The waves due to a submerged sphere moving in a canal

A submerged sphere is moving with uniform horizontal velocity W along a canal of width 2l and infinite depth. The fluid motion is assumed to be irrotational and inviscid, and the condition of constant pressure at the free surface is linearized. The velocity potential is expressed as the double sum of multipole potentials, singular at the centre of the sphere and satisfying Laplace's equation and the boundary conditions on the sidewalls and the free surface. The construction of these multipoles is described and is compared with earlier constructions in this and in an earlier problem.