© 1999 by Oxford University Press
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On the wave motion near a submerged sphere between parallel walls: I. multipole potentials
( Department of Mathematics, Manchester University, Oxford Road, Manchester M13 9PL, UK )
The depth of water is assumed to be infinite. A submerged sphere is placed midway between two parallel vertical walls x = ±l. The normal velocity on the sphere is prescribed and is symmetrical about the mid-plane x = 0. The motion is assumed to have angular frequency 
and small amplitude. The radiated wave field is to be found. The present paper describes a method for solving this problem by means of the method of multipoles. A construction is given for the multipole potentials: for each singularity r-n-1 Pnm (cos 
) cos m
at the centre of the sphere an image potential is constructed so that the boundary conditions are satisfied identically on the two sets of infinite boundaries (the side walls and the free surface) and so that the radiation condition at infinity is also satisfied. It is not difficult to see that the same method will remain applicable when the submerged sphere is placed in any position between the vertical planes. It is shown in a following paper that the resulting infinite processes are convergent. No numerical results are given.