MAGNETRON AS GENERATOR OF CENTIMETER WAVES 217 



of periodicity n of equation (13) may now be seen. Expression (13) may 

 be rewritten in terms of 6 measured from the disturbance eFo in this way: 



Vrf = Z) (^'U - Bk) cos io:t - ke + y) 



k 



+ E 25* COS {^^^ cos L/ + (^^^jj cos ke (28) 



+ E 2Bk sin (^-^\ cos Lot + (^^^)] sin ^0. 



When the degeneracy is removed, the first summation representing the 

 traveling wave vanishes since the amphtudes Ak and Bk are equal for the 

 reasons indicated. Furthermore, the second term involving cos kd terms 

 is then the component of the doublet whose frequency changes, and the 

 third term involving sin kd terms is the component of undeviated frequency. 

 For the latter component the disturbance appears at a node and hence has 

 no effect. The reverse holds for the cos kd solution. For it, the frequency 

 deviation depends on the magnitude of the disturbance through the quan- 

 tity e. The disturbance caused by the coupling loop in an actual magnetron 

 resonator system is sometimes sufficient to split the components into dis- 

 tinguishable resonances. 



Thus an unsymmetrical multicavity resonator system in general has two 

 modes of different frequency for each value of n. With respect to the 

 asymmetry as origin, one of these modes has a cosine-like field pattern, the 



N 

 other a sine-like field pattern. This is true for w=l, 2, •••,-— 1, con- 

 tributing N — 2 modes. The remaining two modes of the resonator sys- 



N 

 tem, for which u = and - , are singlet modes even in the symmetrical 



anode. This may be seen from the analysis demonstrating the splitting of 



the nth mode into two components. For the n = mode, since the anode 



potential wave is independent of azimuthal angle, the solution cj = co,^ for 



which a = —b represents the trivial case of zero amplitude at all points. 



N j^ 



Similarly, for the n = - mode (the tt mode) the w = co„ — , , solution 



2 ** 



for which a = b yields a cosine-like pattern giving zero potentials at each 

 anode segment, an equally trivial case. Thus each of the N modes of the 

 multicavity resonator system have been accounted for. 



As an example, plots of the field configurations for tlie modes of a magne- 

 tron having eight resonators are shown in Fig. 23. For clarity, only the 



