424 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



Considering pi to be positive imaginaiy, as pi -^ rx , (32) becomes 



K 



M 



tpi 



n 



(34) 



Substituting for pi , we have 





riK 



(35) 



Thus, as fj. approaches zero from \'akies greater than zero, the propaga- 

 tion constant tends to become singular. Physically, however, n does not 

 vanish but approaches a small imaginary value caused by ferrite losses. 

 The propagation constant k^ becomes complex and takes on a large 

 imaginary component, signifying large guide attenuation. Since these 

 losses occur in the limited neighborhood of m = 0, we may construe this 

 waveguide behavior as corresponding to a system resonance. 



In the region m < 0, pi becomes real while p2 remains imaginary. The 

 right-hand side of (32) is now composed of onh^ real arguments. Since the 

 zeros of diiferent order Bessel functions alternate, the right side of (32) 

 contains a succession of poles and zeros, leading to an infinite number of 

 branches with each containing a solution pi to the equation. Thus there 

 are an infinite number of propagating modes corresponding to each value 



Fig. 5 — Frequenc}' characteristics of Polder tensor components 



