TM Noo 377 



Since the function c£> (t) is an even function of T * equation (111=17) 

 can be written as s 



$ictf) s ^ \ <k.(r) COS2TTfTcJT . (111-20) 



The inverse transform relationship is given by; 



-'o 



For T-O the relation in equation (111-21) becomes; 



<kc«o =^cc^2 jau*)Jf • (m " 22) 



Thus, the integral of the energy spectrum over all frequencies is equivalent 

 to the total variance of the velocity function., The wave or current motions 

 under examination usually occur within a defined range of frequencies j hence , 

 instead of infinity for the upper frequency limit, an appropriate AT can be 

 chosen which will define a reasonable SJyquist or cutoff frequency* Equation 

 (lll=22) is more realistically represented by; 



4>u.C0)= 2. { $ u 0f)df . (^ 2 3) 



Equation ( 111=21 ) represents the total energy of the record,, <JM-()cjl"f"is 

 the contribution to this total energy from a frequency band of -f _y z d.f. to 

 -f +■ '/t J-f o The function ^ u (&) is the energy density or spectral density* and 

 is in units of the square of the quantity u(t) per unit frequency „ If <$ u Cf) 

 is constant over some range of frequencies, 



Q.(f) - 5 - CONSTANT F06 f ( <-f <fz . (lH-2^) 



Then all frequencies contribute equally to the energy and the spectrum is termed 

 "white" over this range c 



2o Paired Time Series Analysis „ In this situation two variables, U. (t) 

 and Uj (t), were simultaneously recorded. Aside from the individual auto=spectra, 

 the common statistical properties of the two time series must be studied* By 

 computation of the eovariance, the linear correlation coefficient, and the 

 cross=spectra, one may determine a statistical relationship of one time series 

 to another or, in effect, see if definite common periodic fluctuations exist « 

 The phase relationships of any periodic relationships may also be estimated. 



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