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On the existence theory for irrotational water waves

Published online by Cambridge University Press:  24 October 2008

G. Keady  and
J. Norbury
G. Keady
Affiliation:
University of Western Australia
J. Norbury
Affiliation:
University College London
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Abstract

This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.

Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and ⅙π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and

where g is the acceleration due to gravity. Krasovskii's set of solutions is included in the set that we obtain.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 1 , January 1978 , pp. 137 - 157
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1) Amman, H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620.CrossRefGoogle Scholar
(2) Birkhoff, G. and Zarantonello, E. H. Jets, wakes and cavities (Academic Press, 1957).Google Scholar
(3) Copson, E. T. An introduction to the theory of functions of a complex variable (Oxford, 1935).Google Scholar
(4) Courant, R. and Hilbert, D. Methods of mathematical physics, vol. 2 (Interscience, 962).Google Scholar
(5) Dancer, E. N. Global solution branches for positive mappings. Arch. Rat. Mech. Anal. 52 (1973), 181.CrossRefGoogle Scholar
(6) Friedrichs, K. O. and HYers, D. H. The existence of solitary waves. Comm. Pure Appl. Math. 7 (1954), 517.CrossRefGoogle Scholar
(7) Garabedian, P. R. Surface waves of finite depth. J. Analyse Math. 14 (1965), 161.CrossRefGoogle Scholar
(8) Hyres, D. H. Some nonlinear integral equations in hydrodynamics. In Nonlinear integral equations, ed. Anselone, (University of Wisconsin Press, 1965).Google Scholar
(9) Keady, G. Large amplitude water waves and Krasovskii's existence proof. University of Essex Fluid Mechanics Research Institute, Report No. 37 (1972).Google Scholar
(10) Keady, G. and Pritchard, W. G. Bounds for surface solitary waves. Proc. Cambridge Philos. Soc. 76 (1974), 345.CrossRefGoogle Scholar
(11) Krasnoselskii, M. A. Positive solutions of operator equations (Noordhoff, 1964).Google Scholar
(12) Krasovskii, Yu. P. On the theory of steady-state waves of large amplitude. U.S.S.R. Computational Matha. and Math. Phys. 1 (1961), 996.CrossRefGoogle Scholar
(13) Lavrentiev, M. A. On the theory of long waves (1943); A contribution to the theory of long waves (1947). Amer. Math. Soc. Transl. 102 (1954).Google Scholar
(14) Lewy, H. A note on harmonic functions and a hydrodynamical application. Proc. Amer. Math. Soc. 3 (1952), 111.CrossRefGoogle Scholar
(15) Littman, W. On the existence of periodic waves near critical speed. Comm. Pure Appl. Math. 10 (1957), 241.CrossRefGoogle Scholar
(16) Longuet-Higgins, M. S. and Fox, M. J. H. Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80 (1977), 721.CrossRefGoogle Scholar
(17) Milne-Thomson, L. M. Theoretical hydrodynamics (MacMillan, 5th ed., 1967).Google Scholar
(18) Protter, M. H. and Weinberger, H. F. Maximum principles in differential equations (Prentice-Hall, 1967).Google Scholar
(19) Rabinowitz, P. H. Some global results for nonlinear eigenvalue problems. J. Functional Analysis 7 (1971), 487.CrossRefGoogle Scholar
(20) Spielvogel, E. R. A variational principle for waves of infinite depth. Arch. Rat. Mech. Anal. 39 (1970), 189.CrossRefGoogle Scholar
(21) Ter Krikorov, A. M. The existence of periodic waves which degenerate into a solitary wave. J. Appl. Maths. Mech. 24 (1960), 930.CrossRefGoogle Scholar