672 Equation of Progressive Wave. 



The result of eliminating p between this last equation and 



may be regarded as the integral of the original differential 

 equation, satisfying the conditions imposed. 



The two moving planes which we have considered above 

 in connexion with a part of the disturbance at density p have 

 been shown to travel in general with unequal velocities. It 

 is worth noting, however, that the difference in the velocities 

 involves only small quantities of the second order. Thus 



V?VMa/| 



dp 



p 



dp 



V dp + )„/\/ dp P * 



where Mis some mean value of 9 \f 4- between the limits 

 of the integral. Again, 



(OoV dp \/ dp /3 v dp 



= \/% + {p\- l p)p\/ d dp 



The quantities M and p\ / -^-differ by a small quantity, and 



. P / 1 1 \ 



they are each multiplied by the factor ( — ""-) which is 



itself small. V ^o P ' 



If we put the two velocities equal, we are led to the same 

 relation between pressure and density as Rankine found 

 necessary to secure the propagation of a wave of permanent 

 type. The mere fact that the two planes will in general 

 travel with different speeds can be proved by quite elementary 

 •considerations. 



