Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T01:25:05.787Z Has data issue: false hasContentIssue false
  • English
  • Français

On the short-wave asymptotic motion due to a cylinder heavinǵ on water of finite depth. I

Published online by Cambridge University Press:  24 October 2008

P. F. Rhodes-Robinson
P. F. Rhodes-Robinson
Affiliation:
Department of Mathematics, University of Manchester
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is a first investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we present a non-rigorous method for an arbitrary smooth cylinder which intersects the free surface normally. The reduction of the potential problem by the use of two auxifiary potentials, introduced by the subtraction of formal solutions constructed from the limit potential, enables us to find (i) an integral expression, by using a subsidiary approximate Green's function, for the potential in the far field which is evaluated after making plausible assumptions about the asymptotic value of the first auxiliary potential on the cylinder and below the free surface; and (ii) a form for the virtual mass, using the second auxiliary potential whose value on the cylinder is deduced from that of the first to which it is related. Thus we obtain the asymptotic evaluation of the coefficients describing the wave-making and virtual mass of the heaving cylinder, which depend on the limit potential and have no formal dependence on the depth of the bottom.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 67 , Issue 2 , March 1970 , pp. 423 - 442
Copyright
Copyright © Cambridge Philosophical Society 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Havelock, T. H.Forced surface waves on water. Philos. Mag. 8 (1929), 569–576.CrossRefGoogle Scholar
(2)Holford, R. L.On the generation of short two-dimensional surface waves by oscillating surface cylinders of arbitrary cross-section (Technical Report, Department of Mathematics, Stanford University, 1965).Google Scholar
(3)John, F.On the motion of floating bodies. II. Comm. Pure Appl. Math. 3 (1950), 45–101.CrossRefGoogle Scholar
(4)Lamb, H.Hydrodynamics, 6th ed. (Cambridge, 1932).Google Scholar
(5)Rhodes-Robinson, P. F.Asymptotic coefficients for oscillating elliptic cylinders (Dp. Adv. Stud. Sc. Dissertation, University of Manchester, 1964).Google Scholar
(6)Thorne, R. C.Multipole expansions in the theory of surface waves. Proc. Cambridge Philos. Soc. 49 (1953), 707–716.CrossRefGoogle Scholar
(7)Titchmarsh, E. C.The theory of functions, 2nd ed. (Oxford, 1939).Google Scholar
(8)Ursell, F.Short surface waves due to an oscillating immersed body. Proc. Roy. Soc. Ser. A 220 (1953), 90–103.Google Scholar
(9)Ursell, F.Water waves generated by oscillating bodies. Quart. J. Mech. Appl. Math. 7 (1954), 427–437.CrossRefGoogle Scholar
(10)Watson, G. N.Bessel functions, 2nd ed. (Cambridge, 1944).Google Scholar
(11)Writtaker, E. T. and Watson, G. N.Modern analysis, 4th ed. (Cambridge, 1927).Google Scholar
(12)Yu, Y. S. and Ursell, F.Surface waves generated by an oscillating circular cylinder on water of finite depth: theory and experiment. J. Fluid Mech. 11 (1961), 529–551.CrossRefGoogle Scholar