TRAVELING-WAVE TUBES 15 



propagation along the circuit and the electron stream. When we combine 



(2.22) and (2.10) we obtain as the equation which F must satisfy: 



1 = JJM^ , (2.23) 



2 Mr; - r')Ui3. - rf 



Equation (2.23) applies for any electron velocity, specified by ^3^,, and any 

 wave velocity and attenuation, specified by the imaginary and real parts of 

 the circuit propagation constant Fi . Equation (2.23) is of the fourth degree. 

 This means that it will yield four values of F which represent four natural 

 modes of propagation along the electron stream and the circuit. The circuit 

 alone would have two modes of propagation, and this is consistent with the 

 fact that the voltages at the two ends can be specified independently, and 

 hence two boundary conditions must be satisfied. Four boundary conditions 

 must be satisfied with the combination of circuit and electron stream. These 

 may be taken as the voltages at the two ends of the helix and the a-c velocity 

 and a-c convection current of the electron stream at the point where the 

 electrons are injected. The four modes of propagation or the waves given by 



(2.23) enable us to satisfy these boundary conditions. 



We are particularly interested in a wave in the direction of electron flow 

 which has about the electron speed and which will account for the observed 

 gain of the traveling- wave tube. Let us assume that the electron speed is 

 made equal to the speed of the wave in the absence of electrons, so that 



-Fi = -j^e (2.24) 



As we are looking for a wave with about the electron speed, we will assume 

 that the propagation constant differs from /3e by a small amount ^, so that 



-r = -jn. + f 



Using (2.24) and (2.25) we will rewrite (2.23) as 



1 = -Klo^li-^l - 2j0e^ -f- f) . 



Now we will find that, for typical traveling-wave tubes, | is much smaller 

 jthan (Se ; hence we will neglect the terms involving j8e^ and ^^ in the numera- 

 jtor in comparison with /J^- and we will neglect the term ^- in the denominator 

 |in comparison with the term involving l3e^. This gives us 



e = -M ^ (2.27) 



While (2.27) may seem simple enough, it will later be found very convenient 



