50 BELL SYSTEM TECHNICAL JOURNAL 



will be given only to those waves for which Ez does not change sign over a 

 cross-section of the beam. By inspection of equation (A-23), it is evident 

 that this requirement is automatically satisfied if L > 0. On the other 

 hand, if L is negative, one has 



Ezi ^Jo(Vv ^-?- z) ( A-23a) 



Thus attention will be limited to those roots which satisfy 



^■'^l^Z < 2.405 (A-40) 



As 



where 2.405 is the first zero of the Bessel function in equation (A-21a). 



Returning to Fig. 7, portions of three different F2 curves are plotted: one 

 for W^ = 0.01, one for IP = 0.0152 and one for W- = 0.02. All three 

 curves are for (a/A,,) = 0.16. The intersections which represent roots which 

 satisfy the inequality (A-40) are marked with arrows. Evidently there are 

 either four real roots of this type or there are two real roots and a complex 

 conjugate pair, the distinction being determined by the value of W. Thus 

 there is a critical value of IF- (in this case it is 0.0152) for which two of the 

 real roots are identical. This identical pair is indicated by two arrows 

 near the minimum of the Fi curve at Z = 1. 



A pair of conjugate complex roots means that there are an increasing wave 

 and a decreasing wave. Thus for each value of b and (a/Xs) there is a least 

 value of IP below which the tube will have no gain. 



It can be shown that the critical tangency of the Fi and F2 curves occurs 

 at a value of Z which is less than b'^ away from unity. Very little error will 

 be incurred, then, by assuming that this critical point occurs at Z = 1 

 if b is small. 



Letting Z = 1 in equation (A-37), and using equations (A-39) one has 



8(IIV^)^ - 1 = f^^(2WXj/o(V 8aiV^F^ 2WxJ\ (A.41) 

 \A'o(27ra/X.)/i (\/8(IlV^)' - 1 2ira/K)J 



whe:e Wm is the critical value of H'. Equation (A-41) determines (IFV^)^ 

 as af unction of (a/X,). This relationship is plotted in Fig. 4. 



We will find that there will be an increasing wave in the range 

 Wm < IF < oc . The calculation of the gain in this interv-al would be very 

 laborious since Bessel functions of complex argument would be involved. 

 However, a good approximation can be made when b is small. The real 

 part of Z will always be near unity and the imaginary part will be found to 

 be less than b/2. Therefore one can let Z = 1 in equation (A-37) where it 

 multiplies the factor (lira/Xa) in the argument of the Bessel functions and 

 let Z — 1 = U in the right-hand side of Equation (A-37). With these 



