E. J. Skudrzyk 211 
Actually, m and XK are not constant, but increase toward the long wave number 
end. A more accurate determination of K can be obtained for the Kolmogorov 
range by measuring the rms temperature difference between two points sepa- 
rated by a distance that is of the same magnitude as the diameter of the acous- 
tically active thermal patches (about 10 ft apart; see next section). It can be 
shown [8] that the equation 
[ox +p) - e(x)]? =4 f B() (1 - sina) dk =BK x? (15) 
depends predominantly upon the value of E(x) for xp =1, and that the constant 
B is equal to Oe in the range xp ¥1, even if the Kolmogorov law is only an 
approximation. 
12.3. THE SCATTERED PRESSURE AND SCATTERED INTENSITY 
12.3.1. The Rayleigh Integral 
The classical as well as the modern theories of scattering are all based on 
the Rayleigh integral [9]. This integral represents the solution of the wave equa- 
tion for the case in which the properties of the medium deviate slightly from 
their average values. This solution shows that the density changes caused by 
the natural temperature fluctuations have a negligible effect on sound propa- 
gation in comparison to that of the changes in sound velocity. But if the local 
variations of the density are neglected, the derivation of the wave equation is 
the same as that for a homogeneous medium except that the sound velocity now 
is locally variable. Therefore, for periodic vibrations, the wave equation is 
V*p+k*p=0 (16) 
where 9 
K2 -2(1 -2Ac avd Sp G@ = Da) 
In Eq. (16), c= co + Ac, the magnitude cy = < c > represents the space-average value 
of the sound velocity, kg =@/cp is the wave number of the undisturbed medium, 
and 
a= Ac/eo (17) 
is the relative deviation of the sound velocity from the space-average value. 
The variable part of the term that contains the sound velocity may then be trans- 
ferred to the right-hand side, thus: 
Vp + kop = 2kaap (18) 
Since only a small fraction of the incident energy is scattered per unit volume 
of the scatterer, the right-hand side is small and the sound pressure on the 
right-hand side may be replaced by the sound pressure, p;, of the incident wave. 
The solution of the resulting equation is well known; it is given by 
P=P;i+DPsc (19) 
where the scattered pressure 
