GENERAL THEORY OF SOUND WAVES 207 



The law of dependence on distance is here more complicated, 

 but as the wave spreads outwards the first term ultimately 

 predominates ; the velocity at corresponding points of the wave 

 then varies as 1/r, and the kinetic energy per unit volume 

 as 1/r 2 . 



In a diverging wave-system we have, from (9), 



an = -l(r+), .................. (11) 



and similarly, in a converging wave-system. 



These relations correspond to (5) of 60, which is indeed a 

 particular case, since as r increases our spherical waves tend to 

 become ultimately plane. 



The general argument of 23 can be adduced to prove that 

 in a diverging (or a converging) wave-system by itself the 

 energy is half kinetic and half potential. 



The solution (7) can be applied to a region included 

 between concentric spheres, or to a region having only one 

 finite spherical boundary, internal or external. In any case, 

 the conditions to be satisfied at the boundaries, whether finite 

 or infinite, must be given in order that the problem may be 

 determinate. In particular, even when the region is otherwise 

 unlimited, the point r = is to be reckoned as an internal 

 boundary ; . this point might for instance be occupied by a 

 " source " of sound ( 73). When there is no source there, the 

 flux across a small spherical surface surrounding must vanish, 

 i.e. we must have 



(13) 



r =0 \ v 



When applied to (7) this condition gives 



f(ct) + F(ct) = 0, .................. (14) 



for all values of t, and the general solution therefore takes the 

 shape 



r$ = F(ct + r)-F(ct-r) ............. (15) 



This formula may be used to determine the motion con- 

 sequent on arbitrary initial conditions which are symmetrical 



