214 DYNAMICAL THEORY OF SOUND 



the disturbance up to the instant t. The envelopes corre- 

 sponding to successive values of t will form a series of what are 

 known in geometry as "parallel surfaces"; in other words, the 

 boundary of the disturbed region spreads everywhere normal 

 to itself with the constant velocity c. 



As a simple application of the formula (8) we may take the 

 problem already discussed in 71, where an initial uniform 

 condensation s was supposed to extend throughout the interior 

 of a sphere of radius a 

 having the origin as 

 centre. When a spherical 

 surface of radius ct, de- 

 scribed with P as centre, 

 intersects the boundary of 

 the originally disturbed 



region, as in the figure, 



,1 P ,1 ,. Fig. 67. 



the area of the portion 



included within the latter is 2?r . PQ Z (1 - cos OPQ), and the 

 average of the given initial values of s over the whole surface 

 (47r . PQ 2 ) is therefore 



where r = OP. Hence, by the rule, 



in agreement with 71 (24). 



73. Sources of Sound. Reflection. 



The very useful conception of a "point-source" was 

 introduced into the subject by Helmholtz. We may imagine 

 (with Maxwell and Lord Rayleigh) that at such a point fluid 

 is introduced or abstracted at a certain rate, and that the 

 "strength" of the source is measured by the volume thus 

 introduced per unit time. The wave-train due to a source of 

 strength f(t) at the origin is accordingly represented by 



(1) 



