160 



DYNAMICAL THEORY OF SOUND 



correct to the second order of small quantities. 

 states are a finite distance apart we 

 require, to know the manner of transi- 

 tion. For changes along an isothermal 

 line pv = p v we have 



When the two 



dv=p Q v \og ...(7) 

 For variations along an adiabatic 



V % 



Fig. 59. 



59. Plane Waves. Velocity of Sound. 



The theory of plane waves of sound is very similar to that 

 of the longitudinal vibrations of rods ( 43). We assume that 

 the motion is everywhere parallel to the axis of x, and is the 

 same at any given instant over any plane perpendicular to this 

 axis. We denote displacement from the equilibrium position 

 by f . The symbols p, p, % are supposed to refer at the time 

 t to that plane of particles whose undisturbed position is 

 x] they are therefore functions of the independent variables 

 x and t. The constant equilibrium values of p, p are dis- 

 tinguished as PQ, p . 



The dilatation A was defined in 40 as the ratio of the 

 increment of volume to the original volume, viz. 



(1) 



In the present branch of the subject it is usual to introduce 

 a symbol s to denote the " condensation," i.e. the ratio of the 

 increment of density to the original density ; thus 



(2) 



Since v=l/p, we have 



.(3) 



The stratum of air which was originally bounded by the 

 planes x and#+& is at time t bounded by the planes x+% and 



