352 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



where the common factor e^"' is omitted. To = j{co/vo), j = \/— 1 and w 

 is the angular frequency. Ci and C2 are arbitrary constants which will 

 be determined by the boundary conditions at the both ends of the beam. 

 The first two terms are the solutions of the homogeneous equation (or 

 the complementary functions) and are just the cold circuit waves. The 

 third and the fourth terms are functions of electron charge density and 

 are the particular solution of the equation. 



Let us consider a long traveling-wave tube in which the beam starts 

 from z = and ends at 2; = D. The motion of electrons observed at any 

 particular position is periodic in time, though it varies from point to 

 point along the beam. To simplify the picture, we may divide the beam 

 along the tube into small sections and consider each of them as a current 

 element uniform in z and periodic in time. Each section of beam, or each 

 current element excites on the circuit a pair of waves equal in ampli- 

 tudes, one propagating toward the right (i.e., forward) and the other, 

 toward the left. One may in fact imagine that these are trains of waves 

 supported by the periodic motion of the electrons in that section of the 

 beam. Obviously, a superposition of these waves excited by the whole 

 beam gives the actual electromagnetic field distribution on the circuit. 

 One may thus compute the forward traveling wave at z by summing all 

 the waves at z which come from the left. Stated more specifically, the 

 forward traveling energy at z is contributed by the waves excited by the 

 current elements at the left of the point z. Similarly the backward travel- 

 ing energy, (or the backward wave) at z is contributed by the waves 

 excited by the current elements at the right of the point z. It follows 

 obviously from this picture that there is no forward wave at 2 = 

 (except one corresponding to the input signal), and no backward wave 

 at 2 = D. (This implies that the output circuit is matched.) With these 

 boundary conditions, (1) is reduced to 



z) = Finput e " + e ° — -— / e " po,{z) 



Z Jo 



dz 



+ /-^J e-%.(.) 



(3) 



dz 



Equations (2) and (3) have been obtained by Poulter.^ The first term of 

 (3) is the wave induced by the input signal. It propagates as though the ; 

 beam were not present. The second term is the voltage at z contributed 

 by the charges between 2 = and 2 = 2. It is just the voltage of the 

 forward wave described earlier. Similarly the third term which is the 

 voltage at 2 contributed by the charges between z = z and 2 = D is the 

 voltage of the backward wave at the point 2. Denote F and B respec- 



