261-262] EARNSHAW'S. THEORY. 475 



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

 results as above, for the mode of alteration of the wave as it proceeds, forms 

 the subject of a paper by Stokes f. 



262. The conditions for a wave of permanent type have been 

 investigated in a very simple manner by Rankine J. 



If, as in Art. 172, we impress on everything a velocity c equal 

 and opposite to that of the waves, we reduce the problem to one of 

 steady motion. 



Let A, B be two points of an ideal tube of unit section drawn 

 in the direction of propagation, and let the values of the pressure, 

 the density, and the particle-velocity at A and B be denoted 

 by p lt p lt MI &ndp 2 , p 9 , u 2 , respectively. 



Since the same amount of matter now crosses in unit time 

 each section of the tube, we have 



p l (c-u 1 ) = p 2 (c-u 2 ),^m, (1), 



say ; where m denotes the mass swept past in unit time by a plane 

 moving with the wave, in the original form of the problem. This 

 quantity m is called by Rankine the ' mass-velocity ' of the wave. 



Again, the total force acting on the mass included between A 

 and B is p. 2 -pi, and the rate at which this mass is gaining 

 momentum is 



m (c u^) m(c u t ). 



Hence p 2 -pi = m(u 2 -u 1 ) (2). 



Combined with (1) this gives 



p 1 + m 2 /p 1 =p 2 + m*/p 2 (3). 



Hence a wave of finite amplitude cannot be propagated un- 

 changed except in a medium such that 



p + m*/p = const (4). 



This conclusion has already been arrived at, in a different manner, 

 in Art. 259. 



* " Memoire sur la Theorie du Son," Journ. de VEcole Potytechn., t. vii., p. 319 

 (1808). 



t " On a Difficulty in the Theory of Sound," Phil. Mag., NOT. 1848; Math, and 

 Phys. Papers, t. ii., p. 51. 



J "On the Thermodynamic Theory of Waves of Finite Longitudinal Disturb- 

 ance," Phil. Trans., 1870. 



