212 WAVES IN AIR. [CHAP. vm. 



The value of a (=~ Q ) is the same for both these particles, so that 

 we have, by (36) and (37), 



The latter equation may be written 



x x _ 



which shews that any value p of the density is propagated from 



particle to particle with the velocity c . The rate of propagation 



Po 







Po 

 If II C 



in space is given by * , viz. it is c log a + - , or 

 &amp;gt; h a 



+ C- + .M .............................. (39). 



ro 



For a wave travelling in the positive direction we must take the 

 lower signs throughout. If it be one of condensation (i.e. p &amp;gt;/o ), 

 u is, by (35), positive. We see as before that the denser parts of 

 the wave gain continually on the rarer, and at length overtake 

 them, when a bore is formed, and the subsequent motion is beyond 

 the scope of this analysis. 



Eliminating x between (6) and (37), and writing for c log a 

 its value u, we find fora wave travelling in the positive direction, 



which may, in virtue of (35), be written 



This formula is due to Poisson*. Its interpretation, leading of 

 course to the same results as before, was discussed and illustrated 

 by Stokes and Airy, in the Philosophical Magazine, in 1848-9. 



The theoretical result that violent sounds are propagated faster 



* Journ. de VEcole Polyt. t. 12, cahier 14, p. 319. Quoted by Earnshaw and 

 Kankine. 



