204 WAVES IN AIR. [CHAP. VIII. 



we have 



dt r&amp;gt; dr 



a result which might also have been arrived at by direct trans 

 formation of the general equation of continuity. To our order of 

 approximation, (11) gives . 



_ 4- - [r - = 

 dt r 2 dr \ dr) 



and eliminating s between this and (10), we find 



d*(f&amp;gt; _ c 2 d / 2 Jc/A 

 ~de~7*dr( r dr ) 9 

 or 



^? = 2 ^ (12). 



This is of the same form as (6), so that the solution is 



Hence the motion is made up of two systems of spherical waves 

 travelling, one outwards, the other inwards, with velocity c. Con 

 sidering for a moment the first system alone, we have 



s = C ~F (r-ct), 



which shews that a condensation is propagated outwards with 

 velocity c, but diminishes as it proceeds, its amount varying 

 inversely as the distance from the origin. The velocity due to 

 the same train of waves is 



dr r ? &amp;gt;2 



As r increases the second term becomes less and less important 

 compared with the first, so that ultimately the velocity is pro 

 pagated according to the same law as the condensation. 



168. Let us suppose that the initial distributions of velocity 

 and condensation are given by the formula 



where ty, % are any arbitrary functions. The value of &amp;lt;/&amp;gt; at any 



