212 BELL SYSTEM TECHNICAL JOURNAL 



only by the mutual linkage of magnetic lines and which resonate at the same 

 frequency, coo, when uncoupled. 



When such a coupled system is shock excited it is observed that the 

 oscillation amphtude in either of the circuits is modulated at a so-called beat 

 frequency, cob . A fraction or all of the energy in the system, depending on 

 the initial conditions, surges back and forth at this frequency between the 

 circuits similar to the manner in which the energy of motion is exchanged 

 between two coupled pendulums. The total energy in the system is con- 

 stant, the beats differing in phase by 7r/2 radians between the circuits. 



The observation of beats is a manifestation of the fact that the two coupled 

 resonators form a complex system oscillating simultaneously in its two 

 modes for which the frequencies are (coo + ^b) and (coo — cob). The oscilla- 

 tion in either circuit results from the superposition of the two component 

 oscillations in this manner: 



A cos (ojo + wb) t -\- B cos [(wo — cob) t -\- b] = 



( 5\ / 5\ (27) 



{A — B) cos (wo + cob) / + 25 cos ( cob/ — 7 ) cos ( coo/ + ~ 1, 



with a similar expression for the case when A < B. The oscillation may 

 be predominantly of one frequency, that is, almost entirely in one mode, if, 

 for example, A ^ B. In general, the oscillation is a superposition of a 

 steady oscillation in the predominant mode [(wo + wb) ii A > B] and an 

 oscillation whose amplitude varies with the beat frequency, cob . In the 

 special case, when the component oscillations are of equal intensity, A = B, 

 the amphtude of the resultant oscillation in either circuit goes to zero periodi- 

 cally at the frequency cob . This represents the case for which all the energy 

 present in the system is transferred back and forth between the circuits. 



The frequency separation of the two modes arises from the coupled effect 

 of the oscillation in each of the circuits on the oscillation in the other. Thus, 

 in the mode of lower frequency, (wo — wb), the two circuits oscillate in phase 

 and the self induction effect in each circuit is aided by the mutual induction, 

 each circuit behaving as though it were oscillating freely with a greater 

 value of self inductance and hence at lower frequency [equation (17)]. For 

 the mode of higher frequency, (coo + ws), the reverse is true. Here the two 

 circuits oscillate out of phase by tt radians, the mutual induction opposing 

 the self induction and the circuits oscillating as though uncoupled with a 

 smaller value of self inductance and hence at higher frequency. 



If instead of shock exciting the system of two coupled resonators it is 

 forced to oscillate by applying to it a sinusoidal voltage of variable fre- 

 quency, the admittance of the system is found to pass through minima at 

 the mode frequencies (coo + wb) and (coo — wb). Thus it is possible to drive 

 the system and store energy in -^ither of the two modes. For each mode of 



