GENERAL THEOEY OF SOUND WAVES 



211 



shew an initial distribution of s which varies rapidly but 

 continuously from s to in the neighbourhood of r = a, 

 together with the consequent time-variation of s at 0. 



cb 



Fig. 66. 



The problem which we have discussed exhibits a marked 

 contrast with the theory of plane waves, in that the wave 

 resulting from an arbitrary disturbance contains both con- 

 densed and rarefied portions, even when there is no initial 

 velocity and the initial disturbance of density has everywhere 

 the same sign. The statement is easily generalized by means 

 of equations (1) of 69. If we take the integral of the value 

 of s at any point P over a time which covers the whole transit 

 of the wave, so that the values of u, v, w vanish at both limits, 

 we find that its space-derivatives are all zero. The integral 

 has therefore the same value for all positions of P. And by 

 taking P at an infinite distance, so that s becomes infinitely 

 small by spherical divergence, we see that the value is in fact 



zero, i.e. 



sdt = 0. 



.(30) 



The mean value of s at any point is therefore zero. This result 

 is of course not limited to the case of spherical waves. 



142 



