1352 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



become 



The characteristic impedances become 



CO io 

 CO io 



The fact that the characteristic impedances of the two waves are 

 imaginary means that neither of the waves alone has any power flow. 

 Neither of the waves can very well carry power. The amplitudes change 

 with distance; hence for each wave Ui* and iU* mcrease or decrease with 

 distance. But, the circuit and the electron beam are lossless, and the 

 power cannot change with distance. As the ^^'aves do have a group 

 velocity, neither has any stored energy. Does this mean that the beam 

 cannot carry any power? The beam can carry power, just as a filter in 

 its stop band can carry some power from a source at one end to a resistive 

 load at the other end. The power flow is still given properly in terms of 

 the total current i and the total voltage U by the same expression used 

 in section 1. Suppose that the two waves have currents ii and 12 . Then 

 the total power flow is 



P = V2[(ii + ^2){K^*h* + Iu%*) + (zi* + t2*)(Iuii + /v2^2)] 



P = 3^[(i^i + i^l*)(^■l^l*) + (Ko + K2*)i2i2* + iii2*(Ki + A%*) 



+ iiii2*(K, + K2*))*] 



First consider the case in which cog is real and for which the characteris- 

 tis impedances are real and 



7^1 = -lu 

 In this case 



P = K\iii* + /C2i2i2* 



This is the familiar case of unattenuated waves. The total power is the 

 power of each wave calculated individually. 



Let us now consider the case in which the effective plasma frequency 



