Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-21T08:26:09.478Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact solutions of a three-dimensional nonlinear Schrödinger equation applied to gravity waves

Published online by Cambridge University Press:  19 April 2006

W. H. Hui  and
J. Hamilton
W. H. Hui
Affiliation:
Department of Applied Mathematics, University of Waterloo, Ontario, Canada
J. Hamilton
Affiliation:
Department of Applied Mathematics, University of Waterloo, Ontario, Canada Present address: Vickers Offshore, Barrow-in-Furness, Cumbria, England.
Get access Rights & Permissions [Opens in a new window]

Abstract

The three-dimensional evolution of packets of gravity waves is studied using a nonlinear Schrödinger equation (the Davey–Stewartson equation). It is shown that permanent wave groups of the elliptic en and dn functions and their common limiting solitary sech forms exist and propagate along directions making an angle less than ψc = tan−1(1/√2) = 35° with the underlying wave field, whilst, along directions making an angle greater than ψc, there exist permanent wave groups of elliptic sn and negative solitary tanh form. Furthermore, exact general solutions are given showing wave groups travelling along the two characteristic directions at ψc or − ψc. These latter solutions tend to form regions of large wave slope and are used to discuss the waves produced by a ship, in particular the nonlinear evolution of the leading edge of the pattern.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 93 , Issue 1 , 12 July 1979 , pp. 117 - 133
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

Abramowitz, M. & Stegun, I. A. (ed.) 1964 Handbook of Mathematical Functions, p. 446. New York: Dover.
Benjamin, T. B. & Feir, J. E. 1967 J. Fluid Mech. 27, 417.
Chu, V. H. & Mei, C. C. 1970 J. Fluid Mech. 41, 873.
Chu, V. H. & Mei, C. C. 1971 J. Fluid Mech. 47, 337.
Davey, A. & Stewartson, K. 1974 Proc. Roy Soc. A 338, 101.
Hasimoto, H. & Ono, H. 1972 J. Phys. Soc. Japan 33, 805.
Hayes, W. D. 1973 Proc. Roy Soc. A 332, 199.
Hogben, N. 1972 J. Fluid Mech. 55, 513.
Hogner, E. 1923 Ark. Mat. Astr. Fys. 17.
Howe, M. W. 1967 J. Fluid Mech. 32, 779.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lighthill, M. J. 1957 River Waves. Naval Hydrodynamics, publication 515, National Academy of Sciences Research Council.
Lighthill, M. J. 1967 Proc. Roy. Soc. A 299, 25.
Longuet-Higgins, M. S. 1976 Proc. Roy. Soc. A 347, 311.
Newman, J. N. 1971 J. Ship. Res. 15, 1.
Peters, A. S. 1949 Com. Pure Appl. Math. 2, 123.
Stokes, G. G. 1847 Camb. Trans. 8, 441. (Papers 1, 197–229.)
Ursell, F. 1960 J. Fluid Mech. 8, 418.
Warren, F. W. G. 1962 Fluid Mech. 10, 584.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Yuen, H. C. & Lake, B. M. 1975 Phys. Fluids 18, 956.
Zakharov, V. E. & Shabat, A. B. 1972 Soviet Phys. J. Exp. Theor. Phys. 34, 62.