Landweber 



and similarly, from (19), (20), and (21), 



curl (vX w) = e|[j^96 (u^ " ^^^ " "3^ ^^^ " ^^^ " ^4^^^ " u^)] 



^"^aLI^ <^^ - ^^^ - tt; -fe <^^ - ^^) + ^3^^^ - ^^J 



+ K,(we - uU - K2(v; - wti)] (33) 



For the components of curl curl to in the boundary layer we obtain, 

 neglecting small terms, 



curl curl Zi • e, = - |!| - (K, + K,) |i (34) 



curl curl ;; . -i^ = - 1^, - (K, + K^) |^ (35) 



curl curl u • e^ = K, |i + K^l^ + K^K^i + K^K^il . (36) 

 Substituting these results Into (14) yields the vorticity equations 



If=K;^<-^-^) -|^(-e-u;) - K^(w|-uU ^v[|^MK3+K,)|f ](37) 

 |?=^(v^-w.l) .3L^(u,-ve) +K3(v;-w,) +v[|J^ MK3+K^)|3] (38) 



|i =_L^ (we-uC) -^J^ (v^-wti) + K,(we-uU - K2(v;-wti). (39) 



Here u and v are given in terms of the vorticity by (28) and (29); 

 w can then be obtained from the continuity equation. 



In order to start the calculation, conditions at time t = 

 are required. This may be taken to be the vortex sheet for irro- 

 tational flow about the hull In a uniform stream, since this gives the 

 initial vorticity distribution when the body is impulsively accelerated 

 from rest to its constant speed. A procedure for determining this 

 vortex sheet is developed in the following section. 



464 



