Figure 1 — Location of Origin and Volume 
Element in a Turbulent Region 
and 
1 02 ee : 
== — } dz (Equation 15, Reference 1) [2] 
4a J Og, Oz. r 
where F’(¥, 2) is the zth component of the instantaneous applied force or acoustic dipole 
strength per unit volume at time ¢ and at point 7 in the turbulent region, 
T;; (y, ¢) is the 27 component of the instantaneous applied stress or acoustic quadrupole 
strength per unit volume at time ¢ and at point y in the turbulent region, 
F. = F.(y,t-1/c), 
T;; = T;; (y, t-7r/c), * 
ip = |z-y|, and 
Cc is the velocity of sound in the medium. 
If the applied forces are distributed over a surface, * Fis considered to be a surface density 
and the integral is a surface integral instead of a volume integral. 
If the differentiation in Equation [1] is performed, the general expression for the instan- 
taneous sound pressure due to the distribution of dipoles becomes 
ioe Be ay [3] 
4nJ 7 \ re 2 
where r, = a, -—y, and dots indicate partial differentiation with respect to time. The term 
in F; represents the radiated sound pressure, and the term in F, represents induction near-field 
sound pressure. The effect of the geometric near field is manifested in the factor 7,/r and in 
the dependence on r. In other words, if 7, can be approximated by z,, geometrical near-field 
effects are negligible. 
*This is a slight variation of the conventional notation used by Lighthill, necessitated by the need for a short 
symbol. 
