1358 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



as exp (-{-ind). Then from (7b), for any pitch angle t/', 



^ p' v' [tan ^ + 7i(l - vflvvf 



a -n = - 



i 



CXn = 



a 



-n — a„ = 4 



a p2 _ ^2 (1 - v'yi' 1 + tan2 yf, 



^ p V [tan \p — n{l — v^Y' /pvf 



ap^ - n'' {1 - v2)i/2 1 + tan- rp 



^ np V tan ip 



ap"^ — 'n? 1 -\- tan^ yp 



The mode which varies as exp( — zn0) has lower loss if \p and n have the 

 same sign. 



The TM„m attenuation constants are independent of polarization and 

 are given by (7a). 



yp = 90° 



These "helices," with wires parallel to the axis of the waveguide, 

 should propagate TM„m modes with losses approximately the same as 

 in copper pipe. For the TE„„i modes, (7b) gives 



TEnm modes 



2 2 



a + iA(3 = " -T^—2 (^ + ^■'?) 



a(l — j'-)^'- p^ — v} 



IV. NUMERICAL SOLUTIONS FOR ZERO-PITCH HELICES 



The main interest in helix waveguide is for small pitch angles where 

 the TEoi attenuation is very low. The propagation constants of various 

 lossy modes in helix guides of zero pitch have been calculated by solving 

 the characteristic equation (6) numerically. These calculations will now 

 be described. 



Equation (6) is first simplified by setting yp — Q and replacing the 

 Hankel functions with their asymptotic expressions. The condition for 

 validity of the asymptotic expressions, namely 



I r2a I » I {^n - l)/8 I 



is well satisfied in all cases to be treated here. Equation (6) may then be 

 rearranged in the dimensionless form 



Fni^a) = i^oaf [{nhafJn\Ua) - (/5ca)'(fia)V„''(ria)] 



- i{^,af [{nhaf -f (^oa)^(e' - Z6")(^a)V/(fia)/n(fia) (10) 

 = 

 There is no difference between the propagation constants of right and 



