252 



many formulas for W*(x,y,z;t) which represents 

 the pressure induced by an N-bladed propeller moving 

 in infinite space. One of the formulas for W* is 

 given on the assumption of thin blades as follows 

 [jakobs et al. (1972) ]: 



(52), (53), (55), and (56) with the method of Green's 

 function: 



(e) 

 4TrV ^ ' (Q ) = 

 V ^e 



dS 





W* = I W *(x,y,z) e-' 



= vioi7i^%IiJo-;<^''-^°'^^^'""^'^(i), 



q 3n R 



- ^. " (^^' 



where 9^ = ^ (q-1) , (47) 



with j = imaginary unit, fi = angular speed of the 

 propeller, Sp = lifting surface of propeller, L^ ' = 

 pressure jump across Sp, {C',P,9o) = point on Sp, 

 np = normal unit vector at Sp, and R = distance 

 between {£.',p,Sq) and (x,y,z). Kence, using W* and 

 the method of the mirror image , we can obtain a W 

 which satisfies the boundary conditions (43) and 

 (44). Then, we can write W as follows: 



=v£o "v'^'^'^' ^ 



jvfit 



(48) 



Next, let us consider V. Then, we assume that 

 V and ^p can be developed in correspondence with the 

 development (48) as indicated below: 



* 



r„il; (x,y,z)e 



jvS^t 



^l^V^{K,y.z)e 



jvflt 



Hence, using (42) and (44) , we have 



3v^(e) 



On S 



3w 



V 



3n 



(49) 

 (50) 



(51) 



+ V. 



(e) _ v'i')-l- G (Q;Q 



(57) 



where Qg denotes a point outside Sg, Q denotes a 

 point on Sg, and the suffix, (i) , means the inside 

 of the hull surface. Then, we seek a solution of 

 V^(i) which satisfies the boundary conditon on Sg 

 as follows : 



8V 



(i) 



3n 



on S 



3W 



V 



3^ 



(58) 



on S 



Then, using (53) and (58) , we can obtain an internal 

 solution as follows: 



(i) 



(59) 



On S 



Therefore, by substituting (51), (58), and (59) 

 into (57), the external solution v'^'must satisfy 



4TTv(e) (Q ) = dS (v'^' + W ) ^ I G (Q;Qe 



V e I V V V y Q 3 nQ V V e 



S (60) 



Finally, we have the following equation by adding 

 4irW (Q ) to both sides of the Eq. (60) : 



4^ 4;^e)(Q^) = 4^ „^(o^) + I dS *,i^'(Q)3^y(Q;Qe) 



S (61) 



V 



'^' -y when /x^ + y^ + z^ ^ 



(52) 



where the suffix, (e) , means the outside of the 

 hull surface. In the same manner, from Eq. (43) 

 we have 



In this equation, letting Qq be the limit of Q on 



S , we can get 

 s 



^(e) (QJ 



1_ 

 2Tr 



dS Me) (Q)j-— G (Q;Q ) 

 V dng V O 



/ 3V 



V V 

 — + Ko + Ki 



3v 



+ K, V 



3y 



where 



Ko = ^ , Ki 



3x 



2jvt^ 

 U 



y=0 



, K2 = 



(53) 



(54) 



Now, we suppose that we know the functions G (5, 



n,C;x,y,z) (v=0,l,2, ) such that the G are 



harmonic functions for ri<0 except at (x,y,z) where 

 G have a singularity of first order, and G satisfy 

 the boundary conditions: 



d'^G 3g 3g 



V V V 

 + Ko + Ki- — 



+ K2G. 



3C^ 



3n 



35 



V 



, 



n=o 



when /x^ + y^ + z^ ->• ■» 



(55) 



(56) 



Then, we can obtain the following equation by using 



= 2W^(Qo) 



(62) 



because the singularity of first order exists in 

 the G . This ijj'^' (Q ) is exactly the change of 

 pressure on the hull surface caused by the propeller 

 which we intend to calculate. If W and G can be 

 given a priori, Eq. (62) can be considered to be 

 an integral equation for the unknown ^^{e) {Q^) . Thus, 

 the problem of calculating the change of pressure 

 on a hull surface caused by a propeller changes to 

 the problem of solving an integral equation. 



Time-Independent Change of Pressure On the Hull 



Now, we proceed to give Wo and Go for a steady case 

 (v=0) . Go (C,n,C;x,y ,z) can be written as follows 

 based on a wave making theory: 



