HELIX WAVEGUIDE 1351 



II. SHEATH HELIX BOUNDARY VALUE PROBLEM 



Ordinary cylindrical waveguide consists of a circular cylinder of radius 

 a, infinite length, and zero (or very small) conductivity, imbedded in an 

 infinite* homogeneous conducting medium. The sheath helix waveguide 

 has the same configuration plus the additional property that at radius a 

 dividing the tAvo media, there is an anisotropic conducting sheath which 

 conducts perfectly in the helical direction and does not conduct in the 

 perpendicular direction. The attenuation and phase constants are deter- 

 mined by solving Maxwell's equations in cylindrical coordinates and 

 matching the electric and magnetic fields at the wall of the guide. 



The helix of radius a and pitch angle \J/ = tan~^ s/2ira is shown in the 

 upper part of Fig. 1. The developed helix as viewed from the inside when 

 cut by a plane of constant 6 and unrolled is shown in the lower part of 

 the illustration. A new set of unit vectors e^ and Cj. parallel and perpen- 

 dicular respectively to the helix direction is introduced. These are re- 

 lated to er , ee , and Cz by 



er X e\\ — ex 



e\\ = ez sin t^ + ee cos rp 



fij. = ez cos \p — e$ sin xj/ 



The boundary conditions at r = a are 



K = E{ = 



Ej = E/ 



where the superscript i refers to the interior region, ^ r ^ a, and the 

 superscript e refers to the exterior region, a '^ r -^ co . An equivalent set 

 of boundary conditions in terms of the original unit vectors is 



E; tan ^p + Ee' = 



E," tan rp + Ee' = Q 



(1) 



e: = e: 



H; tan ,A -f He' = Hf tan ^p + /// 

 We are looking for solutions which are similar to the modes of or- 



* The assumption of an infinite external medium is made to simplify the mathe- 

 matics. The results will be the same as for a finite conducting jacket which is thick 

 enough so that the fields at its outer surface are negligible. 



