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Two-dimensional oscillations in a canal of arbitrary cross-section

Published online by Cambridge University Press:  24 October 2008

A. M. J. Davis
A. M. J. Davis
Affiliation:
Mining Research Establishment, National Coal Board, Worton Hall, Isleworth, Middx.
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Abstract

An infinitely long canal with uniform cross-section is filled with inviscid fluid. It is required first to show that any small two-dimensional motion of the fluid can be represented as the superposition of normal mode disturbances. A suitable generalized Green's function G(x, y; ξ) is constructed and is used to set up an integral equation (2·9) for the velocity potential on the free surface. It is shown that the eigenfunctions are complete and so are their (possibly time-dependent) extensions to the whole canal, in the sense that an arbitrary disturbance possesses a unique representation. In section 5, it is required to find asymptotic approximations to the large eigenvalues of (2·9). For this purpose a different integral equation (5·5) is set up on the canal, the kernel of which is the sum of a degenerate kernel and a small kernel. The solutions of this equation can therefore be obtained by iteration. The form of the mth eigenvalue is shown to be

for sufficiently large m.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 3 , July 1965 , pp. 827 - 846
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

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