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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

Benjamin., T. B. 1962, “The solitary wave on a stream with an arbitrary distribution of vorticity”, J. Fluid Mech., 12, 97.CrossRefGoogle Scholar
Benjamin, T. B. and Lighthill., M. J. 1954, “On cnoidal waves and bores”, Proc. Roy. Soc. A, 224, 448.Google Scholar
Cokelet., E. D. 1977, “Steep gravity waves in water of arbitrary uniform depth”, Phil. Trans. R. Soc. Lond. A, 286, 183.Google Scholar
Dubreil-Jacotin, L.. 1934, “Sur la d“termination rigoureuse des ondes permanents periodiques d'ampleur finie”, J. Math. Pures et Appl., 23, 217.Google Scholar
Gerstner, F.. 1804, “Theorie der Wellen samt einer abgeleiteten Theorie der Deichprofile”, Abhböhn. Ges. Wiss., 3, 1.Google Scholar
Greenberg., J. M. 1972, “On the existence of standing periodic waves in a perfect fluid”, J. Math. Anal, and Applns., 39, 227.CrossRefGoogle Scholar
Keady, G. and Norbury, J.. 1975, “Water waves and conjugate streams”, J. Fluid Mech., 70, 663.CrossRefGoogle Scholar
Keady, G. and Norbury, J.. 1978, “On the existence theory for irrotational water waves”, Math. Proc. Camb. Phil. Soc, 83, 137.CrossRefGoogle Scholar
Keady, G. and Norbury, J.. 1978, “Comparison theorems for flows with vorticity”, submitted to Q. J. Mech. Appl. Math.Google Scholar
Keady, G. and Pritchard., W. G. 1974, “Bounds for surface solitary waves”, Proc. Camb. Phil. Soc, 79, 345.Google Scholar
Krasovskü, Yu. P.. 1961, “On the theory of steady-state waves of large amplitude”, U.S.S.R. Comp. Math and Math Physics, 1, 996.Google Scholar
Lavrentiev., M. A. 1964, Variational methods (Noorhoff).Google Scholar
Protter, M. H. and Weinberger., H. F. 1967, Maximum principles in differential equations (Prentice-Hall).Google Scholar
Serrin., J. B. 1952, “Two hydrodynamic comparison theorems”, J. Rat. Mech. Anal., 1, 563.Google Scholar
Zeidler, E.. 1973, “Existenzbeweis für permanente Kapillar-Schwerewellen mit allgemeinen Wirbelverteilungen”, Arch. Rat. Mech. Anal., 50, 34.Google Scholar