and 



To solve this set of equations we introduce the shorthand notation 

 s = -JL- . (183 



Multiplying equation (16) by e and adding and subtracting equation (17) 

 result in 



1 + e ikJ?- n Q-^ 



a = —V- e a^ (lyj 



+ 2e t 



and 



a = - 2^ a, , (20) 



— 2e t 



which may be introduced in equation (12) to yield the velocity within 

 the structure 



/g~ rl + c -ikix-i) l-e ik(x-ii)i .„,. 



" = /h ^ ^— ^ ^ — ^ ^ • ^21) 



o 



Adding equations (14) and (15) and introducing a and a_ from 

 equations (19) and (20) yield, after some simple algebraic manipulations, 

 an expression for the complex amplitude of the transmitted wave 



^^ (22) 



I. ,, .2 ik£ ,, ,2 -ikfi, 

 1 (1+e) e -(l-e) e 



Similarly an expression for the complex amplitude of the reflected 

 wave is obtained by subtracting equation (15) from equation (16) and 

 introducing equations (19) and (20) 



a ,, 2, , ik2, -ikii. 

 JL = (l-e ) (e -e ) 



a. ,, ,2 ik£ ,, ,2 -ikJl ' '■^■^-^ 



1 (1+e) e -(l-e) e 



20 



