334 BELL SYSTEM TECHNICAL JOURNAL 



Tio of the dominant wave is obtained by setting m = 1, w = in (1.1-5). 

 The r's corresponding to the higher modes may be obtained from the 

 same formula: 



(5.1-1) 



We shall consider a 90° bend. The approximation (4.2-6) appropriate 

 to gentle bends becomes 



^- = i^''[- .0122 sin (5.03/^) + .0087 cos (5.03/^)] (5.1-2) 



where the exponential terms have been omitted since they are generally 

 negligible. In (5.1-2), ^ = a/ pi and the arguments of the sine and cosine 

 terms arise from 2crio = 7rario/(2^). From (4.1-9) the approximate 

 change in the propagation constant produced by the curvature is obtainable 

 from 



7? - r?o = .294^7^' (5.1-3) 



where 71 is the propagation constant of the dominant mode in the bend. 



The determination of gto by matrix methods will be illustrated for a 90° 

 bend in which pi/a = 0.6. This makes c/a = pnr/{Aa) = .4712, cViq = 

 il.510 and the appropriate equations in (2.3-5") and (2.3-4) become, upon 

 setting /r = ig:ro and ft = gto , 



gto = e'^'^ixi - yi) = (.061 + i.99S)(xi - yO 



gTo = e'''\xi + yi) - e''''''' = (.061 + i.99S){xi + yO (5.1-^) 



+ .993 - i.l21 



Here xi , yi are the top elements in the column matrices x, y. The problem 

 is to compute x and y from the matrix equations (2.3-3) with F replaced 

 by Fa , Fg defined by (1.3-2) with ^ = 0, and h a column matrix whose 

 elements are zero except the top one which is unity. Since the order of the 

 matrices is infinite, an exact solution calls for an infinite amount of work. 

 A compromise must be made between the accuracy desired and amount of 

 work one is willing to do. The following numerical work uses third order 

 matrices. 

 The first step is to compute the square matrix, obtained from (1.3-5), 



aY\ = P~\arl + aS) (5.1-5) 



