418 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



with X dependences as follows: 



Be'"'" 



r\„»*x2(x— a) 



where a is the guide width. Let us assume that kx^ is a sufficiently large 

 imaginary quantity of such sign that 



This assumption will be seen to be consistent with the solution. [(26b) 

 for small size guide.] 



Setting up the boundary conditions for Ey and Ez at x = 0, we have 

 from (14), (15), and (16), 



(A - B) (^ -) sin <pi + iU + B) cos <pi + iCfi^ cos ^i = (21a) 



-1 



K 



m 

 (A + B)^-' + C = (21b) 



let r = B/A. Combining these last two equations we have 



^-^ = " cot <p, (22) 



1 + r M 



Satisfying the boundary conditions at x = a produces an equation 

 similar to (22) with the substitution 



-iikxia — i2X 



r ^ r£ = re 



Thus 



-t2X 1 



LjLZ = ^ - ''' (23) 



1 + r 1 + re""' 



i2X 



Equation (23) is satisfied by the condition X = wtt. Since kxi = 

 i(rmr/b)ix~''' sin <pi , we have 



sm <pi = lu (.24; 



in a 



The assumption that cos ^i is real and less, in magnitude, than unity is 

 realized by the condition 



(_^)^!L^<1 (25) 



771 a 



