480 WAVES OF EXPANSION. [CHAP. X 



Hence, we have 



(r) = 2 r-^ (r) - ^ %i fa 



.(13). 



2c Xl in 

 The complete value of </> 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 formulae 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 



s/s =(r-ct)/2r. 



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 (j> 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), 



General Equation of Sound 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, 



* " On Some Points in the [Received Theory of Sound," Phil. Mag., Jan. 1849 ; 

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



