TM No. 377 



middle and lower limiting curves „ By using these probability curves, one 

 can estimate the confidence limits of a given spectral curve. In subsequent 

 discussions of particular wave velocity spectra, the 80 percent confidence 

 limits are indicated on certain spectra plots,, 



Classical Equations and Definitions ~ The classical descriptions of the 

 auto-covariance, co variance,, and associated spectral functions are given here 

 for both single and paired time series analysis . For a very complete dis- 

 cussion of these parameters see Lee (1960)0 The approximations of these 

 functions suitable for computational procedures are given in the following 

 section, 



1, Single Time Series Analysis . Assume that the process under study 

 is both stationary and ergodic, and that it can be represented by a variable 

 determined as a single time series u(t). It is assumed also that u(t) is 

 bounded in amplitude but unbounded in time durationo The mean or d=c com° 

 ponent of the time series is given by the time average: 



" r-?oo 



The time variable fluctuating quantity is given by 



f i^c*)** . (111-11) 



^ft|-- v(t)-u • (III " 12) 



The mean square or the variance is given by the usual formulation cf the time 



average 



as} - fed? - Lu-tt)-M] z - !&t]j£®-*frt, ■ < m - 13) 



It will be seen later that when u(t) is given as a particle velocity component, 

 the variance <£?= u? is proportional to the turbulent kinetic energy associated 

 with the u velocity component. 



The auto-covariance function of the process u(t) is given by 2 



sometimes written as 





