and SS 
ae 
S -| wiz) [48] 
These expressions also hold for the time derivatives of the quantities under the integrals. 
Since eddies of different sizes have different frequency scales, V and S defined for a narrow 
band of frequencies would be expected to be functions of the frequency. One would expect 
V and S to be smaller for the higher frequencies; that is, that V would be approximately pro- 
portional to the inverse cube of the frequency and S approximately proportional to the inverse 
square of the frequency. This exact proportionality, of course, neglects the effects of sum- 
and-difference frequencies in the velocity fluctuations and ignores the fact that time fluctua- 
tions are not synonymous with the space variations that are usually measured. The frequency 
dependence of V and S partially offsets the higher efficiencies of quadrupole and dipole radia- 
tion at the higher frequencies. As a result, the frequency of the peak in the sound spectrum 
should be about the same as the frequency of the peak in the spectrum of the fluctuating 
velocities or fluctuating forces. The correlation volume and correlation area might also de- 
pend on the particular velocity components and force components used in the integrals. In 
spite of these uncertainties, the effect of integrating over dy’ may be approximated by replac- 
ing dy’ by V or S and removing the primes from the integrals that give the mean-square sound 
pressure and intensity. 
MEAN-SQUARE SOUND PRESSURE IN TERMS 
OF SOUND POWER RADIATED 
The mean-square sound pressure at any point in the sound field outside the geometric 
near field of like-oriented dipoles or quadrupoles may also be given in terms of the sound 
power P radiated by the multipoles. For the four special cases discussed previously (namely, 
dipoles, lateral quadrupoles, longitudinal quadrupoles, and a region of isotropic turbulence), 
the mean-square sound pressure is given, respectively, by 
Lo Py Gl? 2 
2 2 
Da = (3 cos~ @) ( + [49] 
Agr2 reas 
1) MOa@le 2 of 
p? = (15 sin? 6 cos? 6 cos” ¢) (108 +9 ) [50] 
4nr? ro Ge 
pocP 
2 4 
= c 4 1 c 6 1 
p? = (5 cos* 6) + - + )s (s - + | [51] 
Aare rq? cos?@ cos*é ret cos?6@ cos*é 
16 
