IMPEDANCE CONCEPT AND APPLICATION 43 



in the direction normal to the x;y-plane. We need only to rewrite 

 these equations as follows: 



Hy = (Ho cos ^ e^^y ^'° ^)e~''^ ""^ s+io>t^ \^^) 



The relative distribution of the amplitude and the phase of the wave 

 are governed by the factor e-^y ^'° '' and this phase-amplitude pattern 

 is propagated in the direction of the z-axis, the propagation constant 

 being a cos ??. 



The advantages of this point of view are clear. In attempting to 

 find the reaction of the second medium upon the incident wave, it is 

 necessary to satisfy certain boundary conditions at every point of the 

 interface. This can be insured by requiring the reflected and the 

 refracted waves to have the same phase-amplitude patterns at the 

 interface and by adjusting their relative amplitude and phases to 

 secure the fulfilment of the boundary conditions at some one point. 

 In other words, the problem is reduced to that for which the general 

 solution was given in Part II. 



The impedance to the incident wave in the z-direction is found 



from (22) : 



^ Ex Eo _ 



Z^ = jj- = — = 77 sec I?. 



Hy Ho cos ?? 



This impedance is seen to be a function of the intrinsic impedance of 

 the medium and of the angle of incidence. 



For the refracted wave in the second medium the transmission 

 equations are similar to (22) : 



EJ = {Eo'e~'''y ^'" '^)e~'''' ""^ i+io>t^ 



Hy' = {Ho' cos lA e-''^ ^'" ^)e-'''' '=°' ^+-', Eo' = v'Hq. *^^^'* 



The "angle of refraction" \J/ is, in general, different from ??. In our 

 equations we may regard xp merely as a parameter. Its value is 

 obtained from the condition that at the x3'-plane the phase-amplitude 

 pattern of the incident and the refracted waves must be the same, and 

 consequently 



a sin§ — a' sin \p. (24) 



In dielectrics this relation is known as Snell's law of refraction. 

 By (23), the impedance to the refracted wave in the z-direction is 



F ' 



Zt ■'-'X t I 



z =777 = ^ sec lA- 



