PLANE WAVES OF SOUND 181 



only to the corresponding pure tones, but to their octaves, as 

 well as to certain "combination- tones," whose occurrence 

 reminds us again, that the principle of superposition is no 

 longer valid. We shall have occasion to refer to this investi- 

 gation at a later period (Chap. X). 



The analogous phenomenon in tidal theory is the production 

 of "over- tides," which are in fact appreciable, and have to be 

 provided for in the Harmonic Analysis referred to in 39. 



We have seen that the main effect of finite amplitude is 

 that in a progressive wave the gradients, both of pressure and 

 of density, tend to become infinite. This has suggested the 

 question whether a wave of discontinuity might not finally be 

 established, analogous to a "bore" in water-waves. To examine 

 into the possibility of such a wave we take the question in its 

 simplest form, and assume that the circumstances are everywhere 

 uniform, except for the sudden transition at the plane of dis- 

 continuity. Further, by the superposition of a certain uniform 

 velocity, we reduce the problem to one of steady motion in 

 which the plane in question is fixed. 



The symbols p , p , u will then be supposed to refer to the 

 region to the left of this plane, whilst the values of the corre- 

 sponding quantities on the 

 right are denoted by p, p, u. 

 Since in every unit of time 

 the same mass (m) of fluid 

 crosses any unit area normal 

 to the direction of flow, we have 



pu = p u =m, or u = mv, u Q = mv (33) 



Again, since in unit time a mass m has its velocity changed from 

 UQ to u, the momentum of the portion of air included between 

 two planes in the positions indicated by the dotted lines in 

 Fig. 63 is increasing at the rate m (u - u ), whence 



p -p = m(u-u Q \ (34) 



or, in virtue of (33), 



p p = m?(v v ), (35) 



in agreement with (5). If we now superpose a uniform velocity 



