MAGNETRON AS GENERATOR OF CENTIMETER WAVES 215 



mode frequency from that of a single uncoupled resonator. If the coupling 



between resonators were in some way gradually reduced, all mode frequencies 



would converge to the value for a single uncoupled resonator. 



The complete anode potential wave for a mode specified by the parameter 



;; has been given in equation (12). Each of its traveling components is 



represented by a fundamental of periodicity n and a series of so-called 



Hartree harmonics of periodicities k = n -{- pN, /> = 0, ±1, ±2, • • • . 



Any sinusoidal component for which the number of complete cycles around 



N . 

 the anode is greater than - is thus a harmonic of the complete field pattern 



iV 

 for one of the modes whose fundamental is of periodicity n = 1, 2, • • • , — . 



Physically distinguishable modes of oscillation exist only for the values of 



N . . 



n less than or equal to ^ , including zero. However, this accounts for only 



N 



- + 1 of the ;V modes of oscillation which one expects a system of N reso- 

 nators to possess. The reason for this is that in general the frequency of a 

 mode specified by the parameter n (except for the values and - ) is a double 



root for a perfectly symmetrical anode structure. The mode is thus a 

 doublet and is said to be degenerate. One would expect this on mathe- 

 matical grounds from the fact that the general solution of expression (12) 

 has four arbitrary constants, whereas a singlet solution of the system of 

 second order differential equations specifying the oscillations should have 

 no more than two. 



The nature of the degeneracy of the modes of the resonator system is 

 perhaps most clearly seen by investigating what happens when the sym- 

 metry of the system is destroyed by the presence of a disturbance or per- 

 turbation at one point, a coupling loop in one of the cavities for example. 

 Such a disturbance provides the additional boundary condition needed to 

 remove the degeneracy. 



Consider the effect of the perturbation on the nth mode. It represents 

 an admittance shunted across the otherwise uniform closed ring of resonators. 

 This shunt admittance may be represented by eFo , e being a complex num- 

 ber, taken to be small for simplicity, and Yq the characteristic admittance 

 of the closed resonator system. In such a system, a potential wave inci- 

 dent upon the disturbance eFo , having an amplitude a at the disturbance, 

 breaks up into a reflected wave and a transmitted wave which, if e is small, 



have the amplitudes — ^ o and f 1 — - j a, respectively [see equation (30) 



