172, 373.] CHANGE OF TYPE IX ADVAXCIXG WAVES. 213 



than gentle ones is confirmed by the remarkable experiments of 

 Regnault* on the motion of sound in pipes. 



173. The above investigations shew that a plane wave of air 

 experiences in general a gradual change of type as it proceeds. 

 It is worth while to inquire under what circumstances a wave 

 could be propagated without change. 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 l9 p lt u l , and^&amp;gt; 2 , p 2 , u 2 , respec 

 tively. If as in Art. 152 we impress on everything a velocity c 

 equal and opposite to that of the waves we reduce the problem to 

 one of steady motion. Since the same amount of matter now 

 crosses in unit time each section of the tube, we have 



Pi(c - uj = p 2 (c - u 2 ) = const. = m, say (40). 



The quantity m, denoting the mass swept past in unit time by a 

 plane moving with the wave, in the original motion, is called by 

 Rankine-f&quot; the mass- velocity or somatic velocity of the wave. 



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

 and B is p 2 p^ , and the gain per unit time of momentum of this 

 mass is 



m(c- MJ) -m(cu^. 



Hence P*-Pi = ( u z -u l ) (41). 



Combined with (40) this gives 



p^m\=p^ + m\ i (42), 



where, still- following Rankine, we denote by s [= /T 1 ] the bulki- 

 ness/ i.e. the volume per unit mass, of the substance. Hence the 

 variations in pressure and bulk experienced by any small portion 

 of the medium as the wave passes over it must be such that 



p + m*s = const (43). 



This condition is not fulfilled by any known substance, whether at 

 constant temperature, or when free from gain or loss of heat by 



* Htm. de VAcad. t. 37. Quoted in Wiillner s Experimental pliysik, t. 1, 

 p. 685. An abstract of the experiments is printed at the end of the second edition 

 of Tyndall s Sound. 



t Phil. Trans. 1870. 



