Abstract
The present paper describes a method for obtaining local expansions of singularities located on straight boundaries. The ideas are classical and have been applied in recent years to problems in continuum mechanics. The method is here applied to a local source of oblique water waves in the free surface. (For an earlier treatment of this problem see [J. Fluid Mech. 14 (1962) 496].) An integral expression is readily obtained for the potential of such a source. By a deformation of contours this potential is continued into a complete neighbourhood of the source. It is shown that the local expansion must involve the multi-valued functions ∂ ∂ν {I ν(kr) cos νθ} ν=m for m=0,1,2,3,…, and the multi-valued component of the potential is obtained. The complete expansion is then obtained by modifying the integrand in the original representation. For a significant extension of these ideas see [ 2: A. Hulme, The potential of a horizontal ring of wave sources in a fluid with a free surface, Proc. Roy. Soc. Lond. A 375 (1981) 295].
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