354 GENERALIZED COORDINATES 



The solution is then found to be 



E 



^ = A l cos (k^t - l ) + - cos ( Pl t - 7l ), 

 A #! A p l p 1 



or, since a l = ftJQ 



~n< 



^ = A! cos (kj - X ) + - -, - -5T- cos (^ - 7l ). 



Thus the variation hi ^ is now compounded of a simple 

 harmonic motion of frequency k lt and also one of frequency p lt 

 the frequency of the impressed force. 



We notice that if p t is very nearly equal to k 19 then the second 

 vibration is of very large amplitude. In the limiting case in which 

 Pi = ^i> th e amplitude of the second vibration becomes infinite, 

 but now the two vibrations are of the same period, so that they 

 may be compounded, and we cannot say that the resultant vibration 

 is one of infinite amplitude, because we do not know the values of 4i 

 and 1 , and these may just be such as to destroy the infinite ampli- 

 tude of the second term. The result we have obtained may be 

 enunciated in the following form : 



When a system is acted on ~by a periodic force, of frequency 

 very nearly equal to that of one of the principal vibrations of the 

 system, then the forced oscillations will be of very great amplitude. 



This is known as the principle of resonance. 



The principle is one of which many applications appear in nature. For 

 instance, a bridge, not being absolutely rigid, maybe regarded as a system 

 having a number of free vibrations. A body of men marching over the 

 bridge in regular step will apply a periodic force, and if the period of their 

 step happens to nearly coincide with one of the free periods of the bridge, 

 the amplitude of the vibrations forced in the bridge may be so large as to 

 endanger the bridge. For this reason troops are ordered to " break step " 

 when crossing a bridge. 



Again, a ship is not perfectly rigid, and so will possess a number of free 

 vibrations. The motion of its engines will apply a periodic force of period 

 equal to that of its revolution, and if this coincides with that of one of the 

 vibrations of the ship, large pulsations will be set up. This can be reme- 

 died by altering the speed of the engine until it no longer is in resonance 

 with the free vibrations of the ship. 



