FIELD SOLUTIONS 



641 



Near 6 = 6,, the tangent varies with infinite rapidity, making an infinite 

 number of crossings of the axis. 



In Fig. 14.6, the right-hand side of (14.41) has been plotted for a part of 

 the range 6 = 0.90 6eio6 = 1.10 ^^ . The waves corresponding to the inter- 

 sections of the rapidly fluctuating curve with a horizontal line representing 

 P are unattenuated space-charge waves. The nearer 6 is to 6e , the larger 

 (— ci/e) is. The amplitude of the electric field varies with y as 



cosh (j{- 



«l/«) (P 



1/2/ 



,1/2. 



^iry) = COS (i-e,/ey'W - ^tY'^y) (14.45) 



Fig. 14.6— The curve for the central region is not shown completely in Fig. 14.5. A part 

 of the detail around ^ = 1, which means a phase velocity equal to the electron velocity, is 

 shown in Fig. 14.6. The curve crosses the axis, and any other horizontal line, an infinite 

 number of times (only some of the branches are shown). Thus, there is a large number of 

 unattenuated "space charge" waves. For these, the amplitude varies sinusoidally in the y 

 direction. Some of these have no physical reality, because the wavelength in the y direction 

 is short compared with the space between electrons. 



For small values of \6 — 6t\ the field fluctuates very rapidly in the y direc- 

 tion, passing through many cycles between y = and y = d. For very 

 small values oi \d — 6e\ the solution does not correspond to any actual 

 physical problem: spreads in velocity in any electron stream, and ultimately 

 the discrete nature of electron flow, preclude the variations indicated by 

 (14.45). 



The writer cannot state definitely that there are not increasing waves for 

 which the real part of 6 lies between 6e — y/A and 6e -\-'\/A, but he sees 

 no reason to believe that there are. 



There are, however, other waves which exhibit both attenuation and 



