EQUA TIONS FOR TRA VELING-WA VE TUBE 393 



We see that Ci has the form of a capacitance per unit length. We can, for 

 instance, redraw the transmission-Hne analogue of Fig. 2.3 as shown in Fig. 

 7.1. Here, the current / is still the line current; but the voltage V acting on 

 the beam is the line voltage plus the drop across a capacitance of Ci farads 

 per meter. 



Consider as an illustration the case of unattenuated waves for which 



Ti = i/3i (7.5) 



r = i/3 (7.6) 



where /3i and /3 are real. Then 



" = L 203? - If) + cj " (^-^ 



In (7.7), the first term in the brackets represents the impedance pre- 

 sented to the beam by the "circuit"; that is, the ladder network of Figs. 

 2.3 and 7.1. The second term represents the additional impedance due to 

 the capacitance Ci , which stands for the impedance of the nonsynchronous 

 modes. We note that if /3 < ft , that is, for a wave faster than the natural 

 phase velocity of the circuit, the two terms on the right are of the same 

 sign. This must mean that the "circuit" part of the impedance is capacitive. 

 However, for /3 > ft , that is, for a wave slower than the natural phase veloc- 

 ity, the first term is negative and the "circuit" part of the impedance is 

 inductive. This is easily explained. For small values of ^ the wavelength of 

 the impressed current is long, so that it flows into and out of the circuit at 

 widely separated points. Between such points the long section of series 

 inductance has a higher impedance than the shunt capacitance to ground; 

 the capacitive effect predominates and the circuit impedance is capacitive. 

 However, for large values of ^ the current flows into and out of the circuit 

 at points close together. The short section of series inductance between 

 such points provides a lower impedance path than does the shunt capaci- 

 tance to ground; the inductive impedance predominates and the circuit 

 impedance is inductive. Thus, for fast waves the circuit appears capacitive 

 and for slow waves the circuit appears inductive. 



Since we have justified the use of the methods of Chapter II within the 

 limitations of certain assumptions, there is no reason why we should not 

 proceed to use the same notation in the light of our fuller understanding. 

 We can now, however, regard V not as a potential but merely as a convenient 

 variable related to the field by (7.2). 



From (2.18) and (7.3) we obtain 



r rr.(g/<3'P jv^. 

 " = L2(r;-r') - <oC, J • ^^* 



