328 BELL SYSTEM TECHNICAL JOURNAL 



A. In order to study the dominant mode we set / = in the cos {Trly/b) 

 (^A depends on y through this factor) in the formulas of Section 1.3 which 

 involve A and assume the dimensions of the guide to be such that h < a. 

 We wish to determine 71, the first latent root of v\ defined by (1.3-5), 

 from the approximate formula (3.2-2). In our case the elements 5^ of diag- 

 onal matrix To are obtained by putting n{= I) to zero in (1.1-5) : 



bl = rlo = cr' + (ttwA)', w = 1, 2, 3, . . . (4.1-1) 



so that (3.2-2) becomes 



2 

 71 



' r?o + Fii - E F^mFmiairXm^ - 1)"' (4.1-2) 



The first task is to find the elements of the matrix F where, from (3.1-1) 

 and (1.3-5), 



F = tI-tI= (P-' - I)tI + P-'S (4.1-3) 



In the case under consideration P = I -\- R where i? is a square matrix 

 whose elements are very small. In fact, the asymptotic expressions lead- 

 ing to (Al-18) show that 7?ii and Sa are 0(^2), with ^ = a/pi , while Rij 

 and Sij are O(^) iii-\-j is odd and 0(^2) iii+j is even. When the approxi- 

 mate value of P~^ obtained from (A2-2) is set in (4.1-3) and the matrix 

 multiplications carried out it is found that 



Fii = - RiiT]o + Si,- 4- 0{e) 



( 00 \ 00 



— Rii + Z-/ ^im Rmi 1 Tio + Su — 2^ Rim Smi + 

 m»l / m=l 



o{t) 



(4.1-4) 



The ''order of" symbol 0( ) will be omitted in the following equations, it 

 being understood that the terms in the principal diagonal are correct to 

 within 0(^2) ^nd the others to within O(^). 



The values of the F's which enter (4.1-2) may be computed from the 

 asymptotic expressions (Al-18) for the jR's and 6"s. They turn out to be 



Fim = -^m[^V\,T-\m' - i)-' -\- 2>a-\m' - 1)"^] 



Fmi = -4^w[4r5o7r-'(m' - 1)"' + ar\m' - 1)"^] (4.1-5) 



Fn = ^\t\,{\ - 67r-') + ^oT']/!! 



In the expressions for Fim and F„,i w is supposed to have the values 2, 4, 6, 

 • • • . For odd values of m Fim and F„i are 0(^^). When i = 1 in the ex- 

 pression (4.1-4) for Fn, the two series therein reduce to S3 and S4. where 



Sp== E ni\m' - ly (4.1-6) 



