TM No. 377 



equation (111-31) may be written as 



oo 



2Ti£r, 



*-&' LWc ■ 5-p . w 



•UUI 



^> C-f) is the cross-spectrum of the functions u(t) and w(t). 



The complex covariance function may be described in terms of the Fourier 

 transform of its even function (real part) and odd function (imaginary part) 

 as ; 



$^)=[cuu,M-JQ^W] . 



(IH-33) 



The in-phase spectral energy characteristics are given by the value, of the 

 f unction Cuw ("f) s termed the co-spectra; whereas the out-of -phase energy dis- 

 tribution is given by the function QUvjCf); termed the quadrature or qua~speetra<= 



The dimensionless quantity B,- > given hj 



is termed the coherence » This function of frequency is a measure of the phase 

 relation between the two records. The phase difference between the two records 

 u(t) and w(t) is given by- 



6> - ^ Wf » 7 — - (in-35) 



At regions of the spectrum where strong correlation exists, $ -*\ » Where (h) is 

 randomly distributed, f?*-»c . Clearly 9 the reliability of the computed© de- 

 pends upon the coherence between the spectra of the time series. The resemblance 

 between the coherence function and the linear correlation coefficient is note- 

 worthy in that it is a type of correlation coefficient of the spectra of u and w. 



Computation Formulas for Computer Proces sing =- The relations in the pre- 

 ceding section are idealised classical descriptions of the various statistical 

 parameters to be derived from the wave data* In the various integrals occurring 

 in equations (lll<=ll) through (111=34),, u(t) and w(t) are continuous functions. 

 To utilize numerical methods of analysis <, one must first digitize the data over 

 equally spaced time intervals (as already described). In the same manner, the 

 mathematical formulation must be made amenable to digital computation by conver- 

 sion to the appropriate formulas. These formulas 9 which are described by Tukey 

 (19^9), eliminate the difficulties that would occur if an attempt were made to 



5& 



