I may, perhaps, conclude with some remarks concerning multiple scattering. If 

 /? is the proportion of sound energy scattered per unit distance, then at a distance / 

 from the source the scattered sound will be a substantial proportion of the whole if /?/ 

 is not small. In this case one must, and may, consider multiple scattering by successive 

 use of results from single-scattering theory. 



If pi is actually large, then most of the sound that is present will have been 

 scattered many times, and may be thought of as having described a crooked path like 

 the "random walks" treated by statisticians. 



The particular kind of random walk which is involved has been insufficiently 

 studied. However, to one simple question about it there is a rather simple answer. 



If we ask, what change of direction on the average will a ray of sound undergo 

 (due to multiple scattering) in travelling over a total distance (no longer the distance 

 from the source "as the crow flies", but somewhat greater), then we have first that 

 a proportion 



my 



e~& 1 (Poisson's distribution) (6) 



n\ 



of the energy carried by such rays will have been scattered n times. During this 

 process their direction may be regarded as performing a random walk (on the unit 

 sphere). If Vzpin, 9) sin 9 d$ is the probability that after n scatterings the direction 

 will make an angle between 6 and 9 + dO with the initial direction (where the factor 

 Vising has been inserted to ensure that for a uniform directional distribution p = 1), 

 then p satisfies approximately the partial differential equation (of diffusion) 



dp -Id/ dp\ 



— = jk,2 sin £ __ (7) 



dn sin 6d6\ 86/ 



where w 2 is the mean square deviation in direction at each scattering. The solution of 

 (7) such that the direction is initially 9 — is 



p = 2 (2m + l)P m (cos0)e-*" !m{m+ i )re (8) 



Summing the product of (6) and (8) for n = to oo, we obtain the distribution of 

 direction after propagation through a distance /, as 



oo / — la) 2 m(m+l)\ 



p = 2 (2m + l)P m (cos0)e"^i- e ■> (9) 



m=o 



= 1 + 3 cos oe-M-w + • • • 



Equation (9) shows that to make the sound directionally uniform (p = 1) 

 needs a distance 



1 2 



I » __ = __. (10) 



0(1 - e~n /3co 2 

 For homogeneous isotropic turbulence this condition may be written approximately as 



2a 2 

 l» , (11) 



7T fkE(k)dk 

 All 



