1354 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



li n 9^ 0, these conditions correspond to circularly polarized TE„ 

 waves, in which the wall currents follow the direction of the conducting 

 sheath. If n = 0, then i^ = 0, and one has TEom modes with circum- 

 ferential currents only. \ 

 The equations can also be satisfied with 



bn = ttn = bn = 

 ttn 9^ 



yl^ = 90° 

 Jnitia) = 



corresponding to the TM„m modes (either circularly or linearly polarized) 

 of a perfectly conducting pipe, which are associated with longitudinal 

 wall currents only. 



In the general case when the jacket is not perfectly conducting and 

 the helix pitch angle is not restricted to special values, it is necessary to 

 solve (5) simultaneously for the field amplitudes. The equations admit a 

 nontrivial solution if and only if the determinant of the coefficients of 

 the a's and b's vanishes. The transcendental equation which results from 

 equating the determinant of the coefficients to zero is 



f: 



f 1 tan V — r— ) r it^ \ ~ ^ Moeo 



= t. 



(6) 



(2)// 



f 2 tan 1^ - -— I , — CO MoeoCe - te ) , - 



f2ay H^V'{^2a) Hr.^"\^2a) J 



The solution of this equation determines the propagation constant ih 

 and therefore the attenuation and phase constants a and /3. When ih 

 has been obtained, it is a straightforward matter to determine the a and 

 b coefficients from equations (5) and the electric and magnetic fields 

 from (2), (3), and (4). 



"It is well known^ that the only pure TE or TM modes that can exist 

 in a circular waveguide with walls of finite conductivity are the circularly 

 symmetric TEom and TMom modes. The other modes are all mixed modes 

 Avhose fields are not transverse with respect to either the electric or the 

 magnetic vector. In general the modes of helix waveguide are also mixed 

 modes, and no entirely satisfactory scheme for labeling them has been 

 proposed. In the present paper we shall call the modes TE„m or TM„m 

 according to the limits which they approach as the jacket conductivity 

 becomes infinite, even though they are no longer transverse and their 



8 Reference 7, p. 526. 



