THE NATURE OF THE WAVES 399 



we see from Fig. 8.10 that B falls as QC is increased. The gain per wave- 

 length varies as BC and, because Q is constant for a given tube, it varies as 

 BQC. In Fig. 8.11, BQC, which is proportional to the gain per wavelength 

 of the increasing wave, is plotted vs QC, which is proportional to the \ 

 power of current. For very small values of current (small values of QC), 

 the gain per wavelength is proportional to the \ power of current. For 

 larger values of QC, the gain per wavelength becomes proportional to the 

 J power of current. 



It would be difficult to present curves covering the simultaneous eflfect 

 of loss {d) and space charge (QC). As a sort of substitute, Figs. 8.13 and 8.14 

 show dxi/dd for (/ = and b chosen to maximize Xi , and dxi/d{QC) for 

 QC = and h = 0. We see from 8.13 that, while for small values of QC 

 the gain of the increasing wave is reduced by \ of the circuit loss, for large 

 values of QC the gain of the increasing wave is reduced by ^ of the circuit 

 loss. 



8.1 Effect of Varying the Electron Velocity 



Consider equation (7.13) in case d = (no attenuation) and () = 

 (neglect of space-charge). We then have 



b\b^jb)= -j (8.1) 



Here we will remember that 



l^e = WUo (8.2) 



-Fi = -j0e{l + Cb) - ->/^-l (8.3) 



Here z'l is the phase velocity of the wave in the absence of electrons, and Uo 

 is the electron speed. We see that 



«o = (1 + Cb)vi (8.4) 



Thus, (1 + Cb) is the ratio of the electron velocity to the velocity of the 

 undislurbed wave, that is, the wave in the absence of electrons. Hence, b 

 is a measure of velocity difference between electrons and undisturbed wave. 

 For b > 0, the electrons go faster than the undisturbed wave; for Z> < 

 the electrons go slower than the undisturbed wave. For b = the electrons 

 have the same speed as the undisturbed wave. 

 li b = 0, (8.1) becomes 



8' = -j (8.5) 



which we obtained in Chapter II. 

 In dealing with (8.1), let 



d = x+ jy 



