located at the measuring point, the contributions from a given area of uniform boundary layer 
may be easily found by integration. For instance, by using Phillips’ tentative estimate for 
an z-Reynolds number of approximately 108 (sound power radiated per unit area equals approxi- 
2 
ce 
mately 4(1077) Po oP ey and by using Equation [49] with P(8cos? 6) (: + \e 
rT @ 
replaced by 
3 3 , , 
6, 8(10°’)pyU'MS (; sin?) cos? 6 hoy eeaia. ae 
2 
| 1+ —— —— 
; h? 
Oy 
de’ 
cos? 6’ 
the broad-band mean-square sound pressure a perpendicular distance / from the center of a 
circular ring area between 0’°= 6, and 0°= 0, is 
2 
cos 6, a 
, 
5 1 
p? = 6(10-’)p,-U*M [-( ) A (cos? 6, - cos? 6,) + (sin* 0, - sin* 00] [54] 
cos 6, 40h? 
where M, = U/c, 
U_ is the free-stream velocity, and 
6 is the angle with the perpendicular to the plane boundary. 
Phillips has shown further, however, that the dipole contribution is negligible if the boundary 
layer is homogeneous in layers parallel to the boundary. Thus, for this condition the predom- 
inant sound would be from quadrupoles in the surface and in the turbulent boundary layer. A 
procedure similar to the above might be used to estimate the contributions of these quadrupoles. 
The quadrupole radiation from the turbulence proper might be treated approximately as contri- 
butions from isotropic turbulence outside the shear layer and from lateral quadrupoles in the 
shear layer just outside the laminar sublayer. Further, the effects of aconvection velocity 
may be included by using the equations in Appendix C. 
The above discussion of the noise from a turbulent boundary layer holds only for points 
in the fluid outside rigid bodies. In the case of nonrigid bodies, motion of the boundary, in- 
duced by the fluctuating pressure field on it, will create a system of simple sound sources 
over the boundary and will permit propagation of dipole and quadrupole sound through the 
boundary. A thin pliant boundary with turbulent fluid motion along one side and no net fluid 
motion along the other side, for instance, might radiate simple source, dipole, and quadrupole 
sound both intothe moving fluid and intothe motionless fluid, withthe multipole sound radiated 
into the motionless fluid modified by the transmission properties of the boundary. The simple 
sources do not have an induction near field but they can have a geometric near field. The 
transmission and reflection properties of a nonrigid boundary in the induction near field of 
dipoles and quadrupoles are, of course, not as simple as for a plane sound wave. Because 
of their greater efficiency, the simple sources should predominate at low Mach numbers for 
19 
