A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 359 



angular plasma frequency for a beam of infinite extent, and k is 



2 



k = 



a 



0) CO 



— ro — ro 



Uo Wo 



(20) 



In the small C theory, th^e distribution of electrons in time or in time- 

 phase at z is approximately the same as the distribution in z (also ex- 

 pressed in the unit of time-phase) at the vicinity of z. This is, however, 

 not true when C becomes finite. The difference between the time and 

 space distributions is the difference between unity and the factor 

 (1 -}- Cw{y, <po )). We can show later that the error involved in con- 

 ; sidering the time phase as the space phase can easily reach 50 per cent 

 or more, depending on the velocity spread of the electrons. 



6. NUMERICAL CALCULATIONS 



Although the process of carrying out numerical computations has 

 been discussed in Nordsieck's paper, it is desirable to recapitulate here 

 I a few essential points including the new feature added. Using the work- 

 ing equations (15), (16), (17) and (18), 



dai da 2 dw , dcp 



dy ' dy ' dy dy 



\ are calculable from ai , a^ , w and <p. The distance is divided into equal 

 I intervals of A?/, and the forward integrations of Oi , ao , w and (p are per- 

 f formed by a central difference formula 



ax{y + A?/) = ax{y) -f 



dy 



y+y2&y 



■Ay 



In addition. 



d^ai 

 dy^ 



and 



d 02 



df 



in (17) are computed from the second difference formula such that 

 d''ai 



- At/ 



_ dtti da\ 



dy^ j/=j/ \_dy y+l/2i,y dy y-^/2^y_ 



We thus calculate the behavior along the tube by forward integration 

 j made in steps of Ay, starting from y = 0. At ?/ = the initial condi- 

 tions are determined from Pierce's linearized theory. Because of its 

 complications in notation, this will be discussed in detail in Appendix I. 

 j Numerical calculations were carried out using the 701-type I.B.M. 



