Derm 



The approximation h = oO is valid only for type 11 and 

 14 buoys . For type 1 and 15 buoys , this assumption is very .coar- 

 se but it avoids intricate computations . We must recall that N<n<p 

 is experimentally determined (damping tests) . 



We assume that the wave elevation J (t) is a normal , 

 strictly stationary random function . We also assume that the sea 

 is monodirectional and that its spectral density is given by the mo - 

 dified Pierson-Moskowitz formula f 22 1 , namely . 



2 -5 



Sff(f) = 0.11H T ( T f ) exp 



JJ V V V 



o,44 (Tvf) 



It follows that signal s (t) is also gaussian and strictly 



stationary 



III - ROLLING CRITERION 



With the above hypothesis the rolling motion equation 

 takes the form (see paragraph 7 of section II) 



(III -3. 1) 



d 2 P 

 dT2 



+ £ 



d£? 

 d t- 



15 - 26s (t) 



(p =. o 



Now the coefficient of *P is no longer harmonic but varies 

 stochastically . For any given initial conditions which are determi- 

 nistic for example f\ - P anc \7q i = ®J we ma ke now the assumption 

 that the buoy is rolling if and only if equation (III-3. 1) is unstable 

 in the mean square . By definition , equation (III-3. l) is stable in 

 the mean square if , for any (p : 



< (p (t)>— »0 when t— »+ as 

 where < • > denotes ensemble average . 



Now equation (III- 3 . 1) may be written as 

 (III-3.2) -^ + ! 1 + »i (',)] P y (*!> = ° 



dl,2 



where 



Pi 



(t) A 



P. t 

 2 



Pit) 



■■ / 



1064 



