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BELL SYSTEM TECHNICAL JOURNAL 



We see that by means of (24) and (25) we can plot db/wavelength per 

 unit h vs. (U/UmY- This is plotted in Fig. 3. Because U"^ is proportional 

 to current, the variable {U/U mY is the ratio of the actual current to the 

 current which will just give an increasing wave. If we know this ratio, 

 we can obtain the gain in db/wavelength by multiplying the corresponding 

 ordinate from Fig. 3 by b. 



40 60 80 100 



400 600 1000 



(W/Wm)^= (U/Um)' 



Fig. 3— The parameter {W fW mY = ((tVt\\f)-isproportional to current. As the current 

 is increased above a critical value for which {W/Wm)"^ = 1, there is an increasing wave of 

 increasing gain. In this curve the gain per wavelength per unit b, called F{W/WMy, is 

 plotted vs. {IV /Wm)^- For large values of (W/Wm)-, /^(H'/ll'-v/)' approaches 27.3 and the 

 gain per wavelength approaches 27.3 b. 



We see that, as the current is increased, the gain per wavelength at first 

 rises rapidly and then rises more slowly, approaching a value of 27.36 

 db/wavelength for very large values of {U/U m)'- 



We now have some idea of the variation of gain per wavelength with veloc- 

 ity separation b and with current {U/U mY- A more complete theory would 

 require the evaluation of the lower limiting current for gain (or of U m) in 

 terms of physical dimensions and an investigation of the boundary condi- 

 tions to show how strong an increasing wave is set up by a given input 

 signal. The latter problem will not be considered in this paper; the former 

 is dealt with in the third section and in the appendix. 



