Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T03:44:06.536Z Has data issue: false hasContentIssue false
  • English
  • Français

The propagation of a weak nonlinear wave

Published online by Cambridge University Press:  11 April 2006

A. Davey
A. Davey
Affiliation:
School of Mathematics, University of Newcastle upon Tyne
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the propagation of a weak nonlinear wave whose energy is concentrated in a narrow band of wavenumbers in a fluid which is both dispersive and dissipative. We use the small amplitude equations of Whitham's theory of slowly varying wave trains, modified slightly to include dissipation, to show that the modulation of the wave may be described by a nonlinear Schrödinger equation. For long waves which are purely dispersive we obtain the Kortewegde Vries equation, and for long waves which are dissipative we obtain Burgers’ equation by suitable transformations of the nonlinear Schrödinger equation. We mention the problem of Stokes waves in deep water and comment briefly upon invariant far-field theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 53 , Issue 4 , 27 June 1972 , pp. 769 - 781
Copyright
© 1972 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

Asano, N. 1972 Submitted to J. Phys. Soc. Japan.
Asano, N., Taniuti, T. & Yajima, N. 1969 J. Math. Phys. 10, 2020.
Baum, H. R. 1971 J. Fluid Mech. 50, 271.
Benney, D. J. & Newell, A. C. 1967 J. Math. & Phys. 46, 133.
Chu, V. H. & MEI, C. C. 1970 J. Fluid Mech. 41, 873.
Chu, V. H. & MEI, C. C. 1971 J. Fluid Mech. 47, 337.
Diprima, R. C., Eckhaus, W. & Segel, L. A. 1971 J. Fluid Mech. 49, 705.
Feir, J. E. 1967 Proc. Roy. Soc. A 299, 54.
Hocking, L. M. & Stewartson, K. 1972a Proc. Roy. Soc. A 326, 289.
Hocking, L. M. & Stewartson, K. 1972b Mathematika, to be published.
Hocking, L. M., Stewartson, K. & Stuart, J. T. 1972 J. Fluid Mech. 51, 705.
Lighthill, M. J. 1965 J. Inst. Math. Appl. 1, 269.
Matkowsky, B. J. 1970 S.I.A.M.J. Appl. Math. 18, 872.
Midzuno, Y. & Watanabe, T. 1970 J. Phys. Soc. Japan, 28, 738.
Nayfeh, A. H. & Bassan, S. D. 1971 J. Fluid Mech. 48, 463.
Reynolds, W. C. & Potter, M. C. 1967 J. Fluid Mech. 27, 465.
Schrödinuer, E. 1926 Die Naturwissenschaften, 28, 664.
Squire, H. B. 1933 Proc. Roy. Soc. A 142, 621.
Stewartson, K. & Stuart, J. T. 1971 J. Fluid Mech. 48, 529.
Taniuti, T. & Washimi, H. 1968 Phys. Rev. Lett. 21, 209.
Tappert, F. D. & Varma, C. M. 1970 Phys. Rev. Lett. 25, 1108.
Tsuzuki, T. 1971 J. Low Temp. Phys. 4, 441.
Watanabe, T. 1969 J. Phys. Soc. Japan, 27, 1341.
Whitham, G. B. 1965 J. Fluid Mech. 22, 273.
Whitham, G. B. 1967 Proc. Roy. Soc. A 299, 6.