36 BELL SYSTEM TECHNICAL JOURNAL 



in the nomenclature which has become virtually standard. In this system, 

 TE denotes transverse electric modes, or modes whose electric Lines lie 

 in planes perpendicular to the cylinder axis; TM denotes transverse mag- 

 netic modes, or modes whose magnetic lines lie in transverse planes. The 

 first numerical index refers to the number of nodal diameters, or to the order 

 of the Bessel function associated with the mode. The second numerical 

 index refers to the number of nodal circles (counting the resonator boundary 

 as one such) or to the ordinal number of a root of the Bessel function asso- 

 ciated with the mode. On the end plates, the distribution does not depend 

 upon the third index (number of half wavelengths along the axis of the cylin- 

 der) used in the identiiication of resonant modes in a cylinder. This con- 

 siderably simplifies the problem of presentation. The orientation of the 

 field inside the cavity and hence the currents in the end plate depend on 

 other things; thus the orientation of the figures is to be considered arbitrary. 

 The plates also apply to the corresponding modes of propagation in a cir- 

 cular waveguide as follows: The background shading represents the in- 

 stantaneous relative distribution of energy across a cross section of guide. 

 For TE modes, the current streamlines depict the E lines; for the TM 

 modes, they depict the projection of the E lines on a plane perpendicular 

 to the cylinder axis. 



Side Wall: 



The current distribution in the side walls is easily obtained from the 

 field equations of Fig. 1. For TM modes, the currents are entirely longi- 

 tudinal; their magnitudes vary as cos (6 cos nirz/ L. This distribution is so 

 simple as not to require plotting. 



For TE modes, the situation is more complicated, since both Hz and He 

 exist along the side wall. The current streamlines are given by the solu- 

 tions of the differential equation 



dz DHe ,.,. 



de-~2H/ ^^^^ 



By .separation of the variables, the solution is found to be 



Contour lines of constant magnitude of the current are given by 



\k\D 



In the above, C and A' are j)arameters, different values of which correspond 

 to difTerent streamlines or contour lines, respectively. 



log (C cos (6) = 



log cos ksZ. (15) 



2 ^ sin (d cos ksZj -\- (cos fd sin k^z)' = K\ (16) 



