1188 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



50 



C 



Fig. 22 — Real and imaginary parts of Ai/tt^ and ^iJ-k'- for a nonuniform stack 

 whose average properties are constant except at a single, symmetric discon- 

 tinuity. 



iii) f(^) a symmetric rectangular step. Let 



m = 



'Wo' 



2(1 - ^o) ' 



"21;' 



Mo < ^ < 1 - 1^0 



1 - 1^0 < ^ ^ 1, 



(546) 



where ^o is some fixed number between and 1 but not, in the cases of 

 interest, extremely close to either or 1. Inasmuch as /(^) has even sym- 

 metry about ^ = ^, every mode will preserve the (even or odd) sym- 

 metry about 1=2 which it has when C = 0. We shall consider the 

 lowest even mode,* which has the eigenfunction sin x^ when C — 0. 

 Solutions of (532) having even symmetry about ^ = \ (we need consider 

 only the region ^ ^ ^ | on account of the symmetry) and satisfying 

 the boundary conditions (533) are given by 



* For large C the lowest even mode will be confined essentially to 



2sO ^ s ^ 1 2SU ) 



while the lowest odd mode will be confined to the two regions 

 g ? < U^ :ind 1 - Uo < $ ^ 1. 



If ^0 > 2/3, the latter mode will ultimately have a lower attenuation constant 

 than the former; but we shall not take space to investigate it here. 



