PART II — THE UHF NOISE SPECTRUM 871 



In addition, | Vrl (j^hf \ is less than unity, and | 2vr/^ur \ can be neglected 

 entirely. With these assumptions, the boundary and characteristic 

 equations can be combined to solve for ^ : 



^ F{F -R) ^ - U) ^° (28) 



where 



7_ _ _ 



^2 F^ - R F - R 



CObV 



utilizing the low-frequency condition, \ R \ > \ R \ > 1, this equa- 

 tion can be reduced and, after some algebra, solved: 



= 1 - j 



(00 + 5)(±/3,) — '' 2a;6r' 



/3.^/3e + ^« - i^, (29a) 



2ur 



Pf^l3e- ^, + J:^. (29b) 



2ur 



These expressions show that the current in the slow wave (IJ will 

 grow when (vr/r) is negative; i.e., when the beam is contracting, and 

 decrease during its expansion. The fast wave (I/) will do the opposite. 

 In probe measurements along the beam, the detected ac power is pro- 

 portional to the square of the total space-charge current, which has 

 the following dependence on time and distance when the amplitudes of 

 both waves are initially equal: 



(Is + //) = 2/max cos {cot - ^ez) -cos (13 gz) -sinh f ^ j . 



(30) 



In UHF noise-power measurements along beams with long ripple 

 wavelengths, the two planes of maximum dz (vr/r) are separated by 

 only a small fraction of a space-charge wavelength. Therefore, cos ^qZ 

 at the first of these planes is only slightly larger than at the second. 

 Thus, two peaks of current are observed, in agreement Avith (30). By 

 contrast, in rippled-beam amplification at microwave frequencies, 

 shorter ripple wavelengths and smaller ripple amplitudes are employed. 

 Then (iv/r) varies nearly sinusoidally over the ripple wavelength. For 

 maximum net gain per ripple, maximum negative (vr/r) is adjusted to 

 coincide with the plane of cos ^gZ = 1 (maximum current), and maxi- 

 mum positive (vr/r) at the current minimum, half a wavelength bej^'ond. 



