222 BELL SYSTEM TECHNICAL JOURNAL 



By comparing this with (5.2) we see that 



G{i) ^F^t-f^L^ (5.8) 



In other words, the envelope at the output is of the same shape as at the 

 input, but arrives a time r later 



r = 1^ i (5.9) 



This implies that it travels with a velocity Vg 



% = L/r = (^y (5.10) 



This velocity is called the group velocity, because in a sense it is the veloc- 

 ity with which the group of frequency components making up the pulse 

 travels down the circuit. It is certainly the velocity with which the energy 

 stored in the electric and magnetic fields of the circuit travels; we could ob- 

 serv^e physically that, if at one time this energy is at a position x, a time / 

 later it is at a position x + Vg I. 



If the attenuation of the transmission circuit varies with frequency, the 

 pulse shape will become distorted as the pulse travels and the group velocity 

 loses its clear meaning. It is unlikely, however, that we shall go far wrong 

 in using the concept of group velocity in connection with actual circuits. 



We have used earlier the concept of phase velocity, which we have desig- 

 nated simply as v. In terms of phase velocity, 



/? = " (5.11) 



V 



We see from (5.10) that in terms of phase velocity v the group velocity 

 Vg is 



.. = » (l - " fV (5.12) 



\ V dec/ 



For interaction of electrons with a wave to give gain in a traveling-wave 

 tube, the electrons must have a velocity near the phase velocity v. Hence, 

 for gain over a broad band of frequencies, v must not change with frequency; 

 and if v does not change with frequency, then, from (5.12), Vg — v. 



We note that the various harmonic components in a filter-type circuit 

 have different phase velocities, some positive and some negative. The group 



