REFLECTIONS FROM CIRCULAR BENDS 323 



Throughout the remainder of Part III we shall assume that V^, F and V 

 behave as mentioned above. In addition we assume that there is no de- 

 generacy, i.e. all of the b^s are unequal to each other and to zero. 



3.2 Propagation in a Gentle Bend 



Here we assume that the elements of vl and F in the expression (3.1-1) 

 forF- are known. We wish to find the modal propagation constant 7y and 

 the corresponding eigenfunction <pj{x, y) for the^*^ mode. 



After squaring both sides of the collineatory transformation (2.1-11) 

 connecting V and the diagonal matrix [y]d we obtain a relation which may be 

 written as k\y'^]d — V^k = 0. The left hand side is a square matrix having 

 (T;/ — r^)^; as itsy^ column. Here / is the unit matrix and k, is a column 

 matrix having kij, k^i, ... as its elements (kj is the/^ column of k). Thus 

 we have a system of simultaneous Hnear equations in which the coefl&cients 

 are furnished by the square matrix y]l — T^ and in which the unknowns are 

 kij, k2], • • • . Accordingly, 7,- is the j^^ latent root of T^ and kj is its cor- 

 responding modal column just as for the rectangular guide in Section 1.3. 



In order to apply equations (A2-16) of Appendix II we set >;• = 7/ and 

 « = r2 so that, from (3.1-1), 



«yy = 5) + Fa, Uii = Fii, i 9^ j (3.2-1) 



Therefore 



y) = «' + f'yy + S' F„F,M - «') (3.2-2) 



s.l 



k,-^ = 1, ksi = Fsi/{b) - hi), s 9^ j (3.2-3) 



where we have neglected terms of order ^ in (3.2-2) and of order ^^ in ksj, 

 s 7^ j. The prime on the summation indicates that the term 5 = y is to be 

 omitted. 



When ^ly, ^2;, . . . are known the eigenfunction ^y(it:, y) may be written 

 as a series in Qmioo, y) by means of equation (2.1-2). 



In Section 3.3 we shall need the form assumed by the square matrix V 

 tanh cV in a gentle bend. This matrix is used ui computing the reflection 

 from such a bend, as might be inferred from equation (2.3-3). The formula 

 to be used is (A2-18) with « = T^, X; = 7,- and with the elements of the 

 square matrix k given by (3.2-3). In the diagonal matrix of (A2-18) we set 



/(X,) = X/ tanh c\j = 7/ tanh cjj = yjtj 



(3.2-4) 

 /;• = tanh cyj 



and for the elements of k~^ we use (A2-19) together with the line above it. 

 When the three matrices on the right of (A2-18) are multiplied out the ele- 

 ment (r tanh cT) a in the i^^ row and_;'^ column of F tanh cT is found to be 



