Linearized Potential Flow Theory for ACVs in a Seaway 



From the symmetry of the motion with displacements in the 

 longitudinal plane it may be concluded that the lateral force Y 

 is zero. This could in fact, be established as follows : C 



Since 



// P * ^ 



Y p 



*o 

 and p may be assumed to be an even function of y as indeed p 



no doubt is, as may be seen from (I. 1) and s is also an even func- 

 tion of y (i. e. df/dy an odd function), there are surface element 

 pairs on oppposite sides of the longitudinal axis where the integrand 

 is of equal but opposite sign. The surface integral therefore vanishes 

 on account of the symmetry of the domain of integration about the 

 longitudinal axis. 



We are therefore left only with the longitudinal force and 

 the vertical force. 



We may substitute for cj> from (2. 3) and for p from (I. l) 

 * S ^1000 + ^0100 +5/3 *1100 + Re - el0t( ^<i>ioiO + 



+ ^*0110 + c *oooi + 6e *iooi + **oioi 



and 



p = #p + /3 2 r p + bfir p + ^ a e lat r p + 



p s p o v 010 p o M 100 P o 001 P o 



XX X 



+ 5p«e (r s +r s +r p+r r P ) 



v 100 001 001 100 101 P o 100*001 P o ' 



X XX 



A 2 ifft ,- - - - - . 



+ P oc e (r s +r s +r p +r r o ) 



v 010 001 001 010 011 p o 010 001 P o ' 



X XX 



2 - 

 +5p (r s +r s +r p+r r p) 



V 100 010 010 100 110 P o 100 010 P o ' 



x x 



+ 5 ^ (; ioo s ioo + ^" ; ioo 2p o } 



XX 



The integrals in (II. l) are to be taken over the instantaneous position 



175 



