324 BELL SYSTEM TECHNICAL JOURNAL 



(r tanh cV)ij = (7/// - yiti)kii, i 9^ j (3.^-S) 



(r tanh cT)ii = Jiti + Jl' {jiU — ymtm)kimkmi (3.2-6) 



m 



where in (3.2-5) and (3.2-6) terms of 0(^2) (''order of") and 0(^'), re- 

 spectively, have been neglected. This is in line with the fact that the terms 

 in the principal diagonals of our matrices must be accurate to within 0(^2) 

 while the remaining terms need be accurate only to within terms of O(^). 

 The summation with respect to m runs from w = 1 to 00 with 

 m = i omitted. 



d>.?t Reflection from a Gentle Bend 



When the bend is gentle so that V and V behave according to the descrip- 

 tion given in Section 3.1, the matrix expressions for the reflection coefficients 

 given in Section 2.3 may be evaluated. The results stated in Appendix II 

 for ''almost diagonal" matrices furnish the principal tools for this work. 



It is assumed that the incident wave coming in along the straight guide 

 from the left is $» = exp(— s5p) Bp{Xj y) and hence contanis only the />*^ 

 mode. Comparing this with the general expression (2.2-1) for $» shows 

 that hp = l,hm = 0,m 7^ p, and all the elements of the column matrix h are 

 zero except the p^^ which is unity. 



We start by writing the first of equations (2.3-3) as 



(r tanh cT + V-''To)x = V-^ To Z"^" h (3.3-1) 



Since V approaches a unit matrix as ^ — > 0, the element (F~^) ij in the i*^ row 

 andy^*" column of V"^ is 



{V-^)ii = 1 - f a + Z)' Virr. Ymi 

 •m 



where V ii= 1 + ^ti, ^i = 1, 2, 3, . . . and the summation with respect to m 

 runs from 1 to <» with the term form = i omitted (as indicated by the prime 

 on S). In omitting this term we are neglecting va because it is of order 

 ^*. These results follow from equation (A2-2) of Appendix II. As usual, 

 the elements in the principal diagonal are accurate to within 0(^2) and the 

 remaining elements to within O(^). 

 It follows that V~^ Toe'^^° h = ry is a column matrix whose i^^ element is 



Vp 



= (1 - ^PP+Z'l'pn. F;„p)5p A 



(3.3-3) 



