Newman and Tuck 



[(x-^)2 + (y-77)2 + Cz-0']" + [(x-O2 + (y-r,)2 + (z+0']'''', (2-4) 



^1 = ^Kp ^eX(-^)j„(k[(x-a^+(y-^)^]''') 



(2.5) 



The contour of integration in the integral for G j is indented below the singu- 

 larity k = K, in order to satisfy the radiation condition of outgoing waves at 

 infinity. Physically the Green's function G represents the potential of an oscil- 

 latory source, located beneath the free surface at the point x=f, 7=77, z=^; 

 the function G^ is the elementary source function l/R plus its image above the 

 free surface, and the function Gj represents the necessary correction to account 

 for free-surface effects. 



The above statement of the problem is exact, and Eq. (2.2) can be regarded 

 as an integral equation for cj:,^. If the body is slender, however, major reductions 

 can be affected. It can be shown that the term 0g(3Gj/Bn) is small compared to 

 the remainder of the integrand, by a factor l + 0( e log e) , and the surface inte- 

 gral of the term Gi(3<;6g/3n) can, to the same degree of accuracy, be reduced to 

 a line integral over the length. The resulting integral equation, for points (x,y,z) 

 in the near field (i.e., a distance 0(e) from the ship), is then 



4r^jJ 



30g BGq \ (FS) (o P.\ 



-3^-^B-3^°jdS. f^ \x). (2.6) 



where 



( FS) 

 f (X) 



^1^ G,(x,o, 0.^,0,0) 1^1^ ^d-e^d^ 



^ I Gi(x,0, 0,^,0,0) Q(f)d^. (2.7) 



4 



'L 



Since G^ is the Green's function for the rigid free surface problem, it can be 

 shown that (2.6) is equivalent to 



( WALL) ( FS) (n o\ 



Thus, as stated in Eq. (1.1), we can express the velocity potential explicitly in 

 terms of the solution of the corresponding wall problem plus a fvinction f^'^^^Cx) 

 containing the free surface effects. (The f^'^^Vx) of (2.8) is in the form (1.2) 

 with - (1/47t)Gj for the kernel K.) 



It is now a straightforward matter to find the hydrodynamic forces due to 

 the disturbance of the fluid by the body. From Bernoulli's equation the linear- 

 ized hydrodynamic pressure is 



140 



