480 WAVES OF EXPANSION. [CHAP. X 



Hence, we have 



1_ 



2c (13). 



The complete value of c/&amp;gt; is then given by (6), viz. 



As a very simple example, we may suppose that the air is initially at rest, 

 and that the disturbance consists of a uniform condensation S Q extending 

 through a sphere of radius a. The formula) then shew that after a certain 

 time the disturbance is confined to a spherical shell whose internal and 

 external radii are ct-a and ct + a, and that the condensation at any point 

 within the thickness of this shell is given by 



The condensation is therefore positive through the outer half, and negative 

 through the inner half, of the thickness. This is a particular case of a 

 general result stated long ago by Stokes*, according to which a diverging 

 spherical wave must necessarily contain both condensed and rarefied portions. 



We shall require shortly the form which the general value 

 (14) of $ assumes at the origin. This is found most simply by 

 differentiating both sides of (14) with respect to r and then 

 making r = 0. The result is, if we take account of the relations 

 (8), (10), (12), 



(15). 



General Equation of Round Waves. 



265. We proceed to the general case of propagation of ex 

 pansion-waves. We neglect, as before, the squares of small 

 quantities, so that the dynamical equation is as in Art. 263, 



r 8 ?-^ m 



&quot; dt 



* &quot; On Some Points in the Keceived Theory of Sound,&quot; Phil. Mag., Jan. 1849 ; 

 Math, and Pliy. Papers, t. ii., p. 82. 



