In like manner, if the double differentiation in Equation [2] is performed, the general 
expression for the instantaneous sound pressure due to the distribution of quadrupoles becomes 
Meese Mie 8 Tos 8 Te Fea 
p=— eat ee Leys = Oa: i pea dy [9] 
47 r? re? re re INr2g 73 
The term in ih represents the radiated sound pressure, and the terms in T represent the in- 
duction near-field sound pressure. The terms in T;; are Significant only in the transition 
region between the induction near field and the far field, and are conveniently lumped with 
the induction near-field terms. After again neglecting the constants of integration, the veloc- 
ity potential is 
i BVT Se WONT nee S) 6H os OND 
Ga [ep oe -~8,,(—-+—)la7 10 
CPD! || A 5 2?) 2 3 LT \\ 9) 3 
c = @ r TChin: 
where ti = ti (y, t-r/c) and t;, = T;,- This velocity potential, as well as the previous one, 
satisfies the wave equation 
; eryies 
vya=-— © {11] 
pe 
The &th component of the particle velocity is 
1 fs Sih, aD, a 
1) + + — 
kom ae 3 
47pg r3 rc? ro? rc 74 
r PANG T Tanne Bee 
k 
“ Ce 22, | ee | dy [19] 
r TEN 2) 2 noe 74 
By again using primes and different subscripts in the expression for the particle velocity and 
taking the time average of the product of p and v,, the kth component of the sound intensity 
becomes 
ee 
’ ee e eo 
re 2 Ea f i 
} 1 | iil ae ee ts) ( TAOS SET coca! 018 is 
k = a + — + + 
= ——]\ — + + 
16 7p 273 \pG2 r¢ 7 r’e3 2 Q2 r’73¢ 74 
rT r Ben AGE 37. 3 T. T, Bi) Bae 
tj m ij i i lm lm l 
- (0, = +2, 7\(4. —\( + + =| [13] 
r pe r° I \ re? 6 73 r°2g2 9736 mia 
