A General Theory for Marine Propellers 



The induced fluid velocity components given by these equations also have a real 

 and an imaginary part. 



To find the expressions for the complex kernel functions for the gth order 

 of the loading harmonic, for instance, we start with the following equation, cor- 

 responding to Eq. (20), 



Instead of Eqs. (28) we obtain the following expressions for the axial, tangential, 

 and radial components of the pressure dipole: 



1=1- e^q^t , (53a) 



. .. " ' m = m • eiq^t , ' " (53b) 



n = n • eiq^t . (53c) 



From Eq. (31) 



= (IRi +mR^ + nR^) e''^°' 



= Meiq^t . (54) 



By replacing 1, m, n, and M by 1, m, n, and M respectively in Eqs. (32) we 

 obtain 



1 r" • o 7 1 3R,M\ 

 K, = ^^ — e^q^t - — + — —)dk, (55a) 



K = ^— f e^^^^f- ^+ ^dX , (55b) 



K = -^— I e'^^n- ^ + ^^]dk . (55c) 



" 477p,n4,.^ \ R3 r5 y 



Since nt ^ e - 4> - X., 



giqQt _ (,og q(y_0_x.) + i sin q (9 - (p - X.) 



- cos q (0 ~ <ij) cos qk. + sin q(d - 4>) sin q\ 



+ i[sin q(6 - (p) cos q\ - cos q (0 - 4>) sinqX] . (56) 



105 



