CO.\FOR.\rA L TR. I XSFOR.UA TIO.Y 109 



where the upper and lower guide walls are carried into ^ = and d = ir, 

 respectively, and g(v, 6) is given by 



1 + g(^ e) = \J'{v + id) |- iT/b' (2-5) 



„/., n) ^ [7^ V + cos or _ 



^^'' ^ [ch{v - t) - cos 0Hc^(t' + /) - cos 8]- ^ ^ ^ 



Here ch denotes the hN'perbolic cosine, /'(r + id) denotes the first deriva- 

 tive of /(k')) and from Appendix I, lira is the total angle of the bend. / is a 

 parameter which depends upon a and the ratio d/do where d = \ Zi — zq \ 

 and do = | Z4 — ze ] in Fig. 1. A table giving values of / for a 90° bend 

 (a = 1/4) appears in Appendix I. 



That the propagation constant is no longer uniform in the transformed 

 guide shows up through the fact that the coefficient of k'-Q in (2-3) is now a 

 function of the coordinates (i', 9). g(v, 9) measures the deviation of the 

 propagation constant from its value ?Lt v — — x . For example, if we 

 consider a wave front coming down from z-, we expect it to get past S4 before 

 it reaches Zo . In Fig. 2 the same wave front is tilted forward corresponding 

 to a high phase-velocity (or small propagation constant) at Zi where v = 

 and 9 = TT. This is in line with the fact that the coefficient of k-Q in (2-3) 

 vanishes at Z4 by virtue of (2-6). Similar considerations hold at Zi and zo • 



What is our reflection problem in terms of the transformed guide? In 

 addition to satisfying the two equations (2-3) and (2-4) Q must behave 

 properly at infinity. For large negative values of v, Q must represent an 

 incident wave plus a reflected w^ave. The incident wave is of unit amplitude 

 and the reflected wave is of the, as yet, unknown value Re- For large 

 positive values oi v Q must represent an outgoing wave. Thus Q must 

 also satisfy the two equations 



Q ^ g-ikv ^ R^e'''\ ^ ^ - 00 (2-7) 



Q = Tec-''^' , ^ ^ 00 (2-8) 



where the subscript E appears on the "reflection coefficient" Re and the 

 "transmission coefficient" Te to indicate that here we are dealing with an 

 electric corner. 



Our problem is now to take the four equations (2-3, 4, 7, 8) and somehow 

 or other obtain the value of Re ■ We are not so much interested in Te 

 because it does not have the practical importance of the reflection coefficient. 

 There are at least two different ways we may proceed from here. One 

 is to transform the differential equation plus the boundary conditions 

 into an integral equation which may be solved approximately by iteration. 

 Another way is to assume () to be a Fourier cosine series in 9 whose co- 

 efficients are functions of v. Substitution of the assumed series in the 



