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: 



A = Y [ <d' -f Qix + /?2)- - V^'^ + {h, - h2)A 



h{' + h2^ , 3/7. i' + UViiVir + 3//2^ 



_ 4:irhih2 r 



2rf2 



+ 



8d' 



(48) 



The magnitude and phase of the reflection coefficient are given by 

 the following equations. 



R ^ - Ke'^ 



K 



\1 + a 



mx 



, tan ^ = 



nx 



1 + x2 ' '^ 1 - r 



w = o/Vc, w = 6/Vc, 



where 6 is the angle of incidence. 



, sin ^2 = xj^c. 



h = 



g = 2alf = e/Q, r = e - 1 + sin^ ^2, 



- Vr^ + q\ 



1 + 



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, h and c depend on the polarization and ground 

 constants as shown in Table III below. 



For grazing incidence, ^2-^0, \ — K —^ a^z and 1/' -^ b^z. On 

 vertical polarization near normal incidence for e^ -\- q^ ^ I, x^ 1 and 

 the approximations 1 — X — > a/c sin ^2 and yp —^ -k — hjc sin ^2 are 

 useful. These coefficients are given in Table III. 



IT 1 • -^ • ^ 1 V2V7+"e , ^ , V2VF^ 



i^or normal mcidence, sm ^2= 1, « = — ; — and tan \p = — z • 



s-\-\ 1—5 



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

 polarization. For vertical polarization the minimum value of K is 



i 



m 



and occurs when x = 1 and sin ^2 



V? 



For hori- 



2 + w ^^"'^ '^ ^ "' \e2 + g2 



zontal polarization the maximum value of b occurs when q = 4S r 

 and is equal to — 1/V2?'. Under these conditions s = 2r and n = — \. 



