© 1999 by Oxford University Press
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On the wave motion near a submerged sphere between parallel walls: II. notes on convergence
( Department of Mathematics, Manchester University, Oxford Road, Manchester M13 9PL, UK )
In the first part of the present paper the velocity potential was formally expressed as the sum of multipoles placed at the centre of the sphere, and the coefficients C(m, n) in this sum were found to satisfy an infinite system of equations. In the second part of the paper it is now shown that this system has a solution, and that the resulting series for the potential is convergent in the physically relevant domain. A proof of these results will be given in the present paper for all wavenumbers K, except possibly for a discrete set. The proof makes use of bounds for image potentials and for their expansions and depends on the theory of infinite linear systems which is analogous to the theory of integral equations. We shall use extensions of the following fundamental results. Suppose that in the infinite system
X(m, n) + [sum ]M=1
[sum ]N=1 A(m,n; M,N)x(M,N)) = B(m,n), m,n = 1,2,3...
the coefficients A(m,n; M,N) and B(m,n) depend analytically on a parameter 
and satisfy the conditions
[sum ]m=1 [sum ]n=1 [sum ]M=1 [sum ]N=1 |A(m,n; M,N)|2 < and [sum ]m=1 [sum ]n=1|x(m,n)|2 < , except possibly for a discrete set of values of .