Dern 



Finally , the heave equation is 



(II-2.3) (m + m zz )-^- + N zz -|^- + p g S(0)[Z + f(s) ] 



dt 



= Pg S(0) [l - C p kHQ Q (k)]/(t) 



where f (s) is given by (II-2. 2) for type 1 and where 



s (t) = Z (t) -S (t) 



This equation completely determine the buoy heaving beha- 

 viour if the values of m and N are known 



zz zz 



4 3 



m = — pR from ( I" 5 "I p. 200) 



zz 3 o 



N 



[ S (°jf f h - C kHQo(k)] from [l] 



» C p H 



Thus , in this theory , the added mass is independent of 

 the frequency but the damping is frequency dependent . 



The heave equation in J.N.Newman's theory is obtained by 

 setting f (s) = and m = .In fact , in this paper , we mean by 

 J.N.Newman's theory , the theory where f (s)^0 but where m is 

 given by the expression above . 



Ill- EQUATION OF SURGE AND PITCH MOTIONS - 



Equation (II- 1 . l) becomes for coupled surge and pitch motions : 



x d 2 x d 2 e ^ T dx M ae , r , 



(m+m — + m . + N + N = 2p / aox d v 



xx , 2 x0 ,2 xx x9 J 



dt dt d t d t *< 



d 2 6 d 2 X d9 dX r /■__ 



(I +J ) ; +m ft — T + N__— +N — +PJ (GP A "g) &V~=2 P J(GPZZxY 



v yy yy d t 2 xe dt 2 ee dt xe dt -> „ a m ^ a 



where ( ) x designates the projection on the Ox-axis . 



1036 



