130 BELL SYSTEM TECHNICAL JOURNAL 



function corresponding to a guide of width b, we may use equation (3-4) 

 to derive the new integral equation 



de 



Q{v, , do) = e-""-'^ + ^ f dv f 



• [g{v, e)Q(v, 6) - i(v)Tse-''^'}G(vo , do ; v, d) 

 + T,F{v,) 

 in which 



N-(vo) = 2-'k{k, - kV f'g'{i)e''"'''''"dv 

 iV+(ro) = 2-'k{k, + k)-' f g'{i)e-'''^-'''' dv 



(A3-2) 



(A3-3) 



Here ^'{v) denotes d^(v)/dv. Equation (A3-2) and 



Limit Q(v, d) = T^e'"''' (A3A^ 



D— ♦OO 



are to be solved for the unknown function Q{v, d) and the unknown quantity 

 T g . The method of successive approximations may be used in somewhat 

 the same fashion as in the simpler case but we shall not give a general 

 discussion. 



The first approximations are found to be 



ry^ = lAV-(°o), R'-^^ = -N+i-ccyN-i'^) (A3-5) 



where the A^'s may be obtained by setting I'o = ± ^o in equations (A3-3) . 



One of the simplest choices for g{v) is to let it be zero for negative values 



of V and to have the value ^(°o ) = kik~^ — 1 for positive values of v. Then 



Ti'^ = 2k{ki + k)-\ R'^^ = {k- ki) (k + yfei)-i (A3-6) 



These are quite similar to tlie corresponding expressions for a transmission 

 line which have been used extensively in wave guide work. 



In working with these formulas, when k is small, it is sometimes convenient 

 to use the result 



r dv r ddg(v,d) = ir'b-' r dv \ dd\f'{u)f - ds - z-Ot (A3-7) 



Jvi Jo Jfi Jo 



where the evaluation of the double integral on the right is made easier by 

 the fact that it represents the area in the original guide (in the (x, y) plane) 

 enclosed by the lines corresponding to z; = di and v — v^ . r? and vi are 



