﻿196 Mr. R. Moon on the Theory of Sound. 



If x be measured positively to the right, we shall have at all 

 points to the right of the original disturbance 



¥{x + a{l-e)t}=0, f{x + a{\-e)t}=\ 9 



whatever be the value of t; so that for all points to the right 

 of the original disturbance the motion will be represented by 



„= \^l-p{x-a{\ + e)t\ + l\\-f{ x - a {\ + e)t}^ (14) 



Moreover, for all points to the right of the original disturbance 

 we shall have 



F{x-a(I + e)t}=0 } f{x-a{l + e)t} = l 



for all values of t between zero and that value of / which gives 



x — a(l +e)t=l ) 



x — l 

 i. e. between £ = Oand£= -7= r : in other words, at a point x 



a(l+e) r 



to the right of the original disturbance there will be neither ve~ 

 locity nor condensation until t has attained the latter value, at 

 the same time that when t has that value the velocity and con- 

 densation immediately begin to have significance. 



It follows, therefore, that the disturbance has taken the time 



— — - to traverse the space x — /, having been propagated to 



the right with the velocity a(\+e). 



Multiplying (14) by 1— e } and neglecting terms in the result 

 involving e 2 , we get 



(l-e)v=i¥\x-a{l + e)t\ + a ( l ~ e \ l-f\x-a(l + e)ty].{16) 



Also, since 



dx p ~D(l+«) ~ ' 

 substituting in (15) this value of -j-, we get, multiplying by a, 



as=\Y\x-a{l + e)t} + ^—^ [l-f{*-a(I + e)t}] ; 



therefore 



as=(l—e)v. (17) 



Hence, when the motion is represented by the differential equa- 

 tion (12), we shall have a disturbance propagated to the right with 



