212 DYNAMICAL THEOKY OF SOUND 



72. Waves resulting from a given Initial Disturbance. 



We have next to trace the effect of initial conditions in an 

 unlimited region, in the general case. We suppose that at the 

 instant t = we have 



where the functions are arbitrary. To deduce the effect at any 

 subsequent instant, at any assigned point P, we consider in the 

 first instance the average value of < over a sphere of radius 

 r described with P as centre. This will be denoted by 



(2) 



if So> represent the elementary solid angle (SS/r 2 ) subtended at 

 P by any elementary area $S of the sphere. In the same way 

 we write 



* 47T 



//* (3) 



This, like (2), will be a function of the variables r and t only. 

 If in 70 (3) we multiply both sides by 8&)/4?r, and integrate 

 over the aforesaid sphere of radius r, we find 



It is also evident that the average normal velocity over the 

 sphere will be d(j>/dr. The argument by which the rate 

 of change of s was in 71 inferred from the consideration 

 of the total flux out of the region bounded by the spheres 

 r and r + &r can then be applied to prove that in the present 

 case 



_ 



B; t*Sr 



Eliminating s, we have 



Sf-^ 

 dt' 



which is identical in form with (5) of 71. We recognize then 

 that < is the velocity-potential of the system of spherical waves 



