c2 
(3cos* 6 — 4cos2 6 + 1) 
4 
No Fi (c0549 ~ cos24 +0] dy dy’ (35] 
rw 
Li Al Uisenrremms 4 c? 2 = i 
I= alae aa lng COSY tt (1 - 3cos* 6) | dy dy 
16 77p,c° re rw? “ 
eee <(-2sin@ cos 0)| dy ay” [36] 
Ree a ey ee —2sin 6 cos y ay 
16 7*p,c° 7? 1? Gy? 
and 
ly S 0 
If three equal, mutually orthogonal, longitudinal quadrupoles are combined, the mean- 
square sound pressure is 
ao 1 iL 1 20 20"? mee faba 
PAs merase T.,T,, a a [37] 
7 Cc 
As was pointed out by Lighthill,’ these represent the equivalent applied fluctuating pressures 
in the turbulent region. The induction near-field terms in the intensity also disappear. 
Figure 3 gives the polar patterns for the three terms in Equation [35] for all planes con- 
1 
taining the z-axis. The two induction near-field terms have zeros when @ = cos | — = 55 deg. 
3 
The middle term has a negative value for cos” 6 > 1/3 but at no distance or angle from the 
quadrupole does this make py negative. It may be seen that the induction near-field terms 
always predominate at 0 = 90 deg and the far-field term always predominates at @ = 55 deg. 
The polar patterns of the intensity due to a single longitudinal quadrupole are given 
in Figure 4, where the arrows indicate magnitude and direction. The far-field intensity is 
purely radial. The term for the induction near-field intensity, however, indicates a flow of 
sound energy out perpendicular to the axis and in again along the axis of the quadrupole, 
giving no additional net outflow of energy or power radiated. The induction near-field inten- 
sity is perpendicular to the surface of the cone generated by the 6 = 55-deg line. 
The sound field of a region of ¢sotropic turbulence shows no angular variation, but the 
induction near-field effects are still present. The assumption of isotropy permits the elimina- 
tion of many of the terms in such quantities as ih iN . To affect the simplification, it-will 
12 
