SYSTEMS WITH NON-LINEAR REACTANCE 429 



These equations are of the second degree and so are not so simple 

 of solution as are the linear equations of circuit theory which they 

 formally resemble. We note, however, that if, in the last three, we 

 assume 7„ and 0„ to l)e constant, they become linear. We may 

 therefore solve them as linear equations, with this assumption, provided 

 we bear in mind that the resulting impedances will not be linear unless 

 the oscillations are so small that the second and third terms of (13) 

 can be neglected compared with the first. 



Let us make this assumption and explore the properties of the 

 resulting linear system represented by (14), (15) and (16). If we 

 calculate F„,e'*"' and take the ratio F„,e''>""IVme'^'", this will be the 

 analog of the impedance of an analogous electric circuit as measured 

 in the mesh corresponding to vibration of the plate at frequency co^. 

 This ratio, which we shall call ZJ, may be thought of as the me- 

 chanical impedance of the plate when the circuit is activated by the 

 electrical generator. Following circuit theory, as applied to vacuum 

 tubes, let us call Zm' the active impedance of the plate, and Z,„ the 

 passive impedance. The value of the active impedance, when ex- 

 pressed in terms of resistances and reactances, is found to be 



ZJ = {R,n + iX„0 + 



(17) 



We see that the active impedance differs from the passive impedance 

 by two terms, each of which represents the efTect of the impedance 

 of the electric circuit at one of the side frequencies. The second term 

 of (17), which depends on the impedance at the sum frequency, is 

 identical in form with the impedance added to an electric circuit, ■• at 

 a frequency, co, by a transformer of mutual inductance, M, provided 

 that 



MW = -4-^^ ; (18) 



the impedance of the secondary circuit is equal to Zs', and the re- 

 actances of the primary and secondary w^indings are included in X„ 

 and Xs, respectively. The third term which depends on the im- 

 pedance of the electric circuit at the difference frequency, is similar 

 except that the effective resistance is negative. 



It is this negative resistance which makes possible the type of free 

 oscillations here described. To interpret it, let us start with the small 



* Bush, v.; "Operational Circuit Analysis," John Wiley & Sons, Inc., 1929, p. 50, 

 Eq. (66). 



