Proc. R. Soc. London, Ser. A 338, 101 (1974).
E. M. Gromov and V. I. Talanov, Zh. Éksp. Teor. Fiz. 110, 137 (1996) [JETP 83, 73 (1996)].
K. B. Dysthe, Proc. R. Soc. London, Ser. A 369, 105 (1979).
Yu. V. Sedletskii, Zh. Éksp. Teor. Fiz. 124, 200 (2003) [JETP 97, 180 (2003)].
K. Trulsen, I. Kliakhandler, K. B. Dysthe, et al., Phys. Fluids 12, 2432 (2000).
R. S. Johnson, Proc. R. Soc. London, Ser. A 357, 131 (1977).
T. Kakutani and K. Michihiro, J. Phys. Soc. Jpn. 52, 4129 (1983).
E. J. Parkes, J. Phys. A: Math. Gen. 20, 2025 (1987).
A. V. Slyunyaev and E. N. Pelinovskii, Zh. Éksp. Teor. Fiz. 116, 318 (1999) [JETP 89, 173 (1999)].
A. V. Slyunyaev, Zh. Éksp. Teor. Fiz. 119, 606 (2001) [JETP 92, 529 (2001)].
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge Univ. Press, Cambridge, 1997).
R. Grimshaw, D. Pelinovsky, E. Pelinovsky, et al., Physica D (Amsterdam) 159, 35 (2001).
Author information
Authors and Affiliations
Additional information
__________
Translated from Zhurnal Éksperimental’no\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) i Teoretichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) Fiziki, Vol. 128, No. 5, 2005, pp. 1061–1077.
Original Russian Text Copyright © 2005 by Slunyaev.
Rights and permissions
About this article
Cite this article
Slunyaev, A.V. A high-order nonlinear envelope equation for gravity waves in finite-depth water. J. Exp. Theor. Phys. 101, 926–941 (2005). https://doi.org/10.1134/1.2149072
Received:
Issue Date:
DOI: https://doi.org/10.1134/1.2149072