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THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



It will turn out that j3 will be very nearly equal to j3o . Hence, 



^ = ^0 + i5 - j^ = -iiSo + 6 (2.21) 



Then, from (2.16) we have 



i5 = ±i8, 



V'- 



o ieaKMdo + j8) 

 5(-2i/3o + 6)" 



K we neglect 5 with respect to jSo in the sums inside the radical we obtain 

 the equation 



53 = -j^l^^oieaK 



(2.22) 



/ 



/■ 



<f3 A 



-\-(/3p2y3,a.e(rK)'/3 

 / 



V 



Fig. 2.3 — Values of 5 for the three forward waves of a traveling-wave tube when the 

 electron velocity is equal to the velocity of the undisturbed wave. 



This yields the usual three forward waves of the traveling- wave tube. 

 8 = (/82/3Wi^)'^'e^"^-'^^'"'^^' 

 5i = mWaKyfKV3/2 - j/2) 

 82 = mlo>e(TKy'\-Vs/2 - j/2) 



53 = mUaKy'Kj) 



We see that 5i represents an increasing wave slower than the natural 

 phase velocity of the circuit, 82 represents a decreasing wave slower than 

 the natural phase velocity of the circuit, and 83 represents an unattenuated 

 wave faster than the natural phase velocity of the circuit. The 3 5's are 

 illustrated in Fig. 2.3. 



If jSo ?^ i3i , and if /3i is complex (a lossy circuit) the equation for 8 is more 

 complicated, but 8 can be obtained numerically. 



In addition to the three forward waves, that is, waves in the direction of 

 electron motion, there is a backward wave. This is very much out of syn- 

 chronism with the electron stream, and the backward wave is essentially 

 the same as the wave in the absence of electron flow. 



