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Transient development of capillary-gravity waves in a running stream

Published online by Cambridge University Press:  17 April 2009

Kalyan Kumar Bagchi  and
Lokenath Debnath
Kalyan Kumar Bagchi
Affiliation:
Centre of Advanced Study in Applied Mathematics, University of Calcutta, Calcutta, India
Lokenath Debnath
Affiliation:
Department of Mathematics, East Carolina University, Greenville, North Carolina, USA.
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Abstract

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An initial value investigation is made of the propagation of capillary-gravity waves generated by an oscillating pressure distribution acting at the free surface of a running stream of finite, infinite, and shallow depth. The solution for the free surface elevation is obtained explicitly by using the generalized Fourier transform and its asymptotic expansion. It is found that the solution consists of both the steady state and the transient components. The latter decays asymptotically as t → ∞ and the ultimate steady state is attained. It is shown that the steady state consists of two or four progressive capillary-gravity waves travelling both upstream and downstream according as the basic stream velocity is less or greater than the critical speed. Special attention is given to the existence of the critical values associated with the running stream of finite, infinite, and shallow depth. A comparison is made between the unsteady wave motions in an inviscid fluid with or without surface tension.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 3 , December 1973 , pp. 417 - 432
Copyright
Copyright © Australian Mathematical Society 1973

References

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