..wwimvw-wRVTorwsimir'unwnxTO^ 



where oj Is the two-dimensional strength of sources and sinks, (a,b) the 

 location of the vortex, q(x ,z ) a two-dimensional source point, and G„ is the 

 two-dimensional Green function given by Wehausen and Laitone as 



2 (P, ( l) - Re {in [x-x c + i(z-z )] + ln[x-x + i(z+z )] 



(26) 



- e" lk t x - x o * 1 < 2+2 o> ] d u _ 2 Ttl e" ik of x " x o + 1 <« + «o)]i 

 + 2pv / k^~k~ ' 



In Equations (2 5) and (26), Re denotes the real part of a complex quantity and 

 pv the principal-value integral. The vortex strength T in Equation (21) will be 

 determined with the Kutta condition. The pressure around each section of the 

 control plane can be obtained from Equation (16) by substituting <Ji f or a . ^he 

 force and moment acting on the section can be computed with an equation of the 

 saiae form as Equation (15). 



By substituting Equation (24) into either Equation (22) or Equation (23), 

 the unknown strength of the sources and sinks for the two-dimensional case can 

 be determined from the following equation 



— — / Oi(q)G,(p,q)cll(q) - Un. for i - 1 (27) 



2itSn c l 



- - -r-=- for i-3. 

 on 



When the submersible is moving near a wall, the linear free-surface 

 conditions, Equations (3) and (19) are no longer valid. Instead, we should use 

 the wall boundary conditions given by 



•^ - at z - (28) 



dz 



and 



