165 167.] PLANE AND SPHERICAL WAVES. 203 



and therefore 



= (8). 



For a wave travelling in the direction of x negative we should have 



= -w (9). 



It will be noticed that there is an exact correspondence between 

 the above approximate theory, and that of long waves in water 

 (Art. 147). By the substitution of the words condensation of air 

 for elevation of water, and rarefaction for depression, the two 

 questions become identical. 



The analogy becomes however less close when we proceed to 

 a higher degree of approximation. 



Spherical waves. 



167. Let us suppose that the disturbance is symmetrical with 

 respect to a fixed point, which we take as origin. The motion is 

 then necessarily irrotational, so that a velocity-potential &amp;lt;p exists, 

 which is a function of r, the distance of any point from the origin, 

 and t, only. If as before we neglect the squares of small quan 

 tities, we have 



,.i dt 



rJ** 

 Now 



whence, writing p = p (l -f s), and neglecting the square of s, we 

 find 



(10), 



where c has the same value as in (5). 



To form the equation of continuity we remark that, owing to 

 the difference of flux across its inner and outer surfaces, the space 

 bounded by the spheres r and r + dr is gaining matter at the rate 



d ( 2 d&amp;lt;f&amp;gt;\ , 

 - -j- 4nrr 2 p -f- dr. 

 dr\ ^ dr) 



The same rate being also expressed by 



