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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
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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

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