210 WAVES IN AIR. [CHAP. VIII. 



where r lf r z , s lt s 2 are constant. The region within which r is 

 variable advances, that within which s is variable recedes, so that 

 after a time these regions separate, and leave between them 

 a space in which r = r lt s = s 2 , and which is therefore free from 

 disturbance. The original disturbance is thus split up into two 

 waves travelling in opposite directions. In the advancing wave 

 s = s z , and therefore u=f(p}2s 2 , so that both density and 



particle-velocity advance at the .rate \/ ~r + u - This velocity 



of propagation is greater, the greater the value of p. The law 

 of progress of the wave may be illustrated by drawing a curve 

 with x as abscissa and p as ordinate, and making every point of 

 this curve move forward with the above velocity. It appears 

 that those parts move fastest for which the ordinates are greatest, 

 so that finally points with larger overtake points with smaller 

 ordinates, and the curve becomes at some point perpendicular 



to x. The functions -/- , -7- are then infinite, and the above 

 dx dx 



formulae are no longer applicable. We have in fact a bore/ or 

 wave of discontinuity. Compare Art. 152. 



Earnshaivs Method. 



172. The same results follow from Earnshaw s investigation ; 

 which is however somewhat less general in that it embraces waves 

 travelling in one direction only. If for simplicity we suppose p 

 and p to be connected by Boyle s law 



p = c*p, 



the equation (4) is replaced by 



. 



or, writing # = #-ff, so that y denotes the absolute position at 

 time t of the particle x, 



df dx* 



