(19) 



Theory of Unsteady Propeller Forces 

 and the mean surface Sp is represented from Eqs. (5) and (6) by 



6 = ^m(^) + ^(^") V + K' ■ "; (18) 



where 



-l<v<l, ^g<^<l, ~e{^) > , k=l, 2, N. 



From Eqs. (7) and (8) the mean surface Sj^ is expressed as 



i ^ i^ - VqS , l^ ^ ^m('^i) + C (t7j) u , T? = 77j, z* = z* = z|(77j,u) % , 



where 



-l<v<l, 77p<T?j<77^, l{rj^)>0. Cm(^i) - ^(^i) > . 



Since the thickness at the leading and trailing edges of the propeller blades and 

 the rudder are zero, we get 



• ■ t* (^, ±1) = , t|(77j, ±1) = , (20) 



and using Eqs. (15), Eq. (9) is rewritten as 



g^(s^,v,s) = gi(^,v, -S^) . ■ (21) 



Further, the bound vortices must vanish on the trailing edge lines of the pro- 

 peller blades and the rudder, since the extended Kutta's conditions must be sat- 

 isfied there. They are assumed to vanish at the tips and roots of the propeller 

 blades and at the upper and lower ends of the rudder. Thus the following rela- 

 tions are obtained: 



gk(^.l.s) = gjC^.l, -S^) = 0, gR(77j,l,s) = ,. _■ . . . (22) 



g^(l,V,s) = gk(<fB'^'^) = ^' gR('^u'"'^) = gRC^E'^J'S) = . (23) 



Since the surface s^ is closed, the nondimensional source distributions repre- 

 senting the hull must satisfy the relation ■ 



I d^i y <(^.^i--) d77j = 0. ^24) 



The nondimensional velocity potential 4>* at point (^,i7,z*) or {1,^,6) at 

 nondimensional time s is obtained from Eq. (10) by using the results of the pre- 

 vious paper (8) as follows: 



23 



