many actual cases of turbulence on nonrigid boundaries. Any induction near-field correction 
thus will depend significantly on the relative contributions of the sources, dipoles, and quad- 
rupoles in the boundary layer. To facilitate the determination of these relative contributions, 
the effects of the Mach number and other factors should be considered. 
DIMENSIONLESS RATIOS 
A little dimensional analysis facilitates the understanding of some of the properties 
of the induction near-sound field. By using a typical velocity U and a typical linear dimension 
L in the flow, as in Reference 1, and by ignoring the numerical factors, the expressions for 
the ratio of the mean-square sound pressure to the square of the typical Bernoulli pressure 
for the source, dipole, and quadrupole, respectively, may be written 
2 2 
D L 
a 5 
ji V2 5 [55] 
Bo 
p? L2 14 
a a me [56] 
1 2 2 72 0 r* 
— pU 
and 
yD) 2 4 6 
IL L iG 
iA ETE A ete a ease Ng 2 es [57] 
1 9\ 2 7? r4 7? 
Ope 
These expressions indicate that very near the turbulent region, where L/r is of the order of 
one or more, the root-mean-square sound pressure is of the order of the typical Bernoulli 
pressure in the flow itself and independent of the Mach number at low Mach numbers. This 
also indicates that very near the turbulent region the density changes are of the order of the 
density changes in the flow itself. 
It is of interest also to know the ratio of the absolute value of the fluctuating Bernoulli 
pressure to the absolute value of the sound pressure at any point in the field of sources, di- 
poles, and quadrupoles. In the far field one can show that this ratio is always equal to the 
ratio of the particle velocity to twice the sound velocity or 
i 2 
9g Po? | ISK ay| 
ee ee [58] 
p 87 Po re? 
for a source, 
20 
