or 



where 



A General Theory for Marine Propellers 



u(x,r,(p) - u(x,r,0)g + iu ( x , r 



u(x,r,0)3 = J/[APi(P,t5)3Ki^ - APi(p,0)i^Kib] dpd0 



(67) 



(68a) 



u(x,r,0)j^ = J|[Api(p,^)^Kij, + Api(p,d?)t^KiJ ApAO 



Likewise we have 



with 



and 



with 



v(x,r,0) =v(x,r,0)g + iv(x,r,. 



(68b) 



(69) 



s 

 v(x,r,^)j^ = |JtApi(p,^)3K„b + APi(p,5)bK^3] dH5 . : . ., (70b) 



^r- ■• 



(x,r,0) = w(x,r,0)^ + iu(x,r,0). , ^ (71) 



^('^■r,^)^ = J|[Api(p,^)^K„3 - Ap.(p,0)j^K„b] dpde ' C'Sa) 



(x,r,^)t^ = JJ[Api(p,^)^K„b + ^Pi(P-^)bKna] dpde 



(72b) 



In the steady case, the expressions for the velocity components can, of 

 course, be reduced. For instance, considering the expressions for u(x,r,0) in 

 the steady case, Apj(p,t?)^ and Kj,^ are zero. Hence u(x,r,^.)(^ is zero. Also 

 since q is zero, Kj^ is reduced to Kj and Eq. (68a) is reduced to Eq. (19a) with 

 L(f,p,i9) equal to Api(p,f^)g. As a result in the steady case, there is only one 

 surface integration to perform instead of four as in the case of periodic loading. 



109 



