THEORY OF VIBRATIONS 27 



The particle comes asymptotically to rest but does not oscillate; 

 in fact we may easily see that it passes once at most through 

 its zero position. This type of motion is realized in the case of 

 a pendulum swinging in a very viscous liquid, and in "dead-beat" 

 galvanometers and other electrical instruments, but it is ' of 

 little interest in acoustics. 



If k = 2n, exactly, the solution of (3) is of the form 



x = (A +Bt)e~ nt , .................. (16) 



as to which similar remarks may be made. 



12. General Dissipative System with One Degree of 

 Freedom. Effect of Periodic Disturbing Forces. 



The effect of dissipation on the free motion of any system 

 having one degree of freedom is allowed for by the assumption 

 that there is a loss of mechanical energy at a rate proportional 

 to the square of the generalized velocity, so that in the notation 

 of 7 



J <#) = -&?, ............... (1) 



whence aq + bq + cq = Q ................... (2) 



This is of course the same as if we had introduced a frictional 

 force Q = - bq in 9 (3). 



The equation (1) has the same form as 11 (3), and the 

 results will correspond if we put 



n 2 = c/a, r = 2a/6 ................... (3) 



When the dissipation is small, the rate of decay of the 

 amplitude can be estimated by an independent method, due to 

 Stokes*, which we shall often find useful. The period being 

 practically unaffected by vicosity, a considerable number of 

 oscillations can be fairly represented by 



q = G cos (nt + e), ..................... (4) 



provided C and e be gradually changed so as to fit the altering 

 circumstances. The average energy over such an interval will 

 be Jrc 2 a(7 2 , approximately, by 7 (11); and the rate of dissipa- 

 tion will be 



bcf = iw 2 60 2 (1 - cos 2 (nt + e)}, 



* Sir George Gabriel Stokes (18191903), Lucasian Professor of Mathematics 

 at Cambridge (18491903). 



