Tatton T;; T;; ie 67," 15 ih 1507 has 
: + dy dy [13] 
roo p42 G2 r’3e pees 
Similarly, the mean-square sound pressure is given by 
= 1 fj rately Ui ( He) 8 ee 88 =a) i) BIE 8 = 
i = ———— | — + + + + 
16 7? 72’? 2 7? 3 ‘ce? 02 £3 
rC Cc rc r’“o r 
Gee Tj T;; T, ‘ 8 Th 3 Tim 
= 85, — + — — + 
2 ro or? ee p? 46 73 
Bin fTon  BTHe CORN Te aT 
ij ij ij ij m m 
2G 2 ee 
7? rez re r pIa@ Gee 
Te TTC 
l l 
+8558 1m i a ( — + =| dy dy” [14] 
me r’2¢ 3 
SIMPLIFYING ASSUMPTIONS 
These are indeed formidable expressions. A few reasonable assumptions may be made, 
however, to reduce themto more tractable forms. First, it may be assumed that the turbulence 
is sufficiently steady over intervals of time required for sound to travel over distances for 
which the velocity fluctuations are appreciably correlated so that simultaneous correlations 
may be taken at 7 and 7’. This means that the wavelength of the sound in question must be 
long compared to the ‘‘eddy size.’? Observed average eddy sizes are approximately a wave- 
length for frequencies of the order of ten times the frequency of the peak in observed noise 
spectra of turbulent jets. Since it is reasonable to assume that smaller than average eddies 
contribute most to the high-frequency components, the assumption of simultaneous correlations 
seems to be justified. If, in addition, it is assumed that ris much greater than the eddy diam- 
eter, then r= 7. 
Another reasonable assumption that permits a considerable reduction of the equations 
is that the fluctuating forces and fluctuating stresses are approximately stationary random 
functions of time. In other words, the turbulence is assumed to be essentially steady and not 
decaying appreciably.* This assumption still permits the existence of both gradual space 
so proudmanc has shown that, for the case of isotropic turbulence, the ‘‘decay’’ terms are indeed negligible in 
comparison with the ‘‘instantaneous’”’ terms. 
