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Stability of linear shear flows in shallow water

Published online by Cambridge University Press:  26 April 2006

Charles Knessl  and
Joseph B. Keller
Charles Knessl
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of lllinois at Chicago, Chicago, IL 60680, USA
Joseph B. Keller
Affiliation:
Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, CA 94305-2125, USA
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Abstract

The stability or instability of various linear shear flows in shallow water is considered. The linearized equations for waves on the surface of each flow are solved exactly in terms of known special functions. For unbounded shear flows, the exact reflection and transmission coefficients R and T for waves incident on the flow, are found. They are shown to satisfy the relation |R|2= 1+ |T|2, which proves that over reflection occurs at all wavenumbers. For flow bounded by a rigid wall, R is found. The poles of R yield the eigenvalue equation from which the unstable mides can be found. For flow in a channel, with two rigid walls, the eigenvalue equation for the modes is obtained. The results are compared with previous numerical results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 303 , 25 November 1995 , pp. 203 - 214
Copyright
© 1995 Cambridge University Press

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