APPENDIX C 
EFFECT OF CONVECTION OF TURBULENCE 
For the case of fluctuating forces being convected at vector Mach number M the trans- 
formations used by Lighthill! and Equation [1] lead to 
1 rF, Q1-M?)7,F, MAF. 
2 esd ll ees ee oo Smee nome (td [84] 
47) | o(r-M-7F)2 (r-M-F)3— (9 - MF)? 
where 7 = 7 + Mr and F, = F, (j, ¢- 7/c). By taking the time average of the square of the 
instantaneous sound pressure, the mean-square sound pressure becomes 
Ben 1 1 ryt ae 
4 16 72c? an! 
(r—-M-7)? | (r-M-F)? 
1-2)" 2(1-M2)r.M 
c2 - vate - r; j 
+ —_—_ ee 
7)? 
Be 1 fj 1 UU 
16 72c2 
= a = an FF dn dn’ [85] 
(r—-M-F (r—M-7)? (r—M -F) 
where terms involving odd numbers of time derivatives have been neglected as before. 
If the fluctuating forces are being transported in the z-direction, |M| =M = M, and 
M.7/r=Mcos 6. Therefore, the mean-square sound pressure is 
SEN i 
r2(1—M cos 6)? | r2(1-—M cos 6)? 
2 
P (1-M?) 7,7. 2M(1-M?)r, SED 
fp ee | pipe FFs + M?F,F,| (aq dq’ 
2 2 2 ON] b 
r~(1-M cos 9) r~(1-M cos 6) r(1-M cos 6) 
[86] 
For the special case of dipoles having axes in the 2,-direction and being transported at Mach 
number M in the z,-direction, the mean-square sound pressure is 
oo alt 1 2 =e: 
= — cos? 6 F, F’ 
16 72c2JJ r2(1-M cos 6)* 
2 2 
Be) eel O Mah) 2cos 6 M(1-M*) |—— ; 
| PU | IE UB (fC GQ) 
72 (1-Mcos 6)? 1-M cos@ 
[87] 
29 
