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The effect of surface tension on the scattering of waves by a partially immersed vertical barrier in water of infinite depth

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  26 February 2010

P. F. Rhodes-Robinson
P. F. Rhodes-Robinson
Affiliation:
Department of Mathematics, Victoria University of Wellington, P. O. Box 600, Wellington, New Zealand
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Abstract

In this paper time-harmonic surface wave motion for progressive waves incident normally on and scattered by a partially immersed fixed vertical barrier in water of infinite depth is considered in the presence of surface tension. The problem for the velocity potential is solved, as others have been previously, by first supposing that the free-surface slopes at the barrier are prescribed and the formal solution in terms of these is obtained explicitly by complex-variable methods. To simplify the calculation the known solution corresponding to zero free-surface slopes at the barrier is subtracted out first and emphasis is placed on determining the residual potential. Finally, an appropriate dynamical edge condition is imposed on the formal solution to determine the required values of the edge-slope constants and hence fully solve the transmission problem. The problem was first examined some time ago using a complex-variable reduction procedure before the advent of this condition, although an explicit formal solution was not obtained, that earlier work forms a basis for the present investigation. It is noted in conclusion how the solution of the problem for waves generated by a partially immersed non-uniform heaving vertical plate may easily be obtained in a similar manner, since the formal solution required is just the residual potential determined in our main problem.

MSC classification

Secondary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Type
Research Article
Information
Mathematika , Volume 43 , Issue 1 , June 1996 , pp. 182 - 197
Copyright
Copyright © University College London 1996

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References

1.Dean, W. R.. On the reflexion of surface waves by a submerged plane barrier. Proc. Cambridge Phil. Soc, 41 (1945), 231–238.CrossRefGoogle Scholar
2.Evans, D. V.. The influence of surface tension on the reflection of water waves by a plane vertical barrier. Proc. Cambridge Phil. Soc, 64 (1968), 795–810.CrossRefGoogle Scholar
3.Hocking, L. M.. Waves produced by a vertically oscillating plate. J. Fluid Mech., 179 (1987), 267–281.CrossRefGoogle Scholar
4.Hocking, L. M.. Reflection of capillary-gravity waves. Wave Motion, 9 (1987), 217–226.CrossRefGoogle Scholar
5.Rhodes-Robinson, P. F.. The effect of surface tension on the progressive waves due to incomplete vertical wave-makers in water of infinite depth. Proc. Roy. Soc. Ser. A, 435 (1991), 293–319.Google Scholar
6.Rhodes-Robinson, P. F.. The use of Green's theorem in water wave problems. Mathematical Techniques in Water Waves edited by Mandal, B. N., International Series on Advances in Fluid Mechanics (1996), Ch. 4 (Computational Mechanics Publications, Southampton).Google Scholar
7.Ursell, F.. The effect of a fixed vertical barrier on surface waves in deep water. Proc. Cambridge Phil. Soc, 43 (1947), 374–382.CrossRefGoogle Scholar