where p l5 c x are local departures from the uniform mean values p , c , and proceed 

 to linearize in Pl and c x . The equation for p then becomes 



1 d 2 p ( 9x\ 2d 



V 2 p = (Vp) ■ V- V 2 p, (23) 



Co 2 dZ 2 \ po / c 



which is now in the standard form adopted earlier in the lecture, p taking the part of 

 the wave-function $. 



The relative importance of the two terms on the right-hand side of this equation 

 depends on the nature of the medium, and some discussion of the properties of the 

 two common cases of air and water is needed. In the case of air, variations of p and 

 c are likely to be attributable to variations of either temperature or water-vapor con- 

 centration, or both. It happens that the ratio of the specific heats of moist air is about 

 the same as that of dry air, so that, for stationary air at uniform pressure, variations of 

 c and p are related by 



2d pi 

 c 2 oc p -i , i.e. — = , (24) 



Co Po 



irrespective of whether the variations of p and c are a consequence of variations of 

 temperature or of water-vapor concentration. Of the two possible causes of variations 

 in p, variations in temperature seem likely to the more important in the atmosphere, 

 since the difference in density between completely dry air and saturated air at 10°C 

 is no more than that produced by a variation of temperature of about 1.5°C. 

 For air, then, the governing equation becomes 



1 d 2 p / pi\ pi 



- = ( Vp) • ( V — J + — V 2 p, (25) 



Po / Po 



c 2 dt 2 



and from this point onwards the analysis of the scattering follows the path already laid 

 down. The function Q(x) of the general analysis is here given by 



Pi Pi 



Q(x) = ik ■ V k 2 - (26) 



Po Po 



and its Fourier transform is 



r(fc) = (k • k - /c 2 )A(fe) (27) 



= — k 2 cos 6 A(fe), 



where A(fe) is the Fourier transform of pi(*)/p - The function ail) that describes 

 the directional distribution of intensity of the scattered wave is 



a{l) = -k4 C os 2 0*(/c), (28) 



2 



where ^r(k) is the spectrum of the relative density fluctuations pi(*)/p in the sense 

 in which 3>(fc) was defined to be the spectrum of Q(x). 



The method of obtaining more specific results for the case of high frequency 

 sound has already been described; it will be seen to show that the total scattered energy 

 flux is then proportional to « 2 . Another particular case of some interest is that in which 

 the distribution of density is statistically isotropic (as is always likely to be a valid 



417 



