A LARGE SIGNAL THEORY OP TRAVELING-WAVE AMPLIFIERS 355 



member of the equation, we obtain for the first order solution 



iKz, t) ^ 



ZqIo I (^ 



sin <p + — ^^^ cos <p 



40 \ 2(1 + bC) I dij ^ ' dy 



(12b) 



Of course, the solution (12b) is justified only when hi(y) and ?)2(?y) thus 

 obtained are small compared with ai(y) and aoiy). The exact solution 

 of B obtained by successive approximation reads 



Biz, t) 



+ 



ZqIo I c 



4(7 V 2(1 + bC) 



4(1 + hC) 

 It may be seen that the term involving 



dai(y) . , da2(ij) 



-^ sm <p + , cos 

 _ dy dy 



■ ] 



•] 



'^^' cos<p + — f^sm^ + 



(12c) 



dy- dy- 



4(1 + bcy 



and the higher order terms are neglected in our approximate solution. 

 For C equal to few tenths, the difference between (r2b) and (12c) only 

 amounts to few per cent. We thus can calculate the backward wave by 

 (12b) or (12c) from the derivatives of the forward wave. To obtain the 

 complete solution of the backward wave, we should add to (12b) or 

 (12c) a solution of the homogeneous equation. We shall return to this 

 point later. 



4. WORKING EQUATIONS 



With this discussion of the backward wave, we are now in a position 

 to derive the working equations on which our calculations are based. In 

 Nordsieck's notation, each electron is identified by its initial phase. 

 Thus, (p(y, (fo) and Cuow(y, <po) are respectively the phase and the ac 

 velocity of the electron which has an initial phase (fo . It should be remem- 

 bered that y is equal to 



and is used by Nordsieck as an independent variable to replace the vari- 

 al)le z. Let us consider an electron which is at Zo when /, = and is at 

 z (or ?/) when t = /. Its initial phase is then 



OiZo 



<Po = — 



