Integrating the intensities in Equations [18] and [20] over the surface of a sphere gives 
the sound power radiated. The induction near-field terms in Equation [20] average out to zero 
over the sphere and thus do not represent radiated sound. The sound power radiated in the two 
cases is 
1 4n =, 3 
Be cogeallan FR dy dy [29] 
167*poc> 3 
and 
P : { a if, oF i) ay ay’ [23] 
hs era ra Saar, 90 95 aR 4 oe n0 y Yy 
2 5 15 BBW) HSER, 
16 7*po¢ 
(Equation 21, Reference 1) 
To examine the nature of the sound field due to a small volume of turbulence, it is 
helpful to change to spherical coordinates centered in the small turbulent region. For con- 
venience, 6 will be measured from the z,-axis and ¢ will be measured from the z-axis in the 
z,a,-plane. If r;=«a,, the following relations obtain for the components in the directions of 
increasing 7, 0, and ¢, respectively: 
7 0 @ 
Fa Oy oy 
—=sin@cos¢ — =cos¢ cos@ —=-singd 
Tr 0 p 
"9 9, oP) 
—s= sin@sin d —=sin¢g cos 0 —=cos¢ [24] 
r 0 ce) 
"3 05 i $3 
— = cos@ —=-sin@ —=0 
r 0 7) 
where 0 = |6| and ¢ =|¢|. 
These expressions can be used to show that, for the dipole case, the intensity is 
purely radial; namely, 
1 1 valle <8 : 
all cee dg dy [25] 
16 77p,.c° 2 72 
For the quadrupole case, however, the intensity has components in the direction of increasing 
7, 0, and ¢ as follows: 
