m(P)=^V'(s,0)N^£(rQU6^) (33) 



a generalization of (29) . 



In order to improve upon the approximate source distribution (33) , we 

 shall now consider an axial source distribution M(x) . For irrotational 

 flow about the given body, r = f(x), the modified Munk formula of Landweber 

 [3] gives the approximate solution, in terms of the free-stream velocity U , 



Mq(x) =\ (1+kp Uq^ f2(x) (34) 



where k-. is the longitudinal added-mass coefficient. For the thickened 

 body, with r = r^ (x) = f(x) + 6 sec Y> formula (34) gives 



M^(x) =i (1+k^) U^^r^^x) 



=. Mq + I (1+k^) Uq ^ (f6^ sec Y + I 5^^ sec^ y) (35) 



If the alternative displacement thickness 6* is used, the same expression 

 is applicable, with 6^ replaced by 6*. 



Axisymmetric Flow - Second Approximation 



We shall now derive a second-order approximation for an axial source 

 distribution for slender bodies. The given profile is r = r (x) = f (x) , 

 the edge of the boundary layer is r = g(x), and the displacement-thickness 

 profile is r = h(x) . The velocity components in the x- and r-directions 

 for the equivalent irrotational model are u(x,r), v(x,r). 



12 



