494 



BELL SYSTEM TECHNICAL JOURNAL 



Take the simple case of two identical coupled circuits, Fig. 3. The 

 network equations are 





-^§ + ^^ + ^§ = °- ^«) 



It is evident by inspection that there are two possible modes of oscillation. 

 In one mode h = Ii and in the other /i = —h- The natural frequency of 

 the f irst mode is oji = \I\/{L — M)C and that of the second mode C02 = 

 l/-\/{L + M)C. The magnetic energy function is 



T = iLli - Mhh + H/2 



«,_„['_aj£.J,,„,«[W,J. 



(9) 



Thus the sum and the difference of the currents in the two meshes oscillate 

 independently. 



% c+^ 



Fig. 3- 



Il=l2 Il=-l2 



-Two possible modes of oscillation in a symmetric two-mesh circuit. 



More generally a network with n meshes possesses n independent modes of 

 oscillation. In each mode the ratios of the mesh currents Ii, h, • - • In are 

 prescribed by the network parameters and the connections of the network 

 elements, but the relative strength of the oscillation remains arbitrary. 

 When we pass to networks with distributed parameters such as sections of 

 transmission lines and cavity resonators, we find merely that the number of 

 independent modes of oscillation is infinite. In the case of a nondissipative 

 uniform transmission line with both ends shorted, the natural frequencies of 

 the various oscillation modes are proportional to the sequence of integers: 

 1, 2, 3, . . ..The current distribution for the n-ih mode is given by sin {mrx/i), 

 where I is the length of the section ; but the actual amplitude remains arbi- 

 trary. For the gravest mode {n = 1) the middle part of the line section 

 behaves as a capacitor and the ends as inductors. For the higher modes the 

 line is subdivided into sections, some of which act primarily as capacitors 

 and others as inductors. 



In the case of cavity resonators of some simple shapes, such as paral- 

 lelopipedal, cylindrical and spherical, the determination of the oscillation 

 modes is a fairly simple problem. The dynamical equations of the resonator 



