Unstable Motion of Free Spar Buoys in Waves 



(III - 3 . 3) 



S 4 



4 



i- 



co ^ - 



2 6 



7_id_: 



4 



s (t) 



Equation (III- 3 . 2) has been studied by G. F. Carrier I 21 J 

 who has given an expression for<pi ( t, ) > which is valid under 

 certain conditions (see also Keller |_ Z 3 J ) . Carrier's expression and 

 relations (III-3.3) permit us to give the expression o£<p (t) > by 

 the following assertion : 



Assertion - If , in equation (III-3. 1) , the coefficients fl , & , b are 

 constants and if s (t) is a gaussian white noise , then , the asympto- 

 tic expression for < p ^ (t) > is 



t > 



P 



i - 



^2 



Sss ( 2 f p ) 



where S (f) is the spectral density of s (t) and i<p 

 ss r 



— V^ 



p' 



2TT 4 



We note that Carrier's theorem is obtained by setting P =0 



When s (t) is not a gaussian white noise this asymptotic 

 expression does not hold any longer . Nevertheless , Carrier has 

 shown that this formula is a good approximation in the following 

 cases : 



a/ s(t) = s cos ( 4TT iip t + 6 ) where E is a random 

 variable distributed uniformly over the interval (0, 2 7T) 



b/ s (t) is a gaussian process with an auto -correlation 

 function given by 



R (T ) = 2 k e 



- k 



(k>0) 



1065 



