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On the interaction of surface and internal gravity waves: uniformly valid solution by extended stationary phase

Published online by Cambridge University Press:  29 March 2006

B. A. Hughes
B. A. Hughes
Affiliation:
Defence Research Establishment Pacific, Forces Mail Office, Victoria B.C. VOS1BO, Canada
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Abstract

Using simple Fourier transform techniques and extensions to stationary-phase methods, the behaviour of surface gravity waves is determined near triply coalescing roots of the dispersion relation. It is shown that the amplitude of the surface wave is proportional to (∂U/∂x)−¼ at the location of the triple root. Far from the triple root it satisfies conservation of action. The internal wave is modelled simply by its surface current U.

Asymptotic orders of magnitude are also given for the case ∂U/∂x = 0 at the triple root.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 74 , Issue 4 , 22 April 1976 , pp. 667 - 683
Copyright
© 1976 Cambridge University Press

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References

Apel, J. R., Byrne, H. M., Prom, J. R. & Charnell, R. L. 1975 J. Geophys. Res. 80, 865.
Bleistein, N. 1967 J. Math. Mech. 17, 533.
Bretherton, F. P. & Garrett, C. J. R. 1968 Proc. Roy. Soc. A 302, 529.
Chester, C., Friedman, B. & Ursell, F. 1957 Proc. Camb. Phil. Soc. 57, 599.
Gargett, A. E. & Hughes, B. A. 1972 J. Fluid Mech. 52, 179.
Holliday, D. 1973 J. Fluid Mech. 57, 797.
Lafond, E. C. 1962 In The Sea, vol. 1, p. 745. Interscience.
Longuet-Higgins, M. S. & Stewart, R. W. 1961 J. Fluid Mech. 10, 529.
Ludwig, D. 1966 Comm. Pure Appl. Math. 19, 215.
Ludwig, D. & Olver, F. W. J. (eds.) 1970 Special Functions and Wave Propagation. Studies in Applied Math. vol. 6. Philadelphia, Penn.: SIAM.
Pearcey, T. 1946 Phil. Mag. 37, 311.
Perry, R. B. & Schimke, G. R. 1965 J. Geophys. Res. 70, 2319.
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Shand, J. A. 1953 Trans. Am. Geophys. Un. 34, 849.
Smith, R. 1976 Proc. I.U.T.A.M. Sym. on Surface Gravity Waves on Water of Varying Depth (ed. R. Radok).
Ursell, F. 1972 Math. Proc. Camb. Phil. Soc. 72, 49.
West, B. J., Thomson, J. A. & Watson, K. M. 1975 J. Hydronautics, 9, 25.