Coupled Ship Motions 



Pi(0, - ^ . (3-21) 



From these equations combined with Eqs. (3.17) and (3.18) we infer that 



Y'^(cr) - - 1 S. a log a, ct -> (3.22) 



° 77 g 



Y'^(cr) - -S. a , a -> . (3.23) 



° g 



Incorporating these results into the integral of Eq. (3.19) we find, through 

 integrating by parts, the following leading terms at large t: 



CO A 



I Y'^(cr) COS at da- , t -* oo (3.24) 



Y'^(cr) sin at da - — - , 



J ° ^2 



gt 



(3.25) 



whence 



YoCt) = - I YoCO) ^ - - ^ YoCO) ^ , t ^ CO (3.26) 



where a = Ay2 is the half- width of the cylindrical body in the free surface. 



Equation (3.26) gives the large time behaviour of the heave displacement of 

 a cylindrical body released at zero velocity from a position of hydrostatic dis- 

 equilibrium. The expression on the far right of this equation agrees with that 

 obtained by Ursell [6] for a half- submerged circular cylinder of radius a. How- 

 ever we now see that this expression is valid for cylinders of arbitrary cross 

 section having a width 2a in the free surface. 



Three-Dimensional Bodies 



For three-dimensional bodies, we have [5] 



1 A,^ , , (3.27) 



K^ (3.28) 



PdC^) ^ ^^' ^-^■ 



Incorporating these results into Eqs. (3.17) through (3.19), we find after a 

 number of integrations by parts in (3.19) 



419 



