206 DYNAMICAL THEORY OF SOUND 



The case which comes next in importance is that of 

 symmetrical spherical waves. If <j> be a function of the 

 distance r from the origin and of t, only, the velocity is d<t>/dr 

 outwards, in the direction of the radius, and is uniform over 

 any spherical surface having the origin as centre. 



Instead of applying the general equation to the present 

 circumstances it is simpler to form the kinematical relation 

 corresponding to TO (1) de novo. The flux outwards across 

 a sphere of radius r is d<f>/dr . 4-Trr 2 , and the difference of flux 

 across the outer and inner surfaces of a spherical shell of thick- 

 ness Br is accordingly 



The volume of the shell being 47rr 2 8r, this must be equal to 

 A . 4?rr 2 Sr or s . 47rr 2 Sr, whence 



dt c 



bmce c2s== ^i (4) 



dt 



i i. <P& c 2 d / 96\ 

 as usual, we have : = - 5- r 2 ^- (5) 



dc r dr \ d/*/ 



This may also be written 



The solution of this equation, viz. 



represents the superposition of two wave-systems travelling 

 outwards and inwards, respectively, with the velocity c. In 

 the case of a diverging wave- system 



r<t>=f(ct-r) (8) 



we have, by (4), crs=f(ct r) (9) 



Any value of rs is propagated unchanged ; the condensation A- 

 therefore diminishes in the ratio l/r as it proceeds, and the 

 potential energy per unit volume diminishes as l/r 2 . For the 

 particle-velocity we have 



3JL 1 1 



-r) (10) 



