COUPLED HELICES 135 



(2.2.6) 



If we make the same substitutions in (2.2.2) we find 



Fi T ZiL /3(/3i?> + /3o:r) . 

 where the Z's are the impedances of the heUces, i.e., 



Z,. = VXJB, 



2.3 Solution for Synchronous Helices 



Let us consider the particular case where (Si = (S-z = |S. From (2.2.5) 

 we obtain 



r' = -I3\l + xb db (x + b)] (2.3.1) 



Each of the above values of T" characterizes a normal mode of propaga- 

 tion involving both helices. The two square roots of each T" represent 

 waves going in the positive and negative directions. We shall consider 

 only the positive roots of T , denoted Tt and Tt , which represent the 

 forward traveling waves. 



Ttj = i/3Vl + xb ± {x + b) (2.3.2) 



If a: > and 6 > 



I r, I > |/3i, I r,| < 1^1 



Thus Vt represents a normal mode of propagation which is slower than 

 the propagation velocity of either helix alone and can be called the 

 "slow" wave. Similarly T( represents a "fast" wave. We shall find that, 

 in fact, X and b are numerically equal in most cases of interest to us; we 

 therefore write the expressions for the propagation constants 



r. = M^ + H(-^ + b)] 



(2.3.3) 



r. = Ml - Viix + b)] 



If we substitute (2.3.3) into (2.2.6) for the case where /3i = (82 = /3 and 

 assume, for simplicity, that the helix self-impedances are equal, we find 

 that for r = Tt 



Y% _ 



for r = T; 



F2 



-— = -f 1 



Yx ^ 



