1254 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



TMii modes. The finite conductivity of the walls introduces a slight 

 difference in the propagation constants and in a 2-inch pipe at 5.4 mm 

 wavelength a bending radius of a few miles can be tolerated with about 

 double the attenuation constant of the TEoi wave. To get more differ- 

 ence in the phase constants of the TMn and TEoi modes, one might 

 consider a dielectric layer next to the wall of the waveguide. Since the 

 electric field intensity of the TEoi mode goes to zero at the wall but the 

 electric field intensity of the TMn mode has a large value there, one might 

 expect a larger effect of the dielectric layer on the propagation character- 

 istics of the TMn wave than on the TEoi wave. 



In doing this, however, one has to be aware of the influence the 

 dielectric layer will have on the propagation characteristics of the TEi^ 

 modes which also couple to the TEoi wave in curved sections. The TE12 

 wave couples most strongly to the TEoi wave and of the TEi^ family it 

 is the next above the TEoi w^ave in phase velocity. With the dielectric 

 layer one has to expect this difference in phase velocity to be decreased 

 and consequently the mode conversion to the TE12 wave to be increased. 



In the next section we will solve the characteristic eciuation of the 

 cylindrical waveguide with a dielectric layer for the normal modes and 

 arrive at approximate formulas for the phase constants. We will also 

 compute the increase in attenuation of the normal modes as caused by 

 the dielectric losses in the layer. The change in wall current losses as 

 caused by the dielectric layer is of importance here only for the TEm 

 wave and we will calculate it only for this wave. 



In order to know what bending radii can be tolerated with a dielectric 

 coat, we have to evaluate the coupled wave theory for small differences 

 in propagation constants between the coupled waves. This will be done 

 in Section III. Especially we will consider serpentine bends which occur 

 in any practical line with discrete supports. In that situation, at certain 

 critical frequencies, when the supporting distance is a multiple of the 

 beat wavelength between TEoi and a particular coupled wave, we will 

 have to expect serious effects on the propagation constant of the TEm 

 wave. 



In Section IV we will combine the results of Sections II and III and 

 establish formulas and curves for designing circular electric waveguides 

 with a dielectric coat. We will distinguish there between two different 

 applications. The first case is to negotiate uniform bends of as small a 

 radius as possible and the second is to tolerate large bending radii which 

 may occur in a normally straight line. 



At transitions from plain waveguide to the dielectric-coated guide, 

 power of an incident TEoi wave will be partly converted into higher 



