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BELL SYSTEM TECHNICAL JOURNAL 



Here we take both upper signs or both lower signs in (8.21) and (8.22). 

 If we assume d/x « 1 and expand, keeping no powers of d/x higher than 

 the first, we obtain 



x= + (a/3/2)(1 - (l/3((//x)) 



(8.23) 



The plus sign will give .Vi , which is the x for the increasing wave. Let JCjo 

 be the value of .Ti for J = (no loss). 



XiQ 



= V3/2 



(8.24) 



-5-4-3-2-1 1 2 3 4 5 



b 



Fig. 8.4 — The gain of the increasing wave is BCN db, where A'^ is the number of wave- 

 lengths. 



Then for small values of d 



xi = .^10(1 - (l/3)(J/xio)) 



^"1 == ^10 ~ 1/3^/ 



(8.25) 



This says that, for small losses, the reduction of gain of the increasing wave 

 from the gain in db for zero loss is \ of the circuit attenuation in db. The 

 reduction of net gain, which will be greater, can be obtained only by match- 

 ing boundary conditions in the presence of loss (see Chapter IX). 



In Fig. 8.5, B = 54.6 Xi has been plotted vs d from (8.22). The straight 

 line is for Xio = d/3. 



In Fig. 8.6, —Xi , x^ and .T3 have been plotted vs d for a large range in d. 

 As the circuit is made very lossy, the waves which for no loss are unattenu- 

 ated and increasing turn into a pair of waves with equal and opposite small 

 attenuations. These waves will be essentially disturbances in the electron 

 stream, or space-charge waves. The original decreasing wave turns into a 

 wave which has the attenuation of the circuit, and is accompanied by small 

 disturbances in the electron stream. 



