1280 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



I 



Fig. 1 — Serpentine bends. 



uniform bend with the same bending radius. This will be shown in the 

 following analysis. 



A treatment of serpentine bends given previously by Albersheim con- 

 siders only circular and sinusoidal curvatures. Furthermore, it does not 

 show all the effects of serpentine bends that we are interested in. We 

 shall present a more general and complete analysis here. The only re- 

 striction we have to make is that the power exchanged in one section of 

 the serpentine bend from the TEoi mode to any of the coupled modes or 

 vice versa is small compared to the power in the mode from which it has 

 been abstracted. The curvature may be any function of distance along 

 the serpentine bend. 



The general results indicate that normally a serpentine bend causes an 

 additional attenuation to the TEoi mode. Part of the TEoi power which 

 travels temporarily in one of the coupled modes suffers the higher attenu- 

 ation of this coupled mode. Formulae for this increase in attenuation 

 constant are obtained for periodically supported guides from the deflec- 

 tion curve given by the theory of elasticity. 



The couphng between the TEoi and TMu waves causes only a very 

 slight increase in TEoi attenuation, since the difference in propagation 

 constant for these two modes is very small. In fact, if there were no 

 difference in propagation constant, as in the round guide of infinite wall 

 conductivity, coupling between TEoi and TMu waves in serpentine bends 

 would not affect the TEm transmission at all. 



Coupled modes which have a larger difference in phase constant than 

 TMii but still are close to TEqi cause a serious increase in TEoi attenua- 

 tion at certain critical frequencies. If a multiple of the beat wavelength 

 between the TEoi mode and a particular coupled mode is equal to the 

 supporting distance, power is converted continuously into the coupled 



