198 Theory of a Non-Conservative Gas [CH. ix 



Before the time t 0, the coordinate <j> is describing the free harmonic 

 vibration expressed by equation (474). The impulse Udt acting from t = to 

 t = dt, sets up an additional free vibration of initial displacement zero and 



velocity ; the displacement of this additional vibration at any subsequent 



CL 



time is therefore - sin pt. Compounding all these vibrations with the 



original vibration expressed by equation (474), we obtain for the displace- 

 ment < at any instant subsequent to t = T, the well-known solution* 



t'=r 



<f> = Acospt+Bs\npt + - - I U t = t >smp(t-t')dtf (478). 



ap 



f=o 



237. Let us write 



f=T 



ap 



.(479), 



F = - I U t = t ' sin pt' dt' 

 ap J 



ap 



t' = r 



X+iY= t Ut-veWdt' (480), 



ap J 



then equation (478) assumes the form 



<}> = (A Y)cospt + (B + X) siapt (481). 



The energy of vibration is |-(a< 2 + &< 2 ), or 



iZjiYj y\2 i (23-i.Y yi (482) 



If this is written in the form 



the first term represents what would have been the energy of vibration had 

 the encounter not taken pla^e ; the two remaining terms represent the 

 change effected by the encounter. 



At the beginning of the encounter (f> is given by putting t = in 

 equation (474), so tha*t 



<fr = A, <j>=pB. 



On averaging over all the molecules of the gas these quantities may be 

 supposed to be equally likely to be positive as negative, so that on averaging 

 over all encounters we find that the mean value of BX AY may be taken 

 to be zero. Hence from expression (483), the increment in the energy of 



* Rayleigh, Theory of Sound, i. 66. 



