476 WAVES OF EXPANSION. [CHAP. X 



If the variation of density be slight, the relation (4) may, 

 however, be regarded as holding approximately for actual fluids, 

 provided m have the proper value. Putting 



c ............ (5), 



we find c 2 = /c/p .............................. (6), 



as in Art. 255. 



The fact that in actual fluids a progressive wave of finite 

 amplitude continually alters its type, so that the variations of 

 density towards the front become more and more abrupt, has led 

 various writers to speculate on the possibility of a wave of dis 

 continuity, analogous to a bore in water-waves. 



It has been shewn, first by Stokes*, and afterwards by several 

 other writers, that the conditions of constancy of mass and of 

 constancy of momentum can both be satisfied for such a wave. 

 The simplest case is when there is no variation in the values of 

 p and u except at the plane of discontinuity. If, in Rankine s 

 argument, the sections A, B be taken, one in front of, and the 

 other behind this plane, we find 



m 



and, if we further suppose that u 2 = 0, so that the medium is at 

 rest in front of the wave, 



(8), 



p2 \pi - P* 



m -L /(ffl - ff 2 ) (Pi - P 2 )V /QX 



and u, = c = 11 (&quot;/ 



ft \ 



The upper or the lower sign is to be taken according as p l is 

 greater or less than p 2 , i.e. according as the wave is one of 

 condensation or of rarefaction. 



These results have, however, lost some of their interest since 

 it has been pointed out by Lord Rayleigh -f* that the equation of 

 energy cannot be satisfied consistently with (1) and (2). Con 

 sidering the excess of the work done on the fluid entering the 



* I. c. ante p. 475. 



t Theory of Sound, Art. 253. 



