Devn 



When the wave heightvaries while the frequency remains 

 constant the point (p, q) moves along a line parallel to Oq . Thus , 

 when p 7^ 1 , there will be no roll if the wave height is sufficiently 

 small . 



The exact solution of equation (II-5. 3) , taking into 

 account equation ( II- 5 . 1) , has been performed on an analog compu - 

 ter . The results are given in Figures 26 and 27 for the type 1 buoy , 

 and in figures 28 and 29 for type 14 .In figures 27 and 29 , the non , v 

 linear damping term has been taken into account . The coefficient NL/, 

 was obtained by trial and error , and such that the theoretical curve 

 be as close as possible to the experimental points . Unfortunately, 

 the value of N 2 fo^ so obtained corresponds to a value of the coefficient 

 C too large for the type 1 buoy and too small for type 14 . The 

 coefficient N^p^was set equal to zero in figures 26 and 28 for it was 

 not possible to fall back on the experimental points (figures 7 and 8). 

 In particular in the case of buoy type 14 , the curve is a parabola 

 but with a downward concavity as in Kerwin's work . 



It seems therefore that our theory is unsufficient for the 

 prediction of the roll angle . This disagreement may be partly due 

 to the damping coefficient NO)^»pactuall y varying with frequency , 

 while it was assumed (equation II - 5 . 3)to be constant (N^Mty»=N®^f ) . 

 For example for the type 14 buoy , N^p^varies from 1 for f = f fe? to 

 127 for f = 2 iff. 



Conditions for cancelling roll in regular waves - 



We have shown that rolling occurs when 



;n - 7 - 3) \ 4f e + ii A * f *r 



(n-7.3) y At *+—^«i «^ 4 f e 2 - ^T 



Therefore , there is no rolling if and only if 



— ^- = O 



77 2 



which is impossible except for vanishing values of j - . In fact it is 

 necessary to take into account the damping term in pitching . We 

 establish now an approximate condition for no rolling . First we 

 recall that instability occurs in the vicinity of f = 2 f q whenj -. is 



small (see II-7. 3) . Then p~l and q ~ ik. - ZJtk . Unstable 

 solutions of (II -7. 2) grow exponentially as e *■ . Hence, there is 



1056 



