CURVED WAVE GUIDES 17 



Due to the curvature of the guide, this desired mode is coupled to all modes 

 wiiich have the azimuthal index number 1. 



However, for low curvatures, this coupling is very loose and only causes 

 appreciable effects if it can act over a great length of wave guide without 

 {)hase interference. 



This means that the disturbing mode must have nearly the same phase 

 velocity as the desired mode. It so happens that in a perfectly conducting 

 circular cylinder there exists one mode, the TMn , which has exactly the 

 same velocity as the TEoi . Such a coincidence is called "degeneracy." 



In the analysis of very gradual bends, only this TMn mode need be 

 considered. It is characterized in a straight guide by the following equations : 



n =1 3-16 



E, IT i^T-Tnz J\{y) I I N J\{y) , .^ 



-Ev'2 = -£26 • — — COS \ip + ^0) = e-i — — cos ip 3-1/ 



y y 



The TMu mode can be polarized in all directions. But since only the 

 component directed toward 



is excited by the wave guide curvature, <pq has been omitted in the last 

 term of eq. (3-17). 



dJiiy) . T r \ • u T ^J 



Er2 = «2 — T^ sm (p = e2Ji{y) sm (p, where J — -— 



ay dy 



£j2 = ^^ Jiiy) sin <p 

 1 2 



H ^<i — 



~- Jiiy) sin <p 



VT^ 



„ _ _e2J^o Jiiy) 



tiri — :;p; cos if 



jyFa y 



//.2 = 



In a perfectly conducting wave guide the x defined by eq. 3-8 is 



X = X2 = Xo and 



r2 = Ti = i/3i 



In a wave guide with an intrinsic impedance per 3-13, 



(1 - j)Ri , ,,. 



X2 = Xo — and 3-18 



y\av 



\/l — v' at] 

 From 3-15 and 3-19 one finds ai = a-yf- 3-20 



Vl — v or] 



