Energy acquired by Resonators from incident Waves. 189 



The free vibrations are assumed to have considerable per- 

 sistence, and the coefficient of decay is e~ qt , where 



q = 27rakrW(^IW) = 27T*p/cryW, . . . (5) 



We now suppose that the resonator is exposed to pri- 

 mary waves whose velocity-potential is there 



£ = a^ (6) 



The effect is to introduce on the right hand of (3) the term 

 &irr 2 <roi.ipe ipt ; and since the resonance is supposed to be accu- 

 rately adjusted, p 2 = yLt/M'. Under the same conditions icl 2 p/dt 2 

 in the third term on the left of (3) may be replaced by 

 —p dpjdt, whether we are dealing with the permanent 

 forced vibration or with free vibrations of nearly the same 

 period which gradually die away. Thus our equation 

 becomes on rejection of the imaginary part 



M'^+4wpi 



dt 



fip 



= -4< 



2 a ap sin pt, 



(7) 



which is of the usual form for vibrations of systems of one 

 degree of freedom. For the permanent forced vibration 

 M.'d?p/dt 2 + fip = absolutely, and 



dp 

 dt 



a. sin pt 



kr 2 



(8) 



The energy located in the resonator is then 



M* 2 



2&V 



(9) 



and it may become very great when M is large and r small. 

 But when M is large, it may take a considerable time to 

 establish the permanent regime after the resonator starts 

 from rest. The approximate solution of (7), applicable in 

 that case, is 



oc COS pt .. 



(10) 



q being regarded as small in comparison with p ; and the 

 energy located in the resonator at time t 



