Coupled Ship Motions 

 as Ka -» 0, with 



b, = ^a3, a = yV^ (3.10) 



where A^ is the area in which the body intersects the free surface. Hence, at 

 least formally, 



Pd(^) ^ bja — 

 ^PjC^) -' (2h,aa/g) (3.11) 



:^p;(^) - 2bia/g, 



all as cr ^ 0. 



After integrating by parts several times, we may write 



CD 00 



p'(cT) sin (TtdCT = - — (2bia/g) - "^"^^^ p'(a-) da 



-^Q ^ t^ ' Jq t^ Bct3 d 



t3 g 2t 



= -^'x^ + o(i/t3) (3.12) 



as t -»co. Therefore, from Eq. (3.6) the heave force exerted by an arbitrary 

 body is 



Inl-^ ±\ ^ _ 2pA£ (3^13) 



F(t) ^-^p\- 



^ t^ / TTgt 



as t ^00. We see that this force exerted by the body on the fluid is upward if the 

 S -function acceleration is downward. 



We will now find the heave displacement, for large time, of a body released 

 at zero velocity from hydrostatic disequilibrium. The equation of motion, for 

 an arbitrary surface-piercing body, is 



t 



My„(t) = - pgA^yJt) -J F( t - r) y^(-r) dr 



(3.14) 



22] -249 O - 66 - 28 



417 



