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Some applications of Maz'ja's uniqueness theorem to a class of linear water wave problems

Published online by Cambridge University Press:  24 October 2008

Andrew Hulme
Andrew Hulme
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL
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Abstract

This paper describes a criterion which, if satisfied, guarantees uniqueness of solution for the problem involving the radiation or diffraction of small amplitude water waves by a totally submerged body. This criterion was originally stated by a Russian mathematician, V. G. Maz'ja, although his work does not appear to be widely known. For this reason a brief outline of Maz'ja's proof is given here, together with a discussion of the condition to be satisfied by the submerged body.

Maz'ja's criterion is not satisfied by all submerged bodies and so his result does not provide a general proof of uniqueness. However, the criterion is satisfied in some important cases and examples are presented here. The partial nature of this result is due entirely to the method of proof and not to any intrinsic difficulty with the underlying hydrodynamics.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 1 , January 1984 , pp. 165 - 174
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1] Beale, J. T.. Eigenfunction expansions for objects floating in an open sea. Comm. Pure Appl. Math. 30 (1977), 283–313.CrossRefGoogle Scholar
[2] John, F.. On the motion of floating bodies, II. Comm. Pure Appl. Math. 3 (1950), 45–101.CrossRefGoogle Scholar
[3] Lamb, Sir Horace. Hydrodynamics. Cambridge University Press (6th edition) (1932).Google Scholar
[4] Lenoir, M. and Martin, D.. An application of the principle of limiting absorption to the motions of floating bodies. J. Math. Anal. Appl. 79 (1981), 370–383.CrossRefGoogle Scholar
[5] Livchitz, M. L.. On the steady-state oscillations of a sphere submerged in a fluid. Tr. Leningr. Korablestr. Inst. No. 91 (1976), 133–139.Google Scholar
[6] Maz'ja, V. G.. Contributions to the steady problem of small oscillations of a fluid in the presence of a submerged body. Works of the school of S. L. Sobolev, No. 2, Academy of Sciences of the U.S.S.R., Siberian section, Novosibirsk (1977). (In Russian.)Google Scholar
[7] Maz'ja, V. G.. Solvability of the problem on the oscillations of a fluid containing a submerged body. J. Soviet Math. 10 (1978), 86–89. (In English.) (Also, see Math. Revs, vol 58, 6, No. 32354, 1979.)CrossRefGoogle Scholar
[8] Newman, J. N.. Marine Hydrodynamics. MIT Press, 1977.CrossRefGoogle Scholar
[9] Ursell, F.. Surface waves on deep water in the presence of a submerged circular cylinder, II. Prov. Cambridge Philos. Soc. 46 (1950), 153–158.CrossRefGoogle Scholar