G • STATISTICAL THEORIES OF TURBULENCE 

 and 



It is clear that Fi{k) must be even when R{^) is real, and the above 

 equations become the same as Eq. 8-1. 



C,10. Spectral Analysis in Three Dimensions. The one-dimen- 

 sional spectrum, however, does not give an exact representation of the 

 distribution of energy among the scales. Consider a simple harmonic vari- 

 ation with wave number k in a direction making an angle B with the x axis 



Fig. C,10. Diagram illustrating the relationship between one-dimensional 

 and three-dimensional Fourier analysis of a field of turbulence. 



(Fig. C,10). Its period in the x direction would be longer and the wave 

 number in a harmonic analysis in the x direction is 



Kx = K COS 6 (10-1) 



Thus a modified picture is obtained of the energy distribution among the 

 various scales. In the case of isotropic turbulence, as we shall demonstrate 

 below, it is easy to establish the relation between the one-dimensional 

 spectrum Fi{k) and the spectrum function F{k) corresponding to a three- 

 dimensional Fourier analysis. The relation is 



3 ["' dK' 



2L k' 



^iW = ^ / ^3 («'' - •<')F(>^') (10-2) 



or, upon differentiation, 



n^) = iW'Fi'iK) - kF[{k)] (10-3) 



Note that m^ = j;F{K)dK = J^Fi(K)rf/c. 



( 216 > 



