C,10 • SPECTRAL ANALYSIS IN THREE DIMENSIONS 



The analysis of the y component of the motion in the x direction leads 

 to a spectrum 



F,(k) =ll^^s (x' + ^"M'c') (10-4) 



by combining Eq. 10-2 and 10-4, we obtain, after a little calculation, 



(10-5) 





F,{k)+If^{k) 



This relation is more convenient for obtaining F{k) from experimental 

 data. It is numerically more accurate than Eq. 10-3 since only one differ- 

 entiation is involved. 



To estabhsh the relations (Eq. 10-2 and 10-4) let us write^ the three- 

 dimensional Fourier analysis of the velocity in the following form : 



Ui = Y ^4i(/c;)e«^'"-"^ (10-6) 



Then, for a wave in the direction of the vector kj, the equation of con- 

 tinuity gives 



KjAj = (10-7) 



This means that all the motion associated with the vector wave number 

 Kj must be perpendicular to this vector. 



Consider now the contribution to the spectrum of a Fourier analysis 

 in the x direction of a component of turbulent motion with vector wave 

 number kj. In the first place, the motion appears to have a space frequency 

 Kx defined by Eq. 10-1. Secondly, the motion has in general all three com- 

 ponents. ,The X component is (cf. Eq. 10-2 and Fig. C,10).] 



ui{kj) = —A{kj) cos 4> sin d, A^ = AiAi 



where d is the angle between kj and the x axis, and is the angle which 

 the velocity vector Ai makes with the plane containing kj and the x axis. 

 Thus, averaging over the angle 4>, we have (cf. Eq. 10-1) 



M^y)!^ = |A(k,)1^^ = \A{KM'l(i-f) 



Consider now a distribution of energy in the k space. Let the total 

 kinetic energy per unit mass and per unit volume of the /c space be ^^(kj) ; 

 i.e. ^^{Kj)dKidK2dK3 is the energy contained in the range Ki, Ki -f dKi. To 

 obtain the energy per unit mass ^Fi{ki) lying between /ci, /ci -1- dKi and 

 associated with one component of the motion, one must multiply this ex- 

 pression with the factor (1 — Kf//c^)/2 and then integrate for all values of 



* The reasoning here is essentially that used by Heisenberg [23]. 



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