TM No* 377 



where T is the time lag at which the point-for-point self -multiplication of 

 the time series values is carried out. From equation (111-14), for zero lag 

 (T*o): 



4Uo} - tful 2 -- ja'Mj 1 ". (111-15) 



or s stated simply, the auto-covariance function at zero lag is the variance 

 of the function. The auto-covariance function, when normalized by dividing 

 it by <tu(o)^ is often called the autocorrelation function (see Kinsman, 19&5)* 

 Note that: 



4Ut)= 4L(r r ) > (111-16) 



or the auto-covariance is an even function of the lag T « 



Taking the Fourier transform of the auto-covariance produces the spectral 

 function given by; 



The spectrum may actually be defined as ; 



iW4Uwe M, ' f V . (in-i7) 



Jr.. 



This is because the basic function u(t) is defined by; 



t/W = 





(111-19) 



This provides information only over the period ~ T /2 to + ^/2- However, in 

 view of the assumptions regarding stationariness, one may interchange the 

 limits, as with equations (III-17) and (III-18), and not change the meaning 

 of the spectral function, The function in equation (lll"l8) is called the 

 "power spectra" by engineers, because the variance is often referred to voltage 

 squared per unit resistance, which is proportional to power dissipated. Since 

 this study is generally concerned with the variances of velocity components 

 that are proportional to kinetic energy, the "energy spectra" will be used 

 throughout the discussion of the statistics of wave or current motions. 



55 



