42, 43] PHENOMENON OF RESONANCE. 153 



Thus our solution is 



v 



47) s = - ^ cos (pt a). 



y(fc_p)* + xv 



The motion represented by this solution is called the forced 

 vibration, for the system is forced to assume the same period as 



that of the extraneous force F, namely - > of frequency ; while 



the frequency of the free or natural vibration would be 

 or without damping - 



The displacement is not in phase with the force, lagging behind 

 it by less than a quarter -period if tana is positive, that is, if h is 

 greater than p, in other words if the natural frequency is greater 

 than the forced. If on the contrary the natural frequency is less 

 than the forced, tana is negative, and since sin a is positive, the 

 displacement is between a quarter and a half -period behind the force. 

 If the frequencies of the forced and free vibrations coincide, tana 

 becomes infinite, the lag is a quarter period, so that the displacement 

 is a maximum when the force is zero and vice versa. Then 47) becomes 



48) s = sin Jit. 



p* 



and if the damping % is small, the amplitude is very large. This is 

 the case in the phenomenon of resonance, of great importance in 

 various parts of physics, including acoustics, electricity, and dispersion 

 in optics. The equation shows how a very small force may produce 

 a very large vibration if the period coincides nearly enough with 

 the natural one, and explains the danger to bridges from the accu- 

 mulated effect of the measured step of soldiers, the heavy rolling of 

 ships caused by waves of proper period, and kindred phenomena. 



Although in the phenomenon of resonance the excursion and 

 consequently the kinetic energy becomes very large, it is of course 

 not to be supposed that this energy comes from nothing as has been 

 frequently contended by inventive charlatans proposing to obtain vast 

 stores of energy from sound vibrations. 1 ) 



If we form the equation of activity, by multiplying 41) by ^> 

 in x 'd(T-\-W) . (ds\* 1 d i/ds\2 



-dt~ - + *(di) = *di[(di)- 



-E*p 



= ( cos a sin pt cospt + sin a cos s 



^2 



1) Of these the United States has produced more than its share. The 

 ignorance of the above mentioned principle enabled John Keely to abstract in 

 the neighborhood of a million dollars from intelligent (!) American shareholders. 



