203, 204] SPHERICAL WAVES. 547 



2O4. Spherical Waves. If the velocity potential depends only 

 on the distance from a fixed point, using the expression of 135, 

 equation 44), the wave -equation 200) becomes 



qp 2 



= l \w + 7 

 Multiplying by r, this is 



216) ^ 



so that the product rep satisfies an equation like 201), of which the 

 solution is 



r<p = F^(r at) + F< 2 (r -f at). 



Accordingly we have 



217) 9 > = i{F 1 (r-aO + F,(r + at)\, 



of which the first term represents a wave proceeding outwards, the 

 second one proceeding inwards, the magnitude however varying 



according to the factor 



For a periodic solution representing a simple tone proceeding 

 from a single point -source we may take 



218) (p = -- cosJc(at r). 



* 



The physical meaning of the constant A is obtained as follows. Let 

 us find the volume of air flowing in unit of time through the surface 

 of a sphere with center at the source. We will call this the total 

 current, 



219) I 



= A {cos ~k(at r) kr sin ~k (at r)}. 



Accordingly when r = we have I = Aeoshat and A, the maximum 

 rate of emission of air per unit of time, is called the strength of the 

 source, agreeing with the definition of 196. 



In order to obtain the activity of the source, that is the rate 

 of emission of energy per unit of time, we may find the rate of 

 working of the pressure at the surface of a sphere, as explained in 

 188, 



220) P = 



In order to find p, we have, if p Q is the undisturbed atmospheric 

 pressure, by integration of 193), and by 198), 



35* 



