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Short surface waves in a canal: dependence of frequency on curvature

Published online by Cambridge University Press:  29 March 2006

F. Ursell
F. Ursell
Affiliation:
Department of Mathematics, University of Manchester, England
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Abstract

Davis has shown by means of a lengthy calculation that, for two-dimensional oscillations in a canal of width 2a, the mth eigenvalue has the form \[ N_m = {\textstyle\frac{1}{2}}m\pi - \frac{\lambda_1+\lambda_2}{4m\pi}+o\bigg(\frac{1}{m}\bigg), \] where λ1/a and λ2/a are the curvatures of the bounding cross-sectional curve C at its vertical intersections with the free surface. Here the same result is obtained more simply.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 63 , Issue 1 , 18 March 1974 , pp. 177 - 181
Copyright
© 1974 Cambridge University Press

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References

Davis, A. M. J. 1965 Two-dimensional oscillations in a canal of arbitrary cross-section. Proc. Camb. Phil. Soc. 61, 827–846.Google Scholar
Davis, A. M. J. 1969 Short surface waves in a canal: dependence of frequency on curvature. Proc. Roy. Soc. A 313, 249–260.Google Scholar
Lamb, H. 1932 Hydrdynamics, 6th edn. Cambridge University Press.
Ursell, F. 1961 The transmission of surface waves under surface obstacles. Proc. Camb. Phil. Soc. 57, 638–668.Google Scholar