Ogilvie 



phase relationship exists and there are other cases where a phase relationship 

 appears to exist. 



An ensemble of such vector processes as defined by Eqs. (12) can be gen- 

 erated by choosing different sets of the e's at random in a large number of 

 partial sums. One can then compute ten different expected values of various 

 time lagged products of different combinations of these motions. 



E [-r]^(t), 7]g(t + T)] E [z(t), z(t + T)] E [0(t), i/;(t + r)] E [0( t) , 0( t + r)] 

 E [v^(t), z(t + T)] E [7)Jt),0(t + T)] E[n^(t),4>(t + r)] 



E [z(t),^(t+r)\ E [z(t),0(t + r)] 



and 



E [p(t),4>(t + T)] . 



One of these expectations as evaluated is Eq. (13): 



E [^e<^t), z(t + T)] = !S(co^,0^) [c^(c^^,e^) COS c^^r + q^(c^^,0^) sin c^^t] 60 ^. (13) 



The cospectrum between the forcing waves and heaving motion is thus given 

 by Eq. (14) as this is the even part of the Fourier transform of Eq. (13). 



C^z(^e) = !^(^e'(^e) C ,( o;^, 0,) d0^ . (14) 



The quadrature spectrum is given by Eq. (15). 



Q^z(^e) = /S(c.^,0^) q^(a;^,^^)d^^. (15) 



It is to be noted that the cospectrum and the quadrature spectrum still in- 

 volve an integration over 0^. The way in which c^(a)^,0^) and q^(c^^,0^) vary 

 as a function of fii^ for a fixed co^ can evidently have a marked effect on the 

 cross spectra. 



A complete analysis yields four spectra, six co- spectra, and six quadrature 

 spectra. The spectrum for the heave, for example, is given by Eq. (16). 



The co-spectrum between heave and the waves and the quadrature spectrum be- 

 tween heave and the waves are given by Eqs. (17) and (18). 



Sz(-e) = I^C-^e'^e) [^ zC^e ■ ^e)] ^^e (1'^) 



The cross spectra between heave and pitch are given by Eqs. (19) and (20). 



84 



