208 DYNAMICAL THEOKY OF SOUND 



about 0, in an unlimited medium. Suppose that when t = 

 we have 



(16) 



The former of these functions determines the initial distribution 

 of velocity, and the latter that of condensation. The function 

 F must now satisfy the conditions 



F(r)-F(-r)=r^(r) ................ (IV) 



(18) 



It is to be noted that the variable r is essentially positive ; this 

 explains why two equations are necessary to determine F for 

 positive and negative values of the argument. 



Suppose, for example, that there is no initial velocity 

 anywhere, but only an initial condensation, so that </> (r) = 0. 

 From (17) and (18) we deduce 



F' (r)=-J"( -r) = \ r - x .(r) .......... (19) 



The condensation at time t is given by 



Ity F'(ct+r)-F'(ct-r) 



= c *tt- ~^r 



This takes different forms according as ct is less or greater than 

 r. In the former case 



-<*)), ...(21) 



and in the latter 



As a particular case, suppose we have an initial condensation 

 which is uniform ( = s ) throughout the interior of a sphere of 

 radius a, and vanishes for r > a ; and let us examine the 

 subsequent variations of s at points outside the originally 

 disturbed region. Since % (r) vanishes by hypothesis for r > a, 

 the first part of the solution (21) or (22) disappears in the 



