COUPLED HELICES 133 



simple closed expressions of any generality. The best that can be done 

 is in the form of curves, with step-wise increases of particular param- 

 eters. These can be of considerable value in particular cases, and when 

 exactness is essential. 



In this paper we shall proceed by giving the "transmission-line" type 

 theory first, together with the elaborations that are necessary to arrive 

 at an estimate of the strength of coupling possible with coaxial helices. 

 The "field" type theory will be used whenever the other theory fails, or 

 is inadequate. Considerable physical insight can be gotten with the use 

 of the transmission-line theory; nevertheless recourse to field theory is 

 necessary in a number of cases, as will be seen. 



It will be noted that in all the calculations to be presented the presence 

 of an electron beam is left out of account. This is done for two reasons: 

 Its inclusion would enormously complicate the theory, and, as will 

 eventually be shown, it would modify our conclusions only very slightly. 

 Moreover, in practically all cases which we shall consider, the helices are 

 so tightly coupled that the velocities of the two normal modes of propaga- 

 tion are very different, as will be shown. Thus, only when the beam 

 velocity is very near to either one or the other wave velocity, will 

 growing-wave interaction take place between the beam and the helices. 

 In this case conventional traveling wave tube theory may be used. 



A theory of coupled helices in the presence of an electron beam has 

 been presented by Wade and Rynn,^ who treated the case of weakly 

 coupled helices and arrived at conclusions not at variance with our views. 



2.2 Transmission Line Equations 



Following Pierce we describe two lossless helices by their distributed 

 series reactances Xio and A'20 and their distributed shunt susceptances 

 Bio and ^20 . Thus their phase constants are 



/3io = V^ioA'io 



Let these helices be coupled by means of a mutual distributed reac- 

 tance Xm and a mutual susceptance B^ , both of which are, in a way 

 which will be described later, functions of the geometry. 



Let waves in the coupled system be described by the factor 



jut — Tj; 



e e 



\v 



here the F's are the propagation constants to be found. 



