Sound- Waves of Finite Amplitude. 323 



and a plane C fixed in the still air gives 



Pl (V-v)-p v = O; (4) 



while from the principle of momentum, 



p 1 v(V-v)=p l -p ; (5) 



the pressure p being a function of p only, since the tempera- 

 ture is supposed to be constant throughout. If we assume 

 for this case the truth of Boyle's law, so that p = a 2 p always, 

 (5) becomes 



pi ( a 2-Yv + v 2 )=p a 2 , .... (6) 



which together with (4) is sufficient to determine V and p ± 

 when v and p are given. Taking all these quantities to 

 remain constant throughout the motion, we see that at each 

 instant the following conditions are satisfied : — 



(i) Every necessary condition between A and B, since 

 density and velocity are there constant with respect 

 to space and time ; 

 (ii) Every necessary condition to the right of B, since the 



air there is at rest and in a constant uniform state ; 

 (iii) Equality between the velocity of A and that of the 



air in contact with it ; 

 (iv) At B, the conservation of mass and momentum, which 

 are necessary conditions, and which, together with 

 our supposition that the temperature is somehow 

 maintained uniform, are sufficient to determine what 

 takes place at B *. 

 Moreover, if at a time t (reckoned from the instant when 

 A was impulsively started into motion) we take the distance 

 of B from A to be (V — v)t, so that initially B coincides with 

 A, the initial conditions are satisfied. 



Thus the assumed motion satisfies all the necessary con- 

 ditions ; it is therefore the actual motion. 



7. Let us now examine what occurs when no heat is 

 allowed to pass by conduction or radiation ; a state of things 

 much more nearly realized in practice. Suppose the motion 

 of A and the condition of the undisturbed air to be the same 

 as in the last section, while the (constant) velocity of B is 

 now called V', and the density and pressure of the air between 

 A and B (called p', p ! respectively) are also taken to be uni- 

 form and constant. At each instant, in place of (4) and (5), 

 we shall now have 



p'(V'.-i>)- Po »=0, (7) 



p'v(V'-v)=p'-p (8) 



* Energy appears to be lost, because dissipatively produced bent is 

 conducted away isotbermally. 





