E. J. Skudrzyk 227 
of the velocity fluctuations. The relation between the correlation function and 
the power spectrum is 
+0 +0 +0 
R@=f il b(k) eltka E42 7443 Hence (65) 
bal?) -0 -0O 
The scattering integrals have been represented in cylindrical coordinates. It 
is expedient, therefore, to express the correlation function in cylindrical co- 
ordinates too as 
2 2 277 cos ji 1 1 
R(r) = [fe ffs) d(x) e” HIE Bk dk, (66) 
where @ is the angle between the p vector and the projection of the wave-number 
vector on a plane perpendicular to the & axis and 
G . 
k3=k cos@ Ko =k’ sin®@ Ki =K1 
2 
kK? = «3 +3 Ke = Ke + K? (67) 
The 6 integration can then be performed: 
R(p) = 2 ff b(k) cos(ky)K'd'dky 27]p (k’p) (68) 
where the lower limit of integration of x; =- ~ has been replaced by zero, and 
correspondingly, the exponential by the cosine. 
The integral 1, [Eq. (61)] becomes 
L L foo) fo) 2 
boxe |) f J if $(k) cos (k,é)K'dk' dk, (2 sin = 
2 
i sf Jol p)p dn)aé ae, (69) 
where 
g= 5, =§ and aj = k/(é, - €2) and a3 = k[(2L =6 - €) 
The p integral parenthesis is known [10,11]. Its value is cos (x'?/2a2,). Hence 
2 
Lol 2 fo 
ns Me Sf J, i f (k) cos (k1&) kK dk'dk; cos ae dé, dé, (70) 
The value of these integrals depends greatly on the small-wave-number power 
spectrum of the fluctuations of the sound velocity where the Kolmogorov law 
no longer applies. For a square law as a first approximation rather than a of 
power law, and on the assumption that the spectrum is zero at wave numbers 
K<Ko, 
Kx? fork>ko 
E(k) = (71) 
0 for kK <Ko 
The integrals become soluble and integration yields 
ik he, 1,=0 (72) 
16 
