136 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Thus, the slow wave is characterized by equal voltages of unlike sign on 

 the two helices, and the fast wave by equal voltages of like sign. It fol- 

 lows that the electric field in the annular region between two such coupled 

 concentric helices will be transverse for the slow wave and longitudinal 

 for the fast. For this reason the slow and fast modes are often referred 

 to as the transverse and longitudinal modes, respectively, as indi- 

 cated by our subscripts. 



It should be noted here that we arbitrarily chose h and x positive. A 

 different choice of signs cannot alter the fact that the transverse mode is 

 the slower and the longitudinal mode is the faster of the two. 



Apart from the interest in the separate existence of the fast and slow 

 waves as such, another object of interest is the phenomenon of the simul- 

 taneous existence of both waves and the interference, or spatial beating, 

 between them. 



Let V2 denote the voltage on the outer hehx; and let Vi , the voltage 

 on the inner halix, be zero at z = 0. Then we have, omitting the common 

 factor e'" , 



(2.3.4) 



Since at 2 = 0, Fi = 0, Vn = — V(^ . For the case we have considered we 

 have found Fa = — V^ and Vn = V^ . We can write (2.3.4) as 



Fi = I {e~'^' - e-^n 



V, = ^ {e''^' + e-'n 



(2.3.5) 



F2 can be written 



= Ye-"'''''^''^''' cos [-jj^(r, - Vi)z\ 

 In the case when x = 6, and /Si = /32 = /8 



F2 = Ye"'^' cos Wiix + h)^z\ (2.3.6) 



Correspondingly, it can be shown that the voltage on the inner helix is 



y, = j\Tfr^^' sin Wiix + h)^z\ (2.3.7) 



The last tAvo equations exhibit clearly what we have called the spatial 

 beat phenomonou, a wave-like transfer of power from one helix to thc^ 



