PLANE WAVES OF SOUND 175 



being now a function of the space-coordinate x only, the 

 acceleration of the air-particles will be given by udu/dx as 

 in ordinary dynamics. Hence, considering the acceleration of 

 momentum of the mass which at the instant considered lies 

 between the planes x and x + 8x, we have 



du dp 

 pu- r = -/- ...................... (1) 



dx dx 



Also, since the same amount of matter crosses each section in 

 unit time, we have 



pu = const. =??i, ..................... (2) 



say. Hence mdu/dx dp/dx, and 



p= C mu, ........................ (3) 



or p-p = m(u -u) = m* ---, ......... (4) 



\Po pJ 



where the symbols p Q , p , u refer to the parts of the medium 

 which in the original form of the question were undisturbed. 

 This gives the special relation referred to. In terms of the 

 volume per unit mass we have 



p-p =m*(v Q -v), .................. (5) 



which is the equation of a straight line on the indicator 

 diagram. A relation of this type does not hold for any 

 known substance, whether under the adiabatic or the iso- 

 thermal condition, and could in any case only apply to a 

 limited range, since the volume would otherwise shrink to 

 nothing under a certain finite pressure. 



If, however, the range of density be small, the equation (5) 

 can be identified with 59 (6) provided m z =Kp . Since m=p ^o> 

 where u is the wave- velocity in the original form of the problem, 

 this gives u Q 2 = ic/p , in agreement with 59 (8). The process 

 is equivalent to choosing m so that the straight line (5) shall 

 be a tangent at the point (v , p ) to the curve which on the 

 indicator diagram gives the effective relation between p and v. 



The condition (5) was obtained in different ways by 

 Earnshaw (1860) and Rankine* (1870). 



To ascertain the character of the continual change of type 



* W. J. M. Rankine (1820 72), professor of engineering at Glasgow, 

 185572. 



