=> Po 1 Z 4 
p= \{= w* (uu’)? (1 +9 + 12 ay dy’ _—s« ([44] 
16 72c* r2 r2@2 ro 
or 
3 Po 16 2 4 
2= le (win’)? (1 +2 + 12 ag dy’ [45] 
1672c* 7? ra rte 
and 
Po Gis 
I, = a eles (ui)? dy dg’ [46] 
16 mpc 72 
As one would expect, there is no angular variation in the sound field, but the mean-square 
sound pressure has two induction near-field terms. This sound field turns out to be equivalent 
to the individual contributions of a longitudinal quadrupole in each of the coordinate direc- 
tions and two lateral quadrupoles in each of the pairs of coordinate directions. 
The modifications in the equations required when the flow is analyzed with respect to 
a moving frame have been discussed by Lighthill? and Phillips‘ for the far-field radiation. 
Similar modifications may be made for the general sound field. Because of the complexity of 
the equations, however, this has been done only for the instantaneous and mean-square sound 
pressures. These will be found in Appendix C. 
The utility of the foregoing theoretical results depends on the ability of observers to 
estimate certain, up to now, unmeasured quantities. These estimates may be made more realis- 
tic by using a number of useful concepts and by keeping in mind the special properties of the 
near-sound field. These concepts will be discussed in the next section for a few practical 
applications. 
DISCUSSION 
CORRELATION VOLUME AND CORRELATION AREA 
For purposes of estimating some of the integrals introduced previously, the concepts 
of correlation volume and correlation area are useful.1’?’4*> The correlation volume and corre- 
lation area as introduced by Lighthill refer to an average eddy volume and area inside of which 
the fluctuating quantities are considered correlated and outside of which the fluctuating quan- 
tities are considered uncorrelated. The correlation volume and correlation area may be given, 
respectively, as 
TT,” 
L 
V ae dy’ [47] 
15 
