The boundary conditions on p s /p are those appropriate to the circumstances of 

 generation of the incident sound wave. Thus to a first approximation p s , p s and v have 

 the same values as for the incident sound wave in a stationary medium. 



It appears, then, that the interaction between the sound and turbulence fields 

 is weak, and that they each perturb the other. This lack of coupling is essentially a 

 consequence of the big difference in the time scales on which changes take place in the 

 two fields. 



The equation that describes the perturbation of the incident sound wave due 

 to the presence of the turbulence is obtained by going to the second approximation 

 of the above exact equation for p s ; thus 



1 d 2 (p s / Po ) 2 d 2 (u iVj ) 



V 2 ( Ps / P o) = - 



Cn 2 dt 2 



(45) 



[ Ui — ) 



Cq OXj 



to a consistent approximation. Terms on the left-hand side of this equation are of 

 order M s /\ 2 , that on the right-hand side is of order M s M t /\ 2 (assuming, for the sake 

 of definiteness, that the spatial derivatives of u have a magnitude of an order equal to 

 that of the r.m.s. of u divided by A), and the largest term of those neglected is of order 

 M 2 /\ or M s M t 2 /\ 2 , whichever happens to be the larger. 



It should be noted that this equation describing the changes in the incident 

 sound wave contains approximations additional to the conventional assumptions of 

 linearized sound theory, unlike the corresponding equations obtained earlier for the 

 various cases of a non-uniform stationary medium. These additional approximations 

 may have consequences for the perturbation procedure on which the scattering analysis 

 is based. While the equation certainly allows the investigation of single scattering, an 

 investigation of multiple scattering from the same equation would need further justi- 

 fication. 



For the investigation of single scattering, v on the right-hand side can be 

 expressed in terms of p s /p by the ordinary lineary acoustic equation 



dv/dt = — c 2 V(p s /p ), 



and so, for an incident harmonic plane wave of the usual form, the equation becomes 



1 d 2 (p s /p ) / 2u { 2i diii KiKj \ p s 



V 2 (p s /p ) = ( — KK t ) - . (46) 



c 2 dt 2 \ c c dxj k ) po 



This equation is in the standard form used earlier for the general analysis of scattering. 

 The function Q{x) introduced there is here given by 



(47) 



(48) 



where F 4 ,-(Ie) is the spectrum of the turbulent velocity u^x) in the sense in which 

 <£(fc) was defined to be the spectrum of Q(x). -^-F ?; (fc) is simply the spectrum of 



K 2 



All 



