218 Lecture 12 
The last integral is very similar to that which applies to the scattered pressure. 
The deviation, a, of the sound velocity from the mean value has been replaced 
by <a> R(p). The number of scatterers is proportional to the volume T of the 
scattering media, which therefore appears as factors in front of the integral. 
The results obtained for a number of interesting cases are summarized in 
Table 12.II] and plotted as curves in Fig. 12.13. They will be discussed later and 
compared with the corresponding results for the Kolmogorov case. 
12.3.4. Scattering Described by the Power Spectrum of the Sound Velocity Fluctuations 
Frequently the correlation function is unknown or it is too complex to be 
useful in integrals. It is then expedient to describe the fluctuations of the sound 
velocity of the medium by its Fourier space spectrum. The relation between 
the correlation function and the power spectrum ¢4(k,,k2,«3) of the velocity fluc- 
tuations is given by 
KFS I2(EnneanGe) = 10 (ki, Ko, K3) loses +k2&2 +K3&3) devisees (37) 
0 
If the fluctuations are isotropic, $(k:1,k2,x3)= (kx), Where x is the magnitude of 
the wave number, and 
& (k1, Kz, K3) dky dk2 dk3 = $(k) 27K7*dk sin @ dO = E (x) dk sin 0 dé (38) 
= 
where E(k) dk = ®(k)47K7dk (k 20). The exponent in the integral in Eq. (37) can be 
written as x-p =xpcos@, and the angular integration yields 
R(p)= [ eqosineds (39) 
0 (kp) 
If this value is introduced into the scattering integral in Eq. (36), 
L 
<|p2<)> " ao i; J E (w) sin (kp) sin (Ip) dpdk (40) 
TT lb Lo Ik 
and if Lis very large (L + ~), the results becomes [11] 
4p()__k* El2ksin@,/2)l 
k = aE STONOUELE 41 
i EG ey | ER Pee a 
Thus, the scattered intensity becomes proportional to the spectral intensity 
E(x) of the temperature field for a wave number « = 2k sin(@)/2). A similar result 
was obtained in the analysis of X-ray scattering over forty years ago. The 
structure factor was found to be equal to the corresponding Fourier coefficient 
of the atomic structure. 
For back scattering, @)=7 andx=2k. The patch diameter may be crudely 
identified with half the space wavelength of the temperature fluctuations, i.e., 
2R =\,,/2=7/k (where k=7/2R); backscattering is thus caused by patches of a 
diameter 
2R =e =e (42) 
