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

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) S1 = Ʃ∞ m = 1 J0 (k m r )/J1 2 (k m a ) e-k m x , (2) S2 = Ʃ∞ m = 1 J1 (k m r )/k m a J1 2 (k m a ) e-k m x , (3) S3 = Ʃ∞ m = 1 J0 (k m r )/J0 2 (k m a ) e-k m x , (4) S4 = Ʃ∞ m = 1 J0' (k m r )/k m a J0 2 (k m a ) e-k m x , where J0 and J1 denote Bessel functions of orders 0 and 1;k 1 ,k 2 ,k 3 , , are the positive roots of the equation J0 (ka ) = 0;k 1 ,k 2 ,k 3 , , are the positive roots of the equation J1 (ka ) = 0; and, so far as we are concerned,r ,a andx are positive with 0 <r <a . It may be mentioned thata is the radius of the tunnel whiler andx 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 whenx is large; but convergence is slow whenx is small and (in the case of the first and third) is non-existent whenx = 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 whenx 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 ofx andr with coefficients in forms which are fairly well adapted for computation.