CURVED WAVE GUIDES 15 



— may be expressed by a prime : 



dr 



Thus the equations with curl E become 



jnEs Es sin (p RoTE^ „ 



r Kq — r cos <p Kq — r cos (p 



— RoTEr „/ , Es cos <p . „ 



— E, + „ = —joifJt H^ 



Rq — r cos ip Ro — r cos <p 



E^ -\- rE^ = jnEr = —joijxr Us 



For gradual bends 



i?o » a > r 



One may therefore approximate 



Ro . 1 I '' 



p ^ 1 + P cos (^ 



K(j — r cos ^ ico 



It is convenient to introduce the symbol 



a 



which is proportional to the couphng coefficient. All powers of c greater 

 than the first will be neglected. One can now write the approximate field 

 equations in the curved cylinder : 



iflE C T 



— ' + TE^ -\- - E, sin (f -\- cV - E^ cos <p = —jun Hr 2-1 



r a a 



— TEr — Es — cT - Er cos (f -\- - Es cos <p — — jcou H^ 2-2 



a a . 



E^ -f rE^ — jnEr = —joourU, 2-3 



•^'!L^» -f YH^ + - £?. sin ^ -f cr - H^ cos <p = /we £, 2-4 



r G a 



-rZ^, - H[ - cT - Hr cos ^ + - Hs cos ^ = ;«e £, 2-5 



a a ■ 



H^ + rH^ — jnHr = jcoer Es 2-6 



The coupUng terms all contain the factor cos (p or sin tp. This means 

 that every transmission mode is coupled only to modes with an azimuth 

 index differing from its own by ±1. 



