GUIDED WAVE PROPAGATION THROUCill (;YH( )MA(iN10TIC MEDIA. IT 977 



1.0 

 0.9 

 0.8 

 0.7 

 0.6 

 0.5 

 0.4 

 0.3 

 0.2 

 0.1 



Fig. 15 — The function Zx(u) versus x for various w, in the range — % <x 



< h. 



X > 3^ starts from at u = with a downward-directed vertical tangent, 

 achieves a negative minimum, then increases, through its zero, to the 

 asymptotic value unity as w -^ co. The minimum becomes deeper as 

 X approaches %. At x = /4, ^x.i('^) developes a zero at u = 0, which 

 steadily moves to larger u as x increases further towards %. At the 

 same time the zero of TF^.o already discussed moves from 1 to 2 + \/2, 

 and a new zero arises at m = 0, x = %, which increases to 2 — ■\/2 

 as X approaches ^^, but which lags behind the zero of Wx,i(u). The 

 function Zy.(u) now has a pole and two zeros, and behaves as shown in 

 Fig. 14. This process continues; each time x passes (2n + l)/2, a 

 new zero and a new pole appear. (For a detailed list of poles and zeros 

 the reader is referred to Appendix I). To apply these results, we first 

 resort to the Polder relations. 



In terms of a, p, we have, for /3 negative 



and 



1 



x = M 



p 



1 — p(T — a^ 



1 - <7 



^(2x-l) ^^-1 + ^ 

 Mo 



= A, say. 



