IV -26 



We observe that the spectrum in the isotropic case indeed depends only on (k). 

 The integration over 9 can be performed and yields: 



S (k)= -4 



U 4TTk 



/dr 



2 I — (kr)sin(kr) R^(r)<u^> 

 •- o 



4TTk^ 



(IV- 60) 



In the above, we have defined a function E^, which is very closely related to the 

 spectrum S)j and which, in fact, has the property that: 



S , (k) dk = S (k) (4TTk^)dk = E ,(k) dk (IV- 60a) 



In other words, if we wish, in the isotropic case, to deal with a spectrum relative 

 to a wave number interval rather than relative to a wave number volume element 

 (as we must in the nonisotropic case) we should use E^^ as the isotropic spectrum. 

 Henceforth, whenever we refer to the spectrum under conditions of isotropy, we 

 shall always mean E^, , defined according to 



4" 



E (k) - 2 p^ (kr) sin (kr) R^,(r) <u''> (IV-61a) 



The inverse relation to the above may be found by using a spherical coordinate 

 system to permit the integration of (IV- 57a); proceeding in a manner entirely 

 analogous to the above, we obtain: 



J 



<u^>R (r)= I dk £1^1^ E (k) ' (IV-61b) 



u I kr u 



A number of generally valid statements may be deduced from this pair of transforms 

 relating the spectrum and the covariance function. We expect the correlation func- • 

 tion essentially to vanish for values of r greater than some large distance R . 

 Similarly, we expect the spectrum (Eyj) to be cut off at some high frequency (K) 

 corresponding to spatial gradients so steep that the heat conductivity of the water 

 will destroy them. In other words, both the correlation function and the spectrum 

 exhibit cutoffs at large distances and large wave numbers, respectively. Consider 

 now (IV- 61a) for wave numbers much less than _, i.e., wave numbers correspond- 

 ing to spatial wave lengths greater than the maximum correlation distance. It 

 follows that in this case (kr),will always be much less than 1, and the sine in (IV- 61a) 

 may be approximated by its argument. Thus, we find that near k =^ the spectrum 

 has the form: „ 



I 



E.^(k) ---^^-^^ \ dr r^ R(r) (IV-62) 



Arthur a.HittleJitf. 



S-7001-0307 



