Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-28T20:22:43.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of finite-amplitude interfacial waves. Part 1. Modulational instability for small-amplitude waves

Published online by Cambridge University Press:  20 April 2006

R. H. J. Grimshaw  and
D. I. Pullin
R. H. J. Grimshaw
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Vie 3052, Australia
D. I. Pullin
Affiliation:
Department of Mechanical Engineering, University of Queensland, St Lucia, Qld 4067, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

In two previous papers (Pullin & Grimshaw 1983a, b) we studied the wave profile and other properties of finite-amplitude interfacial progressive waves in a two-layer fluid. In this and the following paper (Pullin & Grimshaw 1985) we discuss the stability of these waves to small perturbations. In this paper we obtain anatytical results for the long-wavelength modulational instability of small-amplitude waves. Using a multiscale expansion, we obtain a nonlinear Schrödinger equation coupled to a wave-induced mean-flow equation to describe slowly modulated waves. From these coupled equations we determine the stability of a plane progressive wave. Our results are expressed by determining the instability bands in the (p, q)-plane, where (p, q) is the modulation wavenumber, and are obtained for a range of values of basic density ratio and undisturbed layer depths.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 160 , November 1985 , pp. 297 - 315
Copyright
© 1985 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

Benney, D. J. & Newell, A. C. 1967 The propagation of nonlinear wave envelopes. J. Maths & Phys. 46, 133–139.Google Scholar
Benney, D. J. & Roskes, G. J. 1969 Wave instabilities. Stud. Appl. Maths 48, 377–385.Google Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338, 101–110.Google Scholar
Grimshaw, R. 1981 Modulation of an internal gravity wave packet in a stratified shear flow. Wave Motion 3, 81–103.Google Scholar
Grimshaw, R. 1984 Solitary waves in slowly varying environments: wave packets. In Advances in Nonlinear Waves (ed. L. Debnath), pp. 38–58.
Hayes, W. D. 1973 Group velocity and nonlinear dispersive wave propagation. Proc. R. Soc. Lond. A 332, 194–221.Google Scholar
Mclean, J. W. 1982a Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
Mclean, J. W. 1982b Instabilities of finite-amplitude waves on water of finite depth. J. Fluid Mech. 114, 331–341.Google Scholar
Mclean, J. W., Ma, Y. C., Martin, D. U., Saffman, P. G. & Yuen, H. C. 1981 Three-dimensional instability of finite-amplitude water waves. Phys. Rev. Lett. 46, 817–820.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1983a Nonlinear interfacial progressive waves near a boundary in a Boussinesq fluid. Phys. Fluids 26, 897–905.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1983b Interfacial progressive waves in a two-layer shear flow. Phys. Fluids 26, 1731–1739.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1985 Stability of finite-amplitude interfacial waves. Part 2. Numerical results. J. Fluid Mech. 160, 317–336.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Yuen, H. C. 1983 Instability of finite-amplitude interfacial waves. In Waves on Fluid Interfaces (ed. R. E. Meyer), pp. 17–40. Academic.
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Sov. Phys. J. Appl. Mech. Tech. Phys. 4, 190–194.Google Scholar