Ogilvie 



pCO /» CD ^00 



»'-CD ''-00 •^-m 



X e-J-C'^--) e[0^(/x)-|c/)^(/x)|-0^(i^)-|0^(v)|] d/x 

 - S (c.) - 2R [/3h7^-S (o;)] 



+ /32|H (a;)!' S. . I . .(OJ) • (3.10) 



Now g(t) is the compulsory moment that comes from the waves, and, so, if the 

 waves are Gaussian, g(t) is also a Gaussian process. From Eq. (3.4), 4^^(t) is 

 also Gaussian, and 



^o(t) 



J hg(t- s) g(s)ds . (3.11) 



This shows <P^(t) is also a Gaussian process. 



Here in order to calculate (3.10), we have to evaluate the.expected value of 



[0^(a) -cpjh) • |0„(b)|] and [c/)"„(m) • |0o(m)I • ^^(y^ ' l<^'o(^)l] concerning two 

 Gaussian process 0o(t) and 0o(t). <Po(t) and 0^(t) are correlated to each 

 other, of course by the correlation coefficient p . 



Here, two Gaussian variables i and v, connected by the correlation coef- 

 ficient p are assumed. The two-variable Gaussian probability distribution 

 function is 



p( ^- v) = — HZZ" ^'^P 



2TTa^a^ yi- p^ 



i V- 



2^/0-^2(1-/0') 



(3.12) 



It is necessary to evaliiate E[^, tj-I?]!] and e[^-|^| -t?-!?]! ] by means of this 

 distribution function. The absolute values are what make the problem somewhat 

 intricate. These evaluations could not be found in any text books and papers, 

 and so were calculated by this author. The results are 



E[^-^.hl] = /|-P-f-< = /l-R^. ,(-)-, (3.13) 



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