TM NO. 377 



The analysis of the time series pair involves the usual auto-covarianee 

 and energy spectrum estimates delineated in the previous section on single 

 time series analysis. The cross -co variance functions or simply the covariane.es 

 are defined as : 



<tu, ft) • t-T- £ \ *« wc* r) j* (BM5) 



and 



4UCT) > ££, f (*Ww**-f).te ; (=«*) 



V2_ 



where -co < T £ oo Q The covariance functions are even (symmetric in 7* )l 

 thus, changing ..he order of 'W.(t) and w/(t) changes the value. However,, <J) um »£t) 

 and c£> W(A< (r) can he related by: 



The term (Jj^Cft) has special significance when the functions in the covariance 

 represent orthogonal velocity components (u and w) measured at a point in the 

 plane: 



Here the covariance at zero lag is proportional to the well-known Reynolds 

 stress c Further discussion of the physical meaning of this term is pre= 

 sented later. 



The ordinary linear correlation coefficient between two functions u(t) 

 and w(t) is expressed as: 



r - - ~ , - (ni-29) 



As hefore. the covariance function is expressible as a Fourier integral: 



-eo 



<L to s 



o<5 



, x -zTrif-T nri'-Pt 



arl^Ltoe Jr <s o/f . (in-30) 



L 



-0Q — tO 



Defining a function 5? ffl so that 



-loo 



(IH-31) 



57 



