The use of series of bessel functions in problems connected with cylindrical wind-tunnels

Abstract
1. There are various series of the types associated with the names of Fourier-Bessel and Dini which arise in the discussion of the problem of a body (the Rankine "Ovoid") placed in a cylindrical wind-tunnel. For Such series are (1) S 1 = Ʃ ∞ m = 1 J 0 ( k m r )/J 1 2 ( k m a ) e - k m x , (2) S 2 = Ʃ ∞ m = 1 J 1 ( k m r )/ k m a J 1 2 ( k m a ) e - k m x , (3) S 3 = Ʃ ∞ m = 1 J 0 ( k m r )/J 0 2 ( k m a ) e - k m x , (4) S 4 = Ʃ ∞ m = 1 J 0' ( k m r )/ k m a J 0 2 ( k m a ) e - k m x , where J 0 and J 1 denote Bessel functions of orders 0 and 1; k 1 , k 2 , k 3 , ....., are the positive roots of the equation J 0 ( ka ) = 0; k 1 , k 2 , k 3 , ....., are the positive roots of the equation J 1 ( ka ) = 0; and, so far as we are concerned, r , a and x are positive with 0 < r < a . It may be mentioned that a is the radius of the tunnel while r and x are respectively radial and axial coordinates. Is is not my object to discuss the origin of these series, which will be found elsewhere. It is obvious that the series are rapidly convergent and are well adapted for computation when x is large; but convergence is slow when x is small and (in the case of the first and third) is non-existent when x = 0. Mr. C. N. H. Lock, of the National Physical Laboratory, has asked me whether it is possible to express the series in forms which are suitable for computation when x is small-a problem which is evidently of some physical importance-and the investigation presents various features of mathematical interest. Accordingly in this paper I shall show how to express the series as combinations of elementary functions and convergent series of ascending powers of x and r with coefficients in forms which are fairly well adapted for computation.

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