474 WAVES OF EXPANSION. [CHAP. X 



so that, if u be the velocity of the particle #, we have 



(vii). 



p 



On the outskirts of the wave we shall have w = 0, p = p . It follows that 

 (7=0, and therefore 



P=Po^ w/c ................................. (viii). 



Hence in a progressive wave p and u must be connected by this relation. 

 If this be satisfied initially, the function <p which occurs in (vi) is to be 

 determined from the conditions at time = by the equation 



To obtain results independent of the particular form of the wave, consider 

 two particles (which we will distinguish by suffixes) so related that the value 

 of p which obtains for the first particle at time ^ is found at the second 

 particle at time t. 2 . 



The value of a (=p /p) is the same for both, and therefore by (vi), with 



l , 



J " 



The latter equation may be written 



shewing that the value of p is propagated from particle to particle at the rate 

 p/p . c. The rate of propagation in space is given by 



(xii). 



This is in agreement with Riemann's results, since on the 'isothermal' 

 hypothesis (dpfdp)^=c. 



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

 signs. If it be one of condensation (p>p ), u is positive, by (viii). It follows 

 that the denser parts of the wave are continually gaining on the rarer, and 

 at length overtake them ; the subsequent motion is then beyond the scope of 

 our analysis. 



Eliminating x between the equations (vi), and writing for c log a its value 

 - u, we find for a wave travelling in the positive direction, 



........................... (xiii). 



In virtue of (viii) this is equivalent to 



........................... (xiv). 



