Scattering of water waves by axisymmetric bodies in a channel

Methods are presented for the calculation of wave forces on a vertically axisymmetric body arbitrarily placed within a channel. Integral representations of singular solutions of the Helmholtz equation, called channel multipoles here, are derived and these allow straightforward solution of the scattering problem for a vertical cylinder extending throughout the depth. In contrast to previous methods there is no need to sum series of images. These multipoles are also used in deriving an approximate solution valid when the radius of the cylinder is small relative to the wavelength and channel width.To solve for arbitrary shaped axisymmetric bodies, a plane-wave approximation is developed based on the assumption that the wavelength is much less than the channel width. Comparisons with the accurate solution for a vertical cylinder suggest that this approximate method performs well even when this assumption is clearly violated. The results of calculations of wave forces on a truncated cylinder are also given.All of the methods described may be applied just as easily to the case of an off-centre body as to a centrally-placed body.