CHAPTER VIII 

 THE NATURE OF THE WAVES 



Synopsis of Chapter 



TN this chapter we shall discuss the effect of the various parame- 

 -*• ters on the rate of increase and velocity of propagation of the three 

 forward waves. Problems involving boundary conditions will be deferred 

 to later chapters. 



The three parameters in which we are interested are those of (7.13), 

 that is, b, the velocity parameter, d, the attenuation parameter and QC, 

 the space-charge parameter. The fraction by which the electron velocity is 

 greater than the phase velocity for the circuit in the absence of electrons 

 is bC. The circuit attenuation is 54.6 dC db/ wavelength. Q is a factor de- 

 pending on the circuit impedance and geometry and on the beam diameter. 

 For a helically conducting sheet of radius a and a hollow beam of radius 

 Ui , Q can be obtained from Fig. 8.12. 



The three forward waves vary with distance as 



-JPt(l-VC)z BeXCz 



i3« = - 



Wo 



Thus, a positive value of y means a wave which travels faster than the 

 electrons, and a positive value of x means an increasing wave. The gain in 

 db per wavelength of the increasing waves is BC, and B is defined by (8.9). 



Figure 8.1 shows x and y for the three forward waves for a lossless circuit 

 {d — 0). The increasing wave is described by .vi , vi . The gain is a maximum 

 when the electron velocity is equal to the velocity of the undisturbed wave, 

 or, when b = 0. For large positive values of b (electrons much faster than 

 undisturbed wave), there is no increasing wave. However, there is an in- 

 creasing wave for all negative values of b (all low velocities). For the increas- 

 ing wave, yi is negative; thus, the increasing wave travels more slowly 

 than the electrons, even ivhen the electrons travel more slowly than the circuit 

 wave in the absence of electrons. For the range of b for which there is an 

 increasing wave, there is also an attenuated wave, described by .To = — Xi 

 and 72 = yi • There is also an unattenuated wave described by y3(.V3 = 0). 



For very large positive and negative values of b, the velocity of two 

 of the waves approaches the electron velocity (y approaches zero) and the 



397 



