72 BELL SYSTEM TECHNICAL JOURNAL 



The/„'s are the same functions of / that the c„'s are of k. 



Appendix II 

 The phase angle introduced by the path difference is : 



_ 4irhih2 r 

 ~ Xd [ 



hi' + //2" 3hi' + 10/?!-/? 2- + 3/?2' 



(48) 



The magnitude and phase of the reflection coefficient are given by 

 the following equations. 



R ^ - Ke^'l' 

 K 



\1 + « 



mx , nx 



- 1 + ;,2 ' tan ,A = Y-Z-^i ' ^^" ^2 = x/^^c, 



m = al^c, n = b/^c, ki = 2~ ^ 



where d is the angle of incidence. 



q = 2<T/f = e/Q, r = e - 1 + sin^ ^2, 



-1/2 



5 = Vr- + q-, 



s 



. + if 



where e and a are respectively the dielectric constant and conductivity 

 of the ground in electrostatic units and / is the frequency in cycles per 

 second. The values a, b and c depend on the polarization and ground 

 constants as shown in Table III below. 



For grazing incidence, ^2 — » 0, I — K —^ a^i and yp —>■ b^2- On 

 vertical polarization near normal incidence for e^ -]- q^ ^ l, x ^ 1 and 

 the approximations 1— K —^ a/c sin ^2 and 1/' — > x — &/c sin ^2 are 

 useful. These coefficients are given in Table III. 



17 1 • -^ • ^ 1 V2V^+"e , , , V2VF^ 

 i^or normal mcidence, sm ^2= 1, a = —. — and tan ^f/ = — ;; • 



At Brewster's angle, cot ^2 = Ve and K = when g = for vertical 

 polarization. For vertical polarization the minimum value of K is 



j2 — ffi .Is 



■\ h: — ; and occurs when x — I and sin ^2 = \ hr-, — 5- For hori- 



\2+w "Xe^ + g^ 



zontal polarization the maximum value of b occurs when q = -^3 r 

 and is equal to — l/-'j2r. Under these conditions s = 2r and n = — I. 



