be assumed that: 
T;; = Po%; (y, t — r/c) u; (7, t — 2/¢) = pot, u, [38] 
u,u,=0 if i= (isotropic turbulence)** [39] 
uu; uu, =U; uuu, + U, Uy u; Un + uu,” uu [40] 
(normal probability distribution)? 
and 
u,, =-U,U,, u,u,;=0 ete. [41] 
(stationary random functions of time) 
Also, let & - 7/r = u be the component of the turbulent velocity fluctuation w@ in the 7-direction 
at point 7 and at retarded time (¢ - r/c). The velocity u is also, on the average, equal to the 
component of w in any other direction. 
By using these equations, Equation [21] reduces to 
2 
“Ty. Po 4 —— Tom c? Peery cn ie o Saar, 
p = ff Af farar. ay} +2— wu’ wu’ +3 —(uu’)*fdy dy’ =: [42] 
16 77c* 7? 7? is 
r 
Similarly, the intensity from Equation [26] reduces to 
2 
Po 4 {|__ __, SaanD 5 
I,- ——|| —|uu’ ti’ + 3 (au’)*| dy dy [43] 
1677p ,crry 
The 6 and ¢ components of the intensity are zero. Equation [43] corresponds to the result 
of Proudman, Equations 4.5 and 5.1,3 with the ‘‘decay’’ terms neglected. 
As was pointed out by Lighthill,! the spectrum of the sound radiated is different from 
that of the velocity fluctuations because the radiated sound pressure depends on the square 
of the velocity fluctuations. Therefore, the sound pressure will include sum and difference 
tones of the velocity frequencies and, in general, will have a flatter spectrum. If the veloc- 
ity fluctuations are confined to a narrow band of frequencies around the angular frequency w* 
one might approximate the effect of sum and difference frequencies by assuming that the angu- 
frequency o of the radiated sound is approximately equal to 2w*. The mean-square sound 
pressure and intensity under this assumption then reduce to 
14 
