Putting 



*' =*1 + U? w8^ (280) 



we have the boundary condition at the free surface 



ft-f^'-O (28!) 



while the boundary condition on the hull surface can be written as 



!£" " "i (It +U i) <V"*S> " «V+V afr (^) (282) 



If we omit the second term, we get the usual form of boundary conditions 

 for a heaving cylinder with vertical velocity 



V (f^+u|^j : «\,-,, ) (V85) 



The amplitude of the vertical oscillation is then 



Z A = |v|/u> (284) 



If we write the amplitude of waves generated by the cylinder as £ and 

 define a two-dimensional pulsating source of strength, a e 1 , which 

 generates the radiating wave of amplitude £ there is the relation for 

 the amplitude ratio A as 



k -\'^r <285) 



We have the relation between the density of the three-dimensional source 

 and that of the two-dimensional source as 



98 



