522 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



E.= -J- (^) sin ^ e""" » = 1, 2, 3 ■ . ■ (VII-2) 



E,= -J-(l^) cos ^ e"'"' (VII-3) 



It has been assumed that the wavelength in the dielectric of the frequency 

 under consideration is much greater than d. 



We shall first consider a wave incident upon the boundary from the left. 

 From equations (V-16), (V-17), (VII-1) and (VII-3) we have f or :*: < 



H, = Ae *'^ + B^"" + 2: C„ cos ^ T'^^^" (VlI-4) 



m a 



Be"n 



tO:!€l m \ d / 



(VII-5) 



cos — f- e 

 d 



where w = 1, 2, 3 • • • and k is given by (IV-31). For a: > we have from 

 equations (V-10) and (V-11) 



H.= Ei^ncos^e-^-^"^ (VII-6) 



n la 



£. = y^I:I>ncos^^-•'- (vii-7) 



where n = 1, 3, 5 • • • and kn is given by (V-8). Kt x = the boundary 

 conditions give 



^+5+Zc„cos^=Z/)nCos^^ (VII-8) 



^U-B)-J-Ec.(^)cos??^ 



r ., «... „ V <i / d ^^^^_^^ 



= V^ 



^ niry 



2^ Dn COS —f 



If €1 = €, it can be seen from (VII-8) and (VII-9) that 5 = C^ = 0. Thus 

 there is no reflected wave and no unpropagated waves are needed. Let us 

 consider only this case. The coefficients Dn are determined by 



^ = E^nCOS?^ (VII-10) 



n 2d 



