Ogilvie 



coherencies between ship motions and the forcing waves in long crested waves 

 are an indication of poor experimental design and interpretation, and they are 

 not an indication of the failure of the linear theory.* For short crested waves 

 the coherencies could be low both because of poor experimental design and be- 

 cause they really ought to be low. 



The rapid variation in the cross spectral estimates that are sometimes ob- 

 tained is the first indication that something is amiss. Such rapidly varying 

 cross spectral estimates suggest that the coherency will be low because of the 

 lack of adequate resolution. 



An interesting example of this is given by the study of observations at two 

 points in long crested random waves. Let the long crested random waves be 

 given by Eq. (27). 



](x, t) = I COS f^^^ - wt + ej ^/2S(co)dco 

 J n 



(27) 



By definition the co- and quadrature spectra are given by Eqs. (28) and (29) 

 when (27) is observed at x = and x = L. 



(Z{oS) = cos ^ S(a;) (28) 



(29) 



^2t 



Q(aj) = sin S(c<j) . 



The coherency is one. 



K(a)) = = 1 . (30) 



Pierson and Dalzell (1960) studied two records that were taken in long 

 crested waves. One record was about five feet away from the other record. 

 The spectra and cross spectra were computed. Figure 1 shows the estimated 

 spectrum as indicated by the histogramic presentation. One number centered at 

 the center of each step in the histogram was the number that describe the par- 

 ticular spectrum. There were two spectra and the circle and the diamond indi- 

 cate how these two spectral estimates differed over a separation of five feet in 

 long crested waves. The co- and quadrature spectra were also computed. 

 These spectra are shown in Fig. 2 by the black diamonds. They look fairly rea- 

 sonable in comparison with Eqs. (28) and (29). 



However, the computation of the coherency led to the results shown by the 

 black diamonds in Fig. 3. The coherency is high near a value of h = 7 and it 

 falls of steadily to values of 0.5, 0.4, 0.3 for larger values of h. This is quite 

 disturbing as certainly the model proposed by Eq. (27) for the free surface and 



*In most cases. 



88 



