870 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



por 



-ar/e = - 



__JPQVt 



R{dV/dr) 



(22) 



1 - J 



. Vr 



cohr 



The appropriate potential functions in I and II are reduced by means 

 of the low-frequency, or thin-beam, approximation, as follows: 



']_dV'Y ^ yhjyb) ^ Yb 

 V dr b hiyh) - 2 



']_dV 

 V dr 



~iii 



= /3 



h{yb) ■ 



^% 



_/o(/36)Ko(/3a) - h{l3a)Komj 

 1 



(23) 



~ 2 6 In a/6 

 where the following small-argument approximations have been used; 



Ko{x) ^ -In X, 



KiOr) ^^ln.T-f -, 

 2 X 



(24) , 



The boundary equation thus provides a second relation between 7 and (3: 



6 In 



a 



(25) 



For the smooth beam in Brillouin flow {vr = 0), the boundary equa- 

 tion, to the same low-frequency approximation, is as follows: 



2 



(26) 



Re 



COj 



(« - i3ouy 



a 



im' In ~ 



To see how the beam ripple affects the propagation constant, it is 

 sufficient to find its first-order effect; i.e., to assume relatively small 

 radial velocities and find a solution for /3 which is not very different 

 from its value, /3o , for the smooth beam: 



(27) 



/3 = /3o + 6 = /3e ± ^g + 5, 



where 



5 I « /3o ; ^c 



COt 



u 



^' 



/3e 



CO 



u 



