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Waves and conjugate streams with vorticity

Published online by Cambridge University Press:  26 February 2010

G. Keady  and
J. Norbury
G. Keady
Affiliation:
Department of Mathematics, University of Western Australia.
J. Norbury
Affiliation:
St. Catherine's College, oxford.
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Abstract

Steady plane periodic gravity waves on the surface of an ideal liquid flowing over a horizontal bottom are considered. The flow is rotational with a vorticity distribution ω(ψ) and has flux Q. Let R/g denote the total head and S the flow force of the wavetrain. The diagrams (Fig. 1) show combinations of R and S which are possible in the general case. (We normalise so that Q = 1 throughout. The axes are R/R* and S/S*, where the suffix * refers to the critical flow.) It is proved that no waves are possible below γ+ or to the right of γ here γ+ corresponds to unidirectional supercritical streams, and thus is the best possible barrier, while γ is a barrier to the right of the line of waves of greatest height. Bounds on wave properties are found in the process of establishing the above results. When ω ≡ 0 these bounds were conjectured by Benjamin and Lighthi U (1954) and established in Keady and Norbury (1975). The generalisation to flows with vorticity is accomplished under the condition that , where hc is the height of the crest of the wave.

Type
Research Article
Information
Mathematika , Volume 25 , Issue 1 , June 1978 , pp. 129 - 150
Copyright
Copyright © University College London 1978

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References

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