Here (f) (.r) is the potential due to the propeller which satisfies the 

 condition 



Sf^^(v) ^ |r| ->" (5) 



Inserting this expression for (|)(r) into equation (2) yields an integral 

 equation for the unknown source strengths 



i-^^B^ -hf '^%^ ^(^B>-V jT^ ^^(^B> = ^^-V-t^t+%(^B>^ 



(6) 



"^ IV^B' 



It may be observed that the propeller presents a modified onset flow to the 

 body (right-hand side of equation (6)). Moreover, the change in source 

 strength caused by the presence of the propeller depends only on the com- 

 ponent of induced velocity normal to the body surface. 



A numerical solution of equation (6) is obtained by representing the 

 body surface using planar quadrilateral elements. It is assumed that the 

 source density is constant over each element and the integral equation is 

 replaced by a set of linear algebraic equations. 



L a. C = V. (7) 



J 



where the coefficients, C.., are given by 



C. = 6.. -^ - 7- f n.«V 1^ ^_. I dS. 

 ij iJ 2 4Tr J^, 1 |r^ -r 1 j 



(8) 



th IB. B. 

 element 



and V. is the onset flow evaluated at a selected control point (e.g., cen- 

 troid) of each quadrilateral 



Since the coefficients C.. depend only on the body geometry, the inverse 



-1 -"-J 



matrix C.. need only be derived once for a given hull form. It is then 



