Preston and Lighthill give the solution without hesitation. Their result is 



ZTT ' 2tt ds 1 



for which the argument given by Lighthill is : "This additional outflow is 

 exactly 'as if the irrotational flow around the body were supplemented by 

 the effect of a surface distribution of sources, whose strength ... is ..." 

 Equation (3) does not seem as obvious to the present writer. In order 

 to derive it, one must assume that the velocity U(s) is undisturbed by the 

 distribution m(s) , an assumption which may not be consistent with the 

 uniquely defined solution of the Neumann problem. According to this assump- 

 tion, the body surface is an equipotential for the distribution m(s) (be- 

 cause the tangential velocity due to m(s) is zero) . This immediately yields 



^'' = 



in(s) = ^ V'(s,OJ 



since V'(s,0 ) = on the interior side of the equipotential surface and 

 V'(s,0 ) - V'(s,0~) = 2TTm(s). Then, from the Taylor expression V'(s,6) = 



-^ — j ^ + ... ■=. V'(s,0 ), we obtain the approximate solu- 

 tion (3). 



The special case of a flat plate, (even in a nonzero pressure 

 gradient), however, confirms the approximate solution (3) without requiring 

 the equipotential assumption. Since the plate is the limiting case of a 



<=E 



r.' 



T 



m(s) 



m(s) 

 Figure 1 



very thin body, the distributions m(s) are present on both sides. Applica- 

 tion of the Gauss flux theorem (the usual proof) immediately gives Equa- 

 tion (3) . 



