SYSTEMS WITH NON-LINEAR REACTANCE 427 



if we use the complete exponential expressions for the tiigtjnietric 

 functions, since these are real. 



Accordingly we shall call the electromotive force of the generator 



^. =f [e'"''' + ^-'""']. (5) 



We shall assume that this is accompanied by an alternating current, 



ig = ^^ [^'(".'+9.) -f e-'(«,(+M]. (6) 



\\c shall call the force exerted by the mechanical generator 



and the acc()mi)anying alternating velocity 



Vm = -^ [g '(--'+«".) + g- '{"",'+<'-.']. (8) 



When the corresponding displacements, obtained liy integration of (6) 

 and (8), are substituted in the last term of (3), the resulting electro- 

 motive force is found to consist of components of frequencies. 



Ws = Wg + Wm, (9) 



COd = COg — C0„,, (10) 



which tend to set up currents at the frequencies of the sidebands. 



If such currents flow and we substitute the charges associated with 

 them, together with that from (6), in the last term of (2), we find, in 

 the force on the plate, components of frequency Wm, and a variety of 

 other frequencies including zero, i.e., a steady force. If these produce 

 displacements which are again substituted in (2), and the process is 

 continued, we arrive finally at the entire series of frequencies given by 

 mwg ± noom, where m and n are integers. 



We shall now introduce the limiting assumption that the plate is 

 resonant at or near cj^, and not at any other frequency. The im- 

 pedance at that frequency will then be small and the response to the 

 driving force at that frequency relatively large. At the frequencies of 

 all the other components of the force the mechanical impedance will 

 be relatively very high ; and we will not be making a violent assumption 

 if we say that it is so high that the velocities of response at all the 

 other frequencies are negligible. [There may be some response to the 

 steady force, consisting of a slight change in the position of equilibrium 

 about which the vibrations occur. This can be taken care of by 

 saying that the coefficients in (2) and (3), while constant for any 



