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Small amplitude capillary-gravity waves in a channel of finite depth

Published online by Cambridge University Press:  18 May 2009

M. C. W. Jones
M. C. W. Jones
Affiliation:
Department of Pure Mathematics, The Queen's University of Belfast, Belfast BT7 INN. Northern Ireland., U. K.
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Introductory Remarks. Recently a number of studies (Chen & Saffman [2], Jones & Toland [7,11], Hogan [5]) have been made of periodic capillary-gravity waves which form the free surface of an ideal fluid contained in a channel of infinite depth. However, little work appears to have been done on the corresponding problem when the depth is finite. The most significant contributions appear to be those of Reeder & Shinbrot [9], Barakat & Houston [1] and Nayfeh [8] all of whom confined themselves to Wilton ripples (see §1.3). Yet there are sound reasons why such a study should be made. For quite apart from the unsolved problem regarding the type of capillary-gravity waves which may occur at finite depths, the consideration of the finite depth problem may be regarded as a first step in the study of solitary capillary-gravity waves. In this paper, a new integral equation for the infinite depth problem, due to J. F. Toland and the author, is adapted to be of use in tackling the finite depth problem. Using this we obtain results for the exact equations of motion which answer rigorously the questions of existence and multiplicity of small amplitude solutions of the periodic capillary-gravity wave problem of finite depth.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 2 , May 1989 , pp. 142 - 160
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

1.Barakat, R. and Houston, A., Nonlinear periodic capillary-gravity waves on a fluid of finite depth, J. Geophys. Res., 73 (1968), 6545–6554.CrossRefGoogle Scholar
2.Chen, B. and Saffman, P. G., Steady gravity-capillary waves on deep water—I. weakly non-linear waves, Stud. App. Math., 60 (1979), 183–210.CrossRefGoogle Scholar
3.Crandall, M. G. and Rabinowitz, P. H., Bifurcation from a simple eigenvalue, J. Fund. Anal., 8 (1971), 321–340.CrossRefGoogle Scholar
4.Golubitsky, M. and Schaeffer, D. G., Singularities and Groups in Bifurcation Theory (I). (Springer-Verlag, 1985).CrossRefGoogle Scholar
5.Hogan, S. J., Some effects of surface tension on steep water waves. Parts 1, 2 and 3, J. Fluid Mech., 91 (1979), 167–180; 96 (1980), 417–445; 110 (1981), 381–410.CrossRefGoogle Scholar
6.Hunter, J. K. and Vanden-Broeck, J.-M., Solitary and periodic gravity waves of finite amplitude, j. Fluid Mech., 134 (1983), 205–219.CrossRefGoogle Scholar
7.Jones, M. C. W. and Toland, J. F., Symmetry and the bifurcation of capillary-gravity waves, Arch. Rat. Mech. Anal., 96 (1986), 29–53.CrossRefGoogle Scholar
8.Nayfeh, A. H., Finite amplitude surface waves in a liquid layer, J. Fluid Mech., 40 (4) (1979), 671–684.CrossRefGoogle Scholar
9.Reeder, J. and Shinbrot, M., On Wilton ripples, I: formal derivation of the phenomenon, Wave motion, 3 (1981), 115–135.CrossRefGoogle Scholar
10.Shearer, M., Secondary bifurcation near a double eigenvalue, SI AM J. Math. Anal., 11 (2) (1980), 365–389.CrossRefGoogle Scholar
11.Toland, J. F. and Jones, M. C. W., The bifurcation and secondary bifurcation of capillary-gravity waves, Proc. R. Soc. Lond. A, 399 (1985), 391–417.Google Scholar
12.Zymund, A., Trigonometric series Vol. I (Cambridge University Press, 1959).Google Scholar