the component of turbulent velocity in the direction of propagation of the incident 

 sound wave, showing that only one component of the turbulent velocity is relevant 

 to single-scattering of a plane wave. This expression for a(l) is effectively the same as 

 the results obtained by Blokhintzev. by Lighthill, and by Kraichnan. 



It will be noticed that the single-scattering due to a turbulent velocity field u(x) 

 in a medium of uniform physical properties is exactly the same as that due to a density 

 variation p x (x) in stationary air provided 



K • u(x) Pl(x) 



= - % (49) 



KC p 



This is not an entirely unexpected result, since the two sides of this relation represent 

 the local fractional increases in apparent phase velocity of the incident wave as a result 

 of the non-uniformity of the medium, the increase occurring in one case by bodily 

 convection and in the other by a change in the physical properties of the medium. 

 However, lest this result should begin to seem obvious, may I remind you that it holds 

 only for single-scattering and only for the case of air, for which variations in inertia and 

 elasticity are connected in a particular way. It should be noted, moreover, that 

 k • u/kC and — pi/2p would not have identical distributions (even statistically) in two 

 real cases, since one quantity is subject to the kinematical requirements of the con- 

 tinuity condition and the other is not. A comparison of the relative importance of the 

 scattering effects of turbulent velocities and of variations of density or temperature, as 

 they occur in reality, would require a consideration of turbulent spectra, which lies 

 outside the scope of this lecture. So far as I know, an analytical comparison of this 

 kind has not yet been made. 



Particular cases can now be discussed as before. When the turbulent motion 

 is isotropic, we have 



F«(fc) = ( *<, , (50) 



\ k 2 J 4ttA; 2 



where E(k) is the usual energy spectrum function, and hence 



k 2 cos 2 6 E(2k sin 0/2) 



<r(l) = . (51) 



8 tan 2 0/2 c 2 



When k is very small, E(k) is proportional to A' 4 and so a falls to zero at the straight- 

 ahead direction 9 — (as is also true when the turbulence is not isotropic). For an 

 incident sound wave of very high frequency, and turbulence which is not necessarily 

 isotropic, the total scattered power is given approximately by 



f Fiji™) 

 a(l) dl = 2TKHCJ / — - dm . (52) 



J Co" 



where m is a vector in the plane at right angles to k- This expression can be written 



as 



u 2 K 



a(l) dl = 2k 2 — L K , (53) 



Co 2 



where u K - stands for the mean-square of the turbulent velocity component parallel to 

 k, and L K is the integral scale in this same direction, defined in the usual manner. 



This brings me to the end of my account of scattering of sound waves due to 

 turbulence. It is not a particularly dramatic end, and no startling conclusions or new 



422 



