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The expansion of water-wave potentials at great distances

Published online by Cambridge University Press:  24 October 2008

F. Ursell
F. Ursell
Affiliation:
Department of Mathematics, University of Manchester
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Abstract

The wave function φ(x, y) satisfies the equation

in the infinite region

and satisfies the boundary condition Kφ + φy = 0 on the two line segments (y = 0, −∞ < x < −R) and (y = 0, R < x < ∞). We are not concerned with the properties of φ(x, y) inside the circle r = R. It is shown that outside the circle r = R the wave function φ(x, y) can be written as the sum of a wave source, a wave dipole, regular waves and wave-free potentials. (These basic functions are defined in section 2.) The proof makes use of analytic continuation into the image region (r > R, y ≤ 0) cut along the negative y-axis.

Two methods are given for this step. The first method uses Schwarz's symmetry principle to define the auxiliary wave function Φ = Kφ + φy throughout r > R, and φ can then be found by solving differential equations in the y or x direction. The second method uses Green's theorem. Applications in the theory of water waves are suggested.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 3 , July 1968 , pp. 811 - 826
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

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