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Mathematical aspects of trapping modes in the theory of surface waves

Published online by Cambridge University Press:  21 April 2006

F. Ursell
F. Ursell
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
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Abstract

A horizontal canal of infinite length and of constant width and depth contains inviscid fluid under gravity. The fluid is bounded internally by a submerged horizontal cylinder which extends right across the canal and has its generators normal to the sidewalls. Suppose that the fluid is set in motion by a surface pressure varying across the canal, then some of the energy is radiated to infinity while some of the energy is trapped in characteristic modes (bound states) near the cylinder. The existence of trapping modes in special cases was shown by Stokes (1846) and Ursell (1951); a general treatment, given by Jones (1953), is based on the theory of elliptic partial differential equations in unbounded domains. In the present paper a much simpler treatment is given which uses only the theory of bounded symmetric linear operators together with Kelvin's minimum-energy theorem of classical hydrodynamics.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 183 , October 1987 , pp. 421 - 437
Copyright
© 1987 Cambridge University Press

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