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On forced three-dimensional surface waves in a channel in the presence of surface tension

Published online by Cambridge University Press:  24 October 2008

P. F. Rhodes-Robinson
P. F. Rhodes-Robinson
Affiliation:
Department of Mathematics, Victoria University of Wellington, New Zealand
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Abstract

In this paper wave-maker theory including the effect of surface tension is determined for three-dimensional motion of water in a semi-infinite rectangular channel with outgoing surface wave modes allowed for at infinity; the motion is generated by a harmonically oscillating vertical plane wave-maker at the end of the channel and the cases of both infinite and finite constant depth are treated. The solution of the boundary-value problem for the velocity potential is more complicated in the presence of surface tension due mainly to the additional effect of the channel walls at which the normal free surface slopes are prescribed—as also is the slope at the wave-maker—to ensure uniqueness. The simpler three-dimensional solution for a semi-infinite region—obtained long ago by Sir Thomas Havelock in the absence of surface tension for the case of infinite depth—is also noted.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 3 , May 1974 , pp. 405 - 426
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Evans, D. V.The application of a new source potential to the problem of the transmission of water waves over a shelf of arbitrary profile. Proc. Cambridge Philos. Soc. 71 (1972), 391–410.CrossRefGoogle Scholar
(2)Garrison, C. J.On the interaction of an infinite shallow draft cylinder oscillating at the free surface with a train of oblique waves. J. Fluid Mech. 39 (1969), 227–255.CrossRefGoogle Scholar
(3)Green, M. W.A problem connected with the oblique incidence of surface waves on an immersed cylinder. J. Inst. Math. Appl. 8 (1971), 82–98.CrossRefGoogle Scholar
(4)Greene, T. R. and Heins, A. E.Water waves over a channel of infinite depth. Quart. Appl. Math. 11 (1953), 201–214.CrossRefGoogle Scholar
(5)Havelock, T. H.Forced surface waves on water. Philos. Mag. 8 (1929), 569–576.Google Scholar
(6)Heins, A. E.Water waves over a channel of finite depth with a submerged plane barrier. Canal. J. Math. 2 (1950), 210–222.Google Scholar
(7)Levine, H.Diffraction of surface waves on an incompressible fluid. J. Fluid Mech. 15 (1963), 241–250.Google Scholar
(8)Levine, H.Scattering of surface waves by a submerged circular cylinder. J. Mathematical Phys. 6 (1965), 1231–1243.Google Scholar
(9)Miles, J. W.Surface-wave scattering matrix for a shelf. J. Fluid Mech. 28 (1967), 755–767.CrossRefGoogle Scholar
(10)Rhodes-Robinson, P. F.Fundamental singularities in the theory of water waves with surface tension. Bull. Austral. Math. Soc. 2 (1970), 317–333; and 3 (1970), 432 (Editor's corrigenda).Google Scholar
(11)Rhodes-Robinson, P. F.On the forced surface waves due to a vertical wave-maker in the presence of surface tension. Proc. Cambridge Philos. Soc. 70 (1971), 323–337.Google Scholar
(12)Sebekin, B. I.Diffraction of surface waves by a wedge. Izv. Akad. Nauk S.S.S.B. Ser. Meh. Žhidk. Gaza 1 (1966), no. 5, 148–151 (Russian); translated as Izv. Acad. sci. U.S.S.R. Fluid Dyn. 1 (1966), no. 5, 106–108 (1969).Google Scholar
(13)Ursell, F.Trapping modes in the theory of surface waves. Proc. Cambridge Philos. Soc. 47 (1951), 347–358.Google Scholar
(14)Ursell, F.Edge waves on a sloping beach. Proc. Roy. Soc. Ser. A 214 (1952), 79–97.Google Scholar
(15)Ursell, F.The expansion of water-wave potentials at great distances. Proc. Cambridge Philos. Soc. 64 (1968), 811–826.Google Scholar
(16)Voit, S. S.Diffraction from a half-surface of waves formed on the surface of a liquid by a periodically acting source. Prikl. Mat. Meh. 25 (1961), 370–374 (Russian); translated as J. Appl. Math. Mech. 25 (1961), 545–553 (1962).Google Scholar
(17)Wehausen, J. V. and Laitone, E. V.Surface waves. Handbuch der Physik 9 (1960), 446–778 (Springer).Google Scholar
(18)Williams, W. E.Diffraction of surface waves on an incompressible fluid. J. Fluid Mech. 22 (1965), 253–256.Google Scholar