(33) 



GUIDED WAVK rUorAGATION THROUGH GYHOMAGNKTH' MKDIA. 11 969 



then 



In the uumagnetized case equations (30) and (31) reduce to 



_ ^ii-MCi\ Kijair) ^ _cojXoKi{ocir) , -,. 



j(|8"^ - «-Mo€i) Ko{air) jaiKo{air) ' 



and to 



upo Iijair) 

 jax Io{air) ' 



One might be led to believe that the search for sohitions of Maxwell's 

 equations Anth angular dependence e^'"^ will lead to ri'^ order Whittaker 

 functions (just as in the isotropic case this dependence leads to ri'^ 

 order Bessel functions). Such is not the case, however. Unless n = 0, 

 one is led to two simultaneous second order equations for E^ and H^ , 

 and the character of the problem is changed completely. 



3.2. The Cylindrical Helix 



We are now in a position to derive the characteristic equation for a 

 close wound cylindrical helix and approximated by a helically conducting 

 sheet surrounded by ferrite. We confine the discussion to the case in 

 which the ferrite is in actual contact with the helix, P'ig. 1 1 ; the case of 

 finite separation discussed for the planar hehx (Section 2.3) would be 

 too cumbersome here. Losses are neglected. If they are not excessive, they 

 can be deduced from the curves for the propagation constant in the loss 

 free sample by differentiation, as outlined in Sections 2.1 and 4.15, 

 Part I. 



The boundary' conditions are just the same as in the planar case. In 

 Section 2.3 they were stated in terms of admittances, and it is only 

 necessary to substitute for these the admittances just derived for cy- 

 lindrical geometry. Thus for Hy/Ez we substitute Ftm , and for HJE^ 

 we write Fte = 1/^te • 



If superfixes i and c refer to the inside and outside of the helix (in 

 Section 2.3 on the plane helix i and e were denoted by " — ", and " + "), 

 the characteristic equation is 



{F™^" - FTM^^^lr=.oCOtV = (FtE^" - FTE^'^^ro, (34) 



where ro is the radius of the helix. 



* The ai)peararice of difTerent factors (2x — 1) and (3^^ — x^) is .siini)ly duo to 

 the way the functions H , M are normalized in the literature. 



