negligible for X = 1/2. Jin Wu's measurements [7] indicate that the tur- 

 bulent stresses would contribute about 2 percent to the calculated drag 

 with A = 0; but his actual value was A = 0.5, with which the turbulence 



terms became negligible. We shall assume that the turbulence stresses in 



2 2 

 (87) may be neglected. We shall also neglect the terms - v ~ ^ + A 



2 2 

 (v +w ) since these are small and partly cancel each other. 



Still unknown are the terms u and u in (87). For estimating \i , 

 we shall assume that u^ depends upon z alone, and is given by u = u (z) , 

 the measured value of u at the wake boundary T. According to the defini- 

 tion of u as a mean of u^ in ¥' , we observe that the mean is weighted by 

 the value of the source strength y which, together with u , diminishes to 



zero as X -»- °°. This suggests that the values u = and u^ = u at A 

 can be used to obtain bounds for the drag formula, the "true" value lying 

 closer to the bound given by u., = u . Because u., occurs both in the 

 integrand and in the denominator of the expression for the drag, it is not 

 immediately evident which of the two bounds is the larger. Denoting these 

 bounds by D^ when u = and D„ when u = u » and applying the afore- 

 mentioned approximations, we obtain from (87), 



D, = -^ [2g(H^-H )+u^2-u^^] dS (88) 



I 



m^ 



and, with u„ + u - 2u„ replaced by u - u^ in the last term, 

 E m E m E 



°2=-^ f t2g(Ho-H^)-(uE-uj'] 

 l-u„/U^ J. 



dS (89) 



~ u m am 



"E' "0 "A 



The latter form was given by Tzou and Landweber [8]. 



That D^ < D„ is indicated by the following argument. Since the source 

 strength y represents the displacement effect of the boundary layer and 

 wake, it is a positive quantity of total strength given by (78). Conse- 

 quently, according to (77), u, cannot be of one sign throughout BLW. Over 



33 



