844 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



The amplitude of the a-c. voltage in the diode space is also required, 

 and it is derived from its expression 



Jo 



edx (48) 



where e has its value (47), and the integration can be performed by remem- 



dr 

 bering that — is \/U. This gives 

 dx 



V = Va -\- 



L \^,AE -A^- jBI^ {e-^^' - 1) 



(49) 



We are now in a position to examine the character of the electron stream, 

 and for this purpose we write the conduction current density in its complete 

 form qe^" , that is. 



qe^'-' 



^■§{b-A ^) e-^'-> - -^ t.-' (50) 



The phase angle of the first term involves the d-c. transit time r, which is 

 a function of x, so this term is a wave traveling in the ic-direction. Its ampli- 

 tude involves the d-c. quantities U and E, and its amplitude varies with x. 

 The velocity of the wave is given by 



Wave velocity = (j-) (51) 



and from (34) 



Wave Velocity = U (52) 



That is the velocity of the conduction current wave is equal to the d-c. com- 

 ponent of electron velocity. 



The second term in (50) is an oscillation that has the same phase through- 

 out the diode space, and its amplitude also varies with x. The a-c. conduction 

 current is thus a wave of electric charge traveling at a finite velocity plus an 

 oscillating charge that is in phase over the entire diode space. 



An inspection of equations (46), (47) and (49) shows that the other a-c. 

 components are of the same general form; each of them is a wave traveling 

 in the rc-direction plus an oscillation that is in phase over the entire diode. 

 This clear-cut disclosure of the dual nature of an electron stream is an im- 

 portant contribution of the general theory. 



