158 Mr. T. H. Biivclock on 



Now the potential energy o£ tlu; vibnitions is given by 



Tims, denoting by Yx the density of potential energy at any 

 point, we have 



^^ = -|i.(V..) (3) 



And the mean extra force on any length of the string due to 

 the vibrations is equal to the difference of the mean values of 

 Yx at the ends. 



Hence for a simple progressive wave along the string there 

 is on the average no extra force, and for simple waves in 

 both directions there is a mean force which alternates in 

 direction along the string. 



But consider the case in which there is attenuation in a 

 progressive wave ; so that 



y = Ce--''^ cos (pt—/3x) (4) 



The mean value of V^- now is 



i(^2^/32)W.C2.e-2- (5) 



Hence in this case there is a mean force alono- the strino- in 



the direction of propagation of the waves. 



§ 2. We may consider in detail the following example : — 

 From 07= —CO to .27 = 0, there is no frictional dissipation of 



energy, and the equation of motion of the string is 



But from ^ = to .^' = go there is a frictional term and the 

 equation is of the form 



-dt' ^ -dt ^^' ^^^ 



Then if there is a progressive wave along the first part, 

 there will be reflexion and absorption at ^ = 0. Suitable 

 solutions of (6) and (7) for the two regions are 



y = Acos^(^-3 + ^^"'{H^+3+^}- ' f^) 



i^ = Ce-'^' cos {(pt- fix) +^'}, (9) 



where the first term in (8) represents the incident wave, the 



