Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T02:18:32.891Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Debnath, L., “Effect of surface tension on time dependent gravity waves”, Math. Japan. 16 (1971), 81–90.Google Scholar
[2]Debnath, L. and Rosenblat, S., “The ultimate approach to the steady state in the generation of waves on a running stream”, Quart. J. Mech. Appl. Math. 22 (1969), 221–233.CrossRefGoogle Scholar
[3]Evans, D.V., “The influence of surface tension on the reflection of water waves by a plane vertical barrier”, Proc. Cambridge Philos. Soc. 64 (1968), 795–810.CrossRefGoogle Scholar
[4]Evans, D.V., “The effect of surface tension on the waves produced by a heaving circular cylinder”, Proc. Cambridge Philos. Soc. 64 (1968), 833–847.CrossRefGoogle Scholar
[5]Jones, D.S., Generalised functions (McGraw-Hill, New York; Toronto, Ontario; London; 1966).Google Scholar
[6]Kaplan, Paul, “The waves generated by the forward motion of oscillatory pressure distributions”, Proc. Fifth Midwestern Conf. Fluid Mechanics, 1957, 316–329 (University of Michigan Press, Ann Arbor, Michigan, 1957).Google Scholar
[7]Lighthill, M.J., An introduction to Fourier analysis and generalized functions (Cambridge University Press, Cambridge, 1958).Google Scholar
[8]Packham, B.A., “Capillary-gravity waves against a vertical cliff”, Proc. Cambridge Philos. Soc. 64 (1968), 827–832.CrossRefGoogle Scholar
[9]Rhodes-Robinson, P.F., “Fundamental singularities in the theory of water waves with surface tension”, Bull. Austral. Math. Soc. 2 (1970), 317–333.CrossRefGoogle Scholar
[10]Sneddon, I.N., The use of integral transforms (McGraw-Hill, New York; Toronto, Ontario; London; 1972).Google Scholar
[11]Stoker, J.J., Water waves: The mathematical theory with applications (Pure and Applied Mathematics, 4. Interscience, New York, London, 1957).Google Scholar