1346 THE BELL SYSTEM TECHXICAL JOURXAL, XOVEMBER 1954 



Suppose we make up a radio-freciuency pulse out of various fre- 

 quency components of one wave. The pulse envelope generally travels 

 with a different velocity from that of the rf sinusoids under the envelope. 

 The velocity of the en\'elope is called the group velocity. The group 

 velocity is the velocit}^ with which a signal is transmitted. The direction 

 of the group velocity is the direction in which causality acts (for some 

 waves the phase \'elocit.y and the group velocity have opposite direc- 

 tions). The group \'elocity tells in which direction energy flows, and the 

 power flow P is the stored energy per unit length, W, times the group 

 velocity, Vg . 



P = WVg 



The group velocity is given by 



We see that for our assumption Wq is ec^ual to cop , the group velocity for 

 each wave or mode is «o , the velocity of the electrons in the beam 



Vg = «0 



Thus, of the two waves, the first has a phase velocity slower than that 

 of the electrons, the second has a phase velocity faster than that of the 

 electrons, and each has a group velocity equal to that of the electrons. 



A simple discussion of power flow is given in Appendix B. In describ- 

 ing the excitation of the electron stream we can use the convection 

 current i together with a c^uantity U which is analogous to voltage. In 

 terms of the ac electron velocity v, 



U = — — V 



III 

 m 



The real power flow P is given by 



p = mw* + i*u) 



This relation is justified in Appendix B. 



For each of the two waves the voltage U bears a constant ratio to the 

 current i; this ratio is the characteristic impedance K of the wave. We 

 find that 



K = — = - 9^^ Z? 



ii CO /o 



u CO 7( 



