Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T11:12:26.627Z Has data issue: false hasContentIssue false
  • English
  • Français

The method of multiple scales and non-linear dispersive waves

Published online by Cambridge University Press:  29 March 2006

Ali Hasan Nayfeh  and
Sayed D. Hassan
Ali Hasan Nayfeh
Affiliation:
Aerotherm Corporation, Mountain View, California Present address: Engineering Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia.
Sayed D. Hassan
Affiliation:
Mathematics Department, Stanford, California
Get access Rights & Permissions [Opens in a new window]

Abstract

The method of multiple scales is used to analyze three non-linear physical systems which support dispersive waves. These systems are (i) waves on the interface between a liquid layer and a subsonic gas flowing parallel to the undisturbed interface, (ii) waves on the surface of a circular jet of liquid, and (iii) waves in a hot electron plasma. It is found that the partial differential equations that govern the temporal and spatial variations of the wave-numbers, amplitudes, and phases have the same form for all of these systems. The results show that the non-linear motion affects only the phase. For the constant wave-number case, the general solution for the amplitude and the phase can be obtained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 48 , Issue 3 , 16 August 1971 , pp. 463 - 475
Copyright
© 1971 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

Barakat, R. & Houston, A. 1968 Non-linear periodic capillary-gravity waves on a fluid of finite depth J. Geophys. Res. 73, 6445.Google Scholar
Bogoliubov, N. & Mitropolski, Y. A. 1961 Asymptotic Methods in the Theory of Nonlinear Oscillations. New York: Gordon and Breach.
Brouwer, D. & Clemence, G. M. 1961 Methods of Celestial Mechanics New York: Academic.
Emery, V. J. 1970 Eikonal approximation for non-linear equations. J. Math. Phys. 11, 1893.Google Scholar
Grimshaw, R. 1970 The solitary wave in water of variable depth J. Fluid Mech. 42, 639.Google Scholar
Kamesvara Rav, J. C. 1920 On ripples of finite amplitude Proc. Indian Ass. Cultiv. Sci. 6, 175.Google Scholar
Liepmann, H. W. & Roshko, D. 1960 Elements of Gas Dynamics. New York: John Wiley.
Luke, J. C. 1966 A perturbation method for nonlinear dispersive wave problems. Proc. Roy. Soc. A 292, 403.Google Scholar
Montgomery, D. & Tidman, D. A. 1964 Secular and nonsecular behaviour for the cold plasma equations Phys. Fluids, 7, 242.Google Scholar
Mcgoldrick, L. F. 1970 On Wilton's ripples J. Fluid Mech. 42, 193.Google Scholar
Nayfeh, A. H. 1965 Non-linear oscillations in a hot electron plasma. Phys. Fluids, 8, 1896.Google Scholar
Nayfeh, A. H. 1970a Finite amplitude surface waves in a liquid layer. J. Fluid Mech. 40, 671.Google Scholar
Nayfeh, A. H. 1970b Non-linear stability of a liquid jet Phys. Fluids, 13, 841.Google Scholar
Nayfeh, A. H. 1971b Third-harmonic resonance in the interaction of capillary and gravity waves. J. Fluid Mech. 48, 385.Google Scholar
Nayfeh, A. H. 1971b Second-harmonic resonance in the interaction capillary and gravity waves. (To be published).
Nayfeh, A. H. & Saric, W. S. 1971 Non-linear Kelvin-Helmholtz instability J. Fluid Mech. 46, 209.Google Scholar
Pierson, W. J. & Fife, P. 1961 Some nonlinear properties of long crested waves with lengths near 2·44 centimetres J. Geophys. Res. 66, 163.Google Scholar
Rayleigh, Lord 1945 The Theory of Sound, Vol. II. New York: Dover.
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. Roy. Soc. A 309, 551.Google Scholar
Sturrock, P. A. 1957 Non-linear effects in electron plasma. Proc. Roy. Soc. A 242, 277.Google Scholar
Tidman, D. A. & Stainr, H. M. 1965 Frequency and wave-number shifts for non-linear waves in a hot plasma Phys. Fluids, 8, 345.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. New York: Academic Press.
Wang, D. P. 1968 Finite amplitude effects on the stability of a jet of circular cross-section J. Fluid Mech. 34, 299.Google Scholar
Whitham, G. B. 1965a Non-linear dispersive waves. Proc. Roy. Soc. A 283, 238.Google Scholar
Whitham, G. B. 1965b A general approach to linear and non-linear waves using a Lagrangian. J. Fluid Mech. 22, 273.Google Scholar
Whitham, G. B. 1967a Non-linear dispersion of water waves. J. Fluid Mech. 27, 399.Google Scholar
Whitham, G. B. 1967b Variational methods and applications to water waves. Proc. Roy. Soc. A 299, 6.Google Scholar
Wilton, J. R. 1915 On ripples Phil. Mag. 29, 688.Google Scholar