468 WAVES OF EXPANSION. [CHAP. X 



inversely as the square root of the density, provided the relative 

 densities be determined under the same conditions of pressure and 

 temperature. 



257. The theory of plane waves can also be treated very 

 simply by the Lagrangian method. 



If f denote the displacement at time t of the particles whose 

 undisturbed abscissa is x t the stratum of matter originally in 

 cluded between the planes x and as + &e is at the time t + 8t 

 bounded by the planes 



and *+?+ 



so that the equation of continuity is 



where p is the density in the undisturbed state. Hence if ,9 

 denote the condensation (p /O O )/PO&amp;gt; we nave 



dx 



The dynamical equation obtained by considering the forces 

 acting on unit area of the above stratum is 



o**- dp (S) 



*&quot;~ ; - 



These equations are exact, but in the case of small motions 

 we may write 



........................... (4), 



_ . 



Substituting in (3) we find 



where c 2 = K/p ........................... (7). 



The interpretation of (6) is the same as in Arts. 167, 255. 



