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Long surface waves incident on a submerged horizontal plate

Published online by Cambridge University Press:  12 April 2006

P. F. Siew  and
D. G. Hurley
P. F. Siew
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands
D. G. Hurley
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands
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Abstract

A train of surface gravity waves of wavelength λ in a channel of depth H is incident on a horizontal plate of length l that is submerged to a depth c. Under the assumption that both λ and l are large compared with H, the method of matched asymptotic expansions is used to show that, to first order, the reflexion coefficient R and the transmission coefficient T are given by \[ R = \chi \left\{\frac{\sigma l}{(gH)^{\frac{1}{2}}}\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}-2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\} \] and \[ T =\chi\left\{2i\left[\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\right]\right\} \] where \begin{eqnarray*} \chi &=& 1\left/ \left\{2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)+\frac{\sigma l}{b}\bigg(\frac{H}{g}\bigg)^{\frac{1}{2}}\bigg(1+\frac{c}{H}\bigg)\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}\right.\right.\\ &&\left. +2i\bigg(\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\}, \end{eqnarray*} σ is the angular frequency and g the acceleration due to gravity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 83 , Issue 1 , 1 November 1977 , pp. 141 - 151
Copyright
© 1977 Cambridge University Press

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References

Burke, J. E. 1964 Scattering of surface waves on an infinitely deep fluid. J. Math. Phys. 5, 805–819.Google Scholar
Greene, T. R. & Heins, A. E. 1953 Water waves over a channel of infinite depth. Quart. Appl. Math. 11, 201–214.Google Scholar
Heins, A. E. 1950 Water waves over a channel of finite depth with a submerged plane barrier. Can. J. Math. 2, 210–222.Google Scholar
Hermans, A. J. 1972 A perturbation method for the radiation of surface waves. J. Engng Math. 6, 323–330.Google Scholar
Holford, R. L. 1964 Short surface waves in the presence of a finite dock. I. Proc. Camb. Phil. Soc. 60, 957–983.Google Scholar
Leppington, F. G. 1972 On the radiation and scattering of short surface waves. Part 1. J. Fluid Mech. 56, 101–119.Google Scholar
Rayleigh, Lord 1945 The Theory of Sound. Dover.
Siew, P. F. 1976 Ph.D. thesis, University of Western Australia.
Stoker, J. J. 1957 Water Waves. Interscience.
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic Press.
Wells, L. W. 1953 Some remarks on shallow water theory. N.Y. Univ. Inst. Math. Sci. Rep. IMM-198.Google Scholar