14 BELL SYSTEM TECHNICAL JOURNAL 



The natural transmission modes which satisfy these equations have the 

 form 



E = /„(r) e^'^'^+^o^ . e^'""-^' 2-0 



Each of these modes conforms to the same equations as a wave traveling 

 in a transmission line with an impedance and phase velocity dependent upon 

 the mode. In a straight cylinder with perfectly conducting walls, there 

 exists no coupling between the different modes so that any and all can 

 exist without interacting. If the conductivity of the walls in a straight 

 circular cylinder is finite, it produces a resistive coupling between modes of 

 equal azimuthal index {n in equation 2-0). In copper tubing and at the 

 frequencies now obtainable (co < 10'-) this coupling effect is negligible. 



A stronger coupling may be caused by deviations of the wave guide from 

 the shape of a straight circular cylinder. The deformation considered 

 in the present analysis consists in a circular bend of the axis, as shown 

 schematically on Fig. lb. 



In such a circular bend the longitudinal coordinate is replaced by the 

 product of the bending radius R by the bending angle d: 



z = Re 



This transforms the first two component equations of curl E into 



diREe) dE, 



Rrdip Rdd 

 dEr d(REe) 



= — icon H^ 



Rdd Rdr 



The variable R can be eliminated by the relation 



R = Ro — r cos (f 



where Ro is the bending radius of the cylinder axis. The coordinate 6 can 

 be replaced by a longitudinal coordinate s, measured along the cylinder 

 axis. Hence, 



5 = BRo 



The progressive modes which we investigate have the approximate form 



Hence 



— = Rn — = — Ra V 



dd ds 



— r= jn for all field components. 



