354 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



where ai(y), 02(2/), hi(y) and 62(2/) are functions of y. y and <p are inde- 

 pendent variables and have been used by Nordsieck to replace the vari- 

 ables, z and t, such as 



y = C — Z 



(f = w [ — — t ] 



\Vo / 



Here as defined earlier, I'o is the phase velocity of the cold circuit and Vq 

 the average velocity of the electrons. They are related by the parameter 

 h defined by Pierce as 



Uo 1 



vo (1 - hC) 

 C is the gain parameter also defined by Pierce, 



^3 _ ZqIo 



in which, Vo and 7o are respectively the beam voltage and current. 

 Adding (6) to (7), we obtain an important relation between F and B, 

 that is, 



dFjz, t) _^ 1_ dF{z, t) ^ dBjz, t) _j_ l_ dBjz, t) ^^Q^ 



dz Vo dt dz Vo dt 



Substituting (8) and (9) into (10) and carrying out some algebraic 

 manipulation, we obtain 



'"'^^ = " 2(1 + bC) I ^'^^'> + "'-^"^^ 



(11) 



"'^^'^ = 2(1 + bC) ly '"'^^^ + '"^^^' 



or 



B{z, t) = 



ZqIo C 



dMy) + bM) ,„, ^ + diaM+ b.(,j)) ^.^ - 

 dy dy 



[ 



For better understanding of the problem, we shall first solve (12a) ap- 

 proximately. Assuming for the moment that hiiy) and h^^y) are small 

 compared with ai{ij) and a^iy) and may be neglected in the right-hand 



