Since the sound intensity, defined as the time average of the product of the particle 
velocity and the sound pressure at a point, cannot be approximated in the induction near field 
by D*/po where p, is the mean density and the overbar on the squared quantity signified a 
time average, it is necessary to obtain an expression for the particle velocity. This is most 
easily obtained by taking the gradient of the velocity potential ® which is first obtained by 
integrating the equation 
° 
P=- py? [4] 
After neglecting the constants of integration, the velocity potential for the distribution of 
dipoles is 
1 aff Pa Tg \ 
® =- |e —+—/]dy [5] 
AmpoJj 7 \re 2 
. 
where f, = f; (Y, t—r/c) and f, = F,.* The kth component of the particle velocity v or the 
particle velocity in the z, coordinate direction is then 
1 ope Wy OU | Bile Me i 
On = J Ss | Oe || Se OT [6] 
47 Po r2 \pre? re 73 ro 
where 65,, is the unit diagonal tensor and repeated subscripts are to be summed (z = 1, 2, 3). 
The &th component of the vector sound intensity is then given by 
Ppcrcgil OR NRO a Biaeaiy 4 
1 ij i i j j j 
ee eee et 
16 npg rr \TC 92 I\p%e2 p23 
He Pie te 
= — .)( + ) dy dy’ (7] 
r PO PY 08 Be 
where primes and different subscripts are used in the expression for the particle velocity to 
indicate functions of 7’. Similarly, the mean-square sound pressure is given by 
ser 1 re F. F. Fe Fe 
aoe | euy t say in [8] 
2 , TC 2 , 22 
rT r PRO -— 
1 
* 
-_-—_ ff fi ¥y, t)dy= B; (t), the dipole strength as usually defined. 
™Po 
4 
