438 BELL SYSTEM TECHNICAL JOURNAL 



alternating electromotive force of an unrelated higher frequency, pro- 

 vided that the impedances of the system at these two frequencies and 

 their various combinations satisfy certain relations, and the applied 

 electromotive force exceeds a threshold value. When the oscillations 

 are negligible at all frequencies except these two and their sum and 

 difference, the most favorable condition, lowest threshold voltage, oc- 

 curs when the plate vibrates at its resonant frequency, and the electric 

 circuit is resonant at the applied frequency and at the difference fre- 

 quency, and anti-resonant at the sum frequency. Once the oscillations 

 start, the current of the applied frequency remains constant with in- 

 creasing voltage. Under the most favorable conditions the rates of 

 energy dissipation at the plate and difference frequencies are in the ratio 

 of the frequencies. 



Other Applications; Raman Effect 



While in the case considered above the production of oscillations was 

 associated with a particular type of non-linearity, the application of the 

 principle is much more general. Here the non-linearity occurs in 

 what might be called a mutual stiffness, serving to couple two degrees 

 of freedom. It is not essential, however, that the non-linearity occur 

 in a mutual impedance nor that the impedance be of the stiffness or 

 negative reactance type. So long as the connected system is such as 

 to provide the proper impedances, oscillations may occur in connection 

 with any non-linear reactance. 



A non-linear reactance, as here used, may be defined as any energy- 

 storing element in which the coefficient of inertia is a function of the 

 velocity, or that of stiffness is a function of the displacement, or any 

 mechanical, electrical or electromechanical analog, of such an element. 

 For a non-linear inertia, as in an iron core inductance coil, however, 

 the power varies inversely as the frequency ; instead of directly as for a 

 non-linear stiffness. 



A special case, in which one of the new frequencies is an exact sub- 

 multiple of the driving frequency, has been studied by a number of 

 workers from Rayleigh ^ down to Pedersen.® 



Another special case may be of some interest to physicists because it 

 provides a model of the Raman effect. The transition from the con- 

 denser to the molecular model will be made in two steps. For the 

 first suppose that instead of making the resonant mechanical member 

 one plate of a condenser, we attach the moving part to a point on its 

 support by an elastic string under tension, the direction of the string 



" Rayleigh; "Theory of Sound," Sec. Ed., Vol. 1, p. 81. 



« Pedersen; Jr. Acous. Soc. Anier., Vol. VI, 4, p. 227, April, 1935. 



