470 WAVES OF EXPANSION. [CHAP. X 



the same as the kinetic energy of the whole mass when animated 

 with the maximum velocity &amp;lt;ra. 



The rate of transmission of energy across unit area of a plane moving 

 with the particles situate in it is 



p ~ =p&amp;lt;ra sin (kx -a-t+e) , (i). 



The work done by the constant part of the pressure in a complete period is 

 zero. For the variable part we have 



Ajo = KS= K-~ 



CkOG 



Substituting in (i), we find, for the mean rate of transmission of energy, 



Wa 2 = Po &amp;lt;7 2 a 2 xc (iii). 



Hence the energy transmitted in any number of complete periods is exactly 

 that corresponding to the waves which pass the plane in the same time. 

 This is in accordance with the general theory of Art. 221, since, c being 

 independent of X, the group-velocity is identical with the wave- velocity. 



Waves of Finite Amplitude. 



259. If p be a function of p only, the equations (1) and (3) of 

 Art. 257, give, without approximation, 



d^ = dp d^ 

 dt* pfdp da?&quot; 



On the isothermal hypothesis that 



this becomes - 7 -~ = 



dt PQ 



In the same way, the adiabatic relation 



These exact equations (3) and (4) may be compared with the similar 

 equation for long waves in a uniform canal, Art. 170 (3). 



