The linear operations represented by D n can be carried out explicitly, thus allowing 

 the equation for \j /l to be written as 



i avi 



V-^1 = Ae^- x -^Q{x) (6) 



c 2 dt 2 



say, where Q(x) is independent of t. The formal solution of this equation as a 

 "retarded potential" is 



A f Q(y) 



#i(x,t) = / e ^ K - y -^+ ik l x -y\ - -dy. (7) 



4-n- J \x — y\ 



The sound wave represented by ^ is the first approximation to the effect of the 

 non-uniformity of the medium on the incident wave ij/ . i{/ 1 has the same form as the 

 wave-function appropriate to a volume distribution of acoustic sources in a uniform 

 medium, each source being of frequency > and the density of source strength at posi- 

 tion y being proportional to AQ(y). It is as if the incident wave "polarized" each 

 volume element of the medium with a strength proportional to |^ | and to a quantity 

 representing the local departure of the medium from uniformity, the radiation from 

 these "poles" then being, to this same order of approximation, the same as if the 

 medium were uniform. The usual physical interpretation is that x j /1 represents the col- 

 lection of singly-scattered waves, that is, those waves that would be set up in a uniform 

 medium by the incidence of the wave ij/ on single isolated scattering elements. 



It is not often possible to go to the higher approximation i/, + ^ + xp.,, and 

 most of the available theoretical work concerns single scattering. It is therefore impor- 

 tant that we should know when ijj + ^ is a valid approximation to the wave-function. 

 In the case of a plane wave passing through a medium such that the velocity of sound 

 has different, but uniform, values inside and outside a region of volume V, it has been 

 established that \p + ^ is a good approximation to the wave-function provided 

 eV%/\ <^ 1, where ? is a measure of the magnitude of the fractional variation in the 

 velocity of sound. When the properties of the medium are random, having values 

 above the ambient level in some parts of V, and below it in others, the requirement 

 will be weaker and seems likely to be of the form e(F%L)^/A <^ 1, where L is a 

 length characterizing the scale of variation of the properties of the medium. A numeri- 

 cal example will help to convey the significance of this criterion. Suppose sound 

 waves of wave-length 10 cm pass through either water or air in which the temperature 

 varies, with an r.m.s. fluctuation of 1°C and a length scale of a metre. Single scattering 

 will then be a reasonable approximation provided the region containing the non-uni- 

 formities has linear dimensions small compared with a kilometre. The restriction on 

 the size of the volume V can be thought of as ensuring that the chances of a scattered 

 wave being itself scattered before leaving V are slight. 



The wave-function that describes the waves associated with a point singularity 

 of some kind takes a simpler form at large distances from the singularity. The wave 

 motion is purely radiative at these large distances, with an intensity that is a measure 

 of the rate of output of energy from the source. There is thus both analytical con- 

 venience and physical significance in a consideration of the radiation fields of the 

 scattered waves emanating from the various elements of the volume V in which the 

 medium is non-uniform. If we wish to describe the fluctuations in sound intensity at 

 some point within the scattering volume V, we must of course take into account the 

 "near" fields of the scattered waves from neighbouring elements. Investigations of this 

 kind have been made for certain choices of the governing quation (Ellison 1951, 

 Mintzer 1953 & 1954). In order to keep this lecture within bounds, only the algebraically 

 simpler quantities depending on the radiation fields will be considered. 



412 



