PLANE WAVES OF SOUND 177 



no additive constant being necessary if we assume that f = 

 in the parts of the medium not affected by the wave. This 

 may also be written 



?= + clog(l+s), .................. (16) 



by 59 (3). Another form is 



P/P* = e^' c ...................... (17) 



When s is infinitesimal the formula (16) reduces to f = + cs, in 

 agreement with 60. 



To find the rate at which any particular value of s is 

 propagated, in either of these cases, we note that the value of 

 9f/9a? which is associated with the particle x at the instant t 

 will have been transmitted to the particle x + $>x at the instant 

 t + $t, provided 



|f 8M.ga.-o, 



oxdt da? 

 i.e. by (12) and (14), 



&c c(l+s)& = ................ (18) 



The phase s is therefore propagated with the velocity 



J = + c(l + S ) .................. (19) 



relative to the undisturbed medium. To find the rate of 

 propagation in space we have to take account of the total 

 variation of x + , which is 



The required velocity is therefore 



('+DM S ........ ; 



The lower sign relates to a wave travelling in the direction of 

 #- positive. It appears from (16) that positive values of f are 

 then associated with positive values of s, as in the approximate 

 theory of 60 ; but the formula (20) shews that the velocity 

 of propagation is greater, the greater the value of s. The parts 

 of the wave where the density is greater therefore gain con- 

 tinually on those where it is less. Thus if the relation between 

 s and x be exhibited graphically, the curve A in the annexed 

 L. 12 



