The scattered wave: general theory 



We shall see later that the propagation of sound waves can be described, in all 

 the situations under consideration, by a linear modified wave-equation of the form 



i av 



V¥ = 2P.(f)D^). (1) 



c ' 2 dt' 2 n 



(It is desirable to use a complete wave theory, rather than the methods of geometrical 

 optics, in a discussion of scattering, because the relative scattered acoustic power is 

 greatest under conditions for which a treatment by geometrical optics would not be 

 valid.) ijj is here a scalar wave-function, such as the fluctuating pressure or density, 

 the exact interpretation being left unspecified for the moment. c is a constant, D n 

 is a linear differential or integral operator with respect to space or time, and P„(£) 

 is a function of the group of variables £ that specify the (departures from the mean 

 of the) relevant local properties of the medium. The summation is over all terms of 

 the type shown, the usual number being not more than two. The variables £, of 

 which temperature is a typical example, are themselves random functions of position 

 and time as a result of the action of turbulent motion in the medium. The physical 

 problems that we have in mind are such that the medium is only slightly non-uniform, 

 and P n is small compared with unity when all quantities are made dimensionless with 

 the aid of parameters characteristic of the sound wave. 



We wish to find a solution of this equation representing the passage of a sound 

 wave, generated by given external means, through a region in which the random 

 variables £ are known in the statistical sense. It is important for the theoretician that 

 he be allowed to assume that the incident sound wave has a frequency which is high 

 by comparison with the fractional rates of change of the variables £, and fortunately 

 this is nearly always true in practice. The consequence of this assumption is that ^ 

 is effectively the only quantity in the above equation that depends on t, and, since xp 

 occurs linearly in all terms, the dependence on / plays only a minor role in the problem. 



Since the right-hand side of the equation is small, whereas terms on the left- 

 hand side are not, it is appropriate to seek a solution by a perturbation procedure. The 

 first approximation to \p, represented by i[/ , is given by 



i avo 



Wo = 0. (2) 



Co 2 dt 2 



We need consider only one Fourier component (with respect to t) of the incident 

 wave, and we shall suppose the incident wave that is generated externally to be har- 

 monic, with frequency w , and to be approximately plane; thus 



(3) 

 where 



a; 2?r 



-• (4) 



A 

 The second approximation to ^ is then ip + ^ l5 where 



1 avi 



Wi = 2 P„(f) DnMo) . (5) 



c 2 dt 2 



411 



