Sound- Waves of Finite Amplitude. 319 



Then, since A and B are fixed in the fluid, they are ap- 

 proximately moving with the respective velocities u ly u 2 ; in 

 and n being taken sufficiently small. On the same under- 

 standing, the mass of fluid between A and B (referred to 

 unit surface) = mp ± + np 2 ; and since this mass must remain 

 constant, 



^(m Pl + np 2 )=0; 



i. e. in the limit, when m and n are infinitesimal, 

 dm dn ~ 



*-5- + **:* ' 



or 



pi(V-u 1 )=p 2 (V-u 2 ). ..... (I, 



Similarly, if p x and p 2 are the pressures corresponding to 

 pi and p 2 , the principle of momentum gives:— 



p x —p 2 = rate of change of momentum between A and B 



= Uipi(V— u x ) — u 2 p 2 (V — u 9 ) (2) 



If the energy per unit volume corresponding to density p 

 (in the absence of bodily motion) is called xO 3 )? the principle 

 of energy would further give 



p 1 u 1 —p 2 u 2 = rate of change of energy between A and B 



= 3£ M i^i 2 + x(pO) + n (i p* u * 2 + X W ) \ 

 = {W+xWKV-tO- {iP2^ 2 +x(p2)}(y-^l (3) 



Since (1), (2), and (3) involve only the instantaneous values 

 of w 1? jo 1? m 2 , p2^ an( i V, together with explicit functions of such 

 values, while the space- and time-variations of all these quan- 

 tities are absent from the equations, it is evident that the 

 conditions to be satisfied at the surface S are the same as if 

 u v Pi? u 2, /°2? V were absolute constants. We conclude then, 

 that, with our assumptions, a surface of discontinuity cannot 

 be propagated through a fluid with any velocity, uniform of 

 variable, except under that special law of pressure for which 

 progressive waves are of accurately permanent type. 



3. What, then, becomes of waves of finite amplitude after 

 discontinuity has set in ? We may emphasize this difficulty, 

 and at the same time obtain a clue to its solution, by con- 

 sidering the following case (fig. 2): — A is a piston fitting a 



Z2 



