Interference Phenomena. 539 



quantity is not more than a certain fraction of a period, which 

 we may roughly take to be a tenth, the energy gained is prac- 

 tically the same in both cases. But as n increases, the first 

 pendulum will gradually accumulate an amount of energy 

 decidedly greater than the second, provided there is no fric- 

 tion. If the friction is sufficiently large so that in the 

 interval nr, at which the non-agreement of phase between 

 the blows and the pendulum comes to be important, the effects 

 of a single blow are practically obliterated, the two pendu- 

 lums will show no appreciable difference in energy, whatever 

 the time. If, on the other hand, not knowing beforehand 

 whether there is any friction or not, we observe that the two 

 sets of blows delivered at intervals t and t + r produce a 

 markedly different effect on the pendulum, we may fix a 

 lower limit to the possible amount of friction that can exist, 

 and may say that a single blow must produce an effect wdiich 

 lasts through a number n of periods, where the value of n 

 must be greater than that at w T hich the difference in phase 

 becomes detrimental. 



From the effects produced by a succession of blows we may 

 pass to those produced by a continuous periodic force. A 

 vibrating system without friction, whose natural period is 

 27r/n, will, if acted on from rest by a force E cos fct, show T a 

 displacement 



E 



« = 



n 2 —K 2 



2 (cos Kt — cos ni), 



2E (n + K\ ± . /n — fc\ 



The latter form of the equation shows that if n and k are 

 nearly equal, the motion is approximately represented by a 



vibration of amplitude 2 _ 2 sin — ^— t, the maximum velocity 

 being n k 



u 



E . n — K ft (n — K) 2 t*\ 



if (n — tc)t is sufficiently small. If n is the maximum velocity 

 in the special case that n = K, 



\2r2 



u — u _ (n — fc) 2 t 2 

 uo 24 



Supposing we are dealing with luminous vibrations, and 

 that our eye can easily appreciate a difference of intensity of 

 2 per cent., that is a difference in amplitude of 1 per cent., 

 we find that the time t at which we begin to draw a clear 



