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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



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well's equations and the boundary conditions for this structure is ob- 

 tained by integrating plane waves of the form of (27) traveling at all 

 possible angles \l/, the integration being subject to a weighting factor 

 G(\l/) to obtain the most general field. The coordinates (r, cp) refer to the 

 physical system and the coordinate \{/ identifies a plane wave traveling 

 along a particular // axis. We have thus in an (/-, (p, z) coordinate frame: 



2t Jo 



GirP) 



Recognizing that 



Ey 



E. 



-i(kyy+kzz) 



(28) 



dxl/ 



and 



an integration results over the variable f . Because of the uniqueness of 

 the field as a function of (p, the only term containing (p, G(^ — (p), must 

 be a periodic function in its argument. A typical mode is formed by 

 choosing one of the terms of the Fourier series of G(^ — <p), namely 



in(.!;—(p) 



We find from (28) that 



Er^ 



n 



E/-Jn{p) + EyJn'ip) 

 P 



n 



EjJip) + Ey-J,Xp) 



p 



EJ,Xp) 



-i(kzZ+n<(>) 



(29) 



i 



where En is that partial expansion of the total field E, corresponding to 

 the number of angular variation «, and p = kyr. There are two values of 

 p corresponding to the two A-alues of A^ , and each leads to a partial wave. 

 Let .4 and B be the respective partial wave amplitude; satisfying the 

 boundary conditions on E^ and E^ , we have from (29) : 



1 



./,/(pi) 



nA 

 Pi 



Jnipi) - Jnip-d = 



Afx V„(pj) + BJ^ip.^ = 



(30a) 

 (30b) 



where pi and p-y are defined for r = A', the radius of the cylinder 



Recog- 



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