VACUUM TUBE ELECTRONICS 65 



where 



„ = (l8,^ /.)'"• (9) 



The solution of (7) is more complicated. We assign a particular 

 value to /i, namely, Ji = A sin pt and find the corresponding value of 

 Ui. To do this, it is convenient to change the variable x to a new 

 variable ^, which will be called the transit angle. This new variable 

 is equal to the product of the angular frequency p and the time r 

 which it would take an electron moving with velocity Uo to reach the 

 point X and is given as follows: 



^ = ^r=^xi/3. (10) 



a 



Upon changing the dependent variable from Ui to w, where Ui = co/^, 

 we find from (7) 



-^ + i>^)'^ = ^/3sin^/, (11) 



where 



This has the solution 



^ = 47r- A. 

 m 



f/i = 



sin pt + 7 cos pt + 7^,(^ - pt) + 7 i^2(^ - pt) 



(12) 



This equation contains two arbitrary functions of (^ — pt) which must 

 be evaluated by the boundary conditions selected for Ui. Thus the 

 boundary conditions for the alternating-current component make their 

 first appearance. 



From the form of (7) which is linear in Ui, it is evident that Ui 

 must be a sinusoidal function of time having an angular frequency p 

 in order to correspond with the form of /i. It follows, then, that the 

 most general form which can be assumed for the steady state functions 

 Fi and F^ is as follows: 



Fi(^ - pt) = a sin (^ - pt) -{-b cos (^ - pt)\ .^^. 



Fi{^ - pt) = c sin (^ - pt) + d cos (^ - ^/) j 



Now for the boundary conditions. As pointed out, there is no 

 mathematical necessity for the boundary conditions imposed upon Ui 

 to correspond with those which were imposed upon Uq. At an actual 

 cathode consisting of an electron emitting surface it would be appro- 

 priate to assume that the initial velocities are in no way dependent 

 upon the current, but we shall have to deal not only with actual 



k 



