are relevant, and we therefore calculate the mean square of ^ : , this being the most 

 significant of these properties. The mean intensity of the scattered wave at position x, 

 expressed as a fraction of the intensity of the incident wave, is 



i^i* r(fc)r*(fc) 



A 2 167r 2 .r 2 



1 6t V 



Q(y)Q(s) e ik -(y~^ dy ds. (9) 



V V 



When the physical properties of the medium are approximately stationary random 

 functions of position, and the volume V is large enough to contain many fluctuations of 

 Q, this relation can be simplified to 



iMi* V 



, Q(y)Q(y + Z)e- ik -Zd£ 

 A 2 16t 2 x 2 J y 



tV (10) 



£S — $(fe), 

 2x 2 



where <J>(fc) is the function usually referred to as the spectrum of the quantity Q. 



The form of this expression for the mean intensity of the scattered wave at 

 position x is worthy of note, not only for its simplicity. It reveals the prominent role 

 played by the spectrum of the relevant property of the medium, showing that this is one 

 of the functions that must be supplied from turbulence theory. A good many authors 

 have analysed the scattering of sound and radio waves in terms of distributions of 

 temperature, etc., in physical space. This makes the mathematical work cumbersome, 

 and conceals the dependence of the intensity of the scattered wave on a very particular 

 aspect of the turbulence. If, for instance, we wish to calculate the scattering of incident 

 waves of high frequency, in directions not too close to that of the incident wave, we 

 see from the above formula that a knowledge of the spectrum function cf>(fe) at large 

 wave-numbers is required. Quite large variations in the form of <3>(fc) at large k 

 correspond to relatively slight changes in the form of the correlation function 

 Q(y) Q(y + V near £ = 0, so that the latter function would need to be known 

 very accurately indeed if it were to be used in a calculation of the scattering. 



The flux of energy in the scattered wave in direction I is sometimes expressed 

 in terms of a quantity a(l) called the "scattering cross-section." The interpretation of 

 our wave-function if, has been left unspecified, but it will bear the same relation to the 

 energy flux for both the incident and scattered waves. 



Thus cr(l) = (mean flux of energy in scattered wave per unit solid angle in direction 

 I per unit scattering volume) ~ (energy flux in incident wave, at posi- 

 tion of scattering volume, per unit area of wave-front) 



x°- 



VA* 



«-*(ft). (11) 



2 



The fraction of the energy of the incident wave that is lost by scattering, per unit 



414 



