E. J. Skudrzyk 217 
by a single scatterer and the number of scatterers. For constant density of the 
scatterers, the scattered intensity is proportional to the total scattering vol- 
ume. A further elaboration of this procedure, however, is of little practical value 
since the true temperature field is of a statistical nature and cannot be described 
by a collection of individual scatterers. 
12.3.3. Scattering Described by the Correlation Function 
If the temperature field is not made up of simple temperature patches, or 
if a detailed description of the scattering inhomogeneities is not available, the 
scattering properties of the fluid can be described by the correlation function 
R(p) or the space power spectrum E(x) of the sound velocity fluctuations, x being 
the Fourier space wave number. The Rayleigh integral can be transformed into 
a form based on the correlation function by multiplying p,. by its conjugate com- 
ples value, p.., and taking the time EXISTE Cc If the variables are denoted by ¢’, 
7’, and ¢’ in the first integral and by &", n’, and ¢” in the second, the product of 
the see can be written as 
ri>-eom ae [ [oe 
P:.| = bacPD Ga ees <a a> 
x etKl(E' "ae (=n B+ (E' - 8) (y-DI gé"an" do dé'dn' do’ (33) 
where the integral sign denotes triple integrations over the primed and double- 
primed coordinates, and L is the linear extension of the scattering medium in 
the three coordinate directions. If the medium is statistically homogeneous, the 
time average 
al(E'n'L') al (En £") = a?R(p) (34) 
is a function of the distance p ofthe two points é'7'¢' and &"n"¢" only. This time 
average is then assumed to be the same as the space average <a*> (Ergodic hy- 
pothesis). The time average a’a” has therefore been replaced by the space 
average <a’a’> in the above integral. The magnitude R(p) is called the normalized 
correlation function. If, the two points are coincident, p=0, <a‘a> — <a’>, and 
R(0)=1. If the points €'7'¢' and €"n"¢" are far away, a’ and a” can be assumed 
to be independent of one another; then <a’a"> = <a'><a">= 0 because of a well- 
known theorem of statistics. The above ets can then be written as follows: 
(pid>= ce Lars La Bcf I R71, 0) ike +6n+¥-) 1 géandtdt'dn'dl' (35) 
where é'- €", 7'~7", and ¢'-¢" have been replaced by €, 7, and ¢. The €', 7, ¢’ 
integration leads to a factor oe to the volume ; of the scattering region. The 
é, n, ¢ integration is performed in exactly the same way as in Eq. (27). The re- 
sult is 
2 k*po <a?> . 
A) £20, ,<2> | Rip)sin (Cp) pdp (36) 
L 
