154 BELL SYSTEM TECHNICAL JOURNAL 



that k is given by (1-2) and c by (2-2), and finally the difference (not the 

 sum) of Re and Rh must be taken. In (6-1) Re is the retlection coetBcient 

 of the component of the magnetic vector normal to the (.v, y) plane (which 

 is proportional to Q), while in (7-1) Rh is the reflection coefficient of the 

 transverse electric vector (which is proportional to P) and there is a difference 

 in sign just as in the case of voltage and current reflection coefficients. If 

 a > b and Xo is the wavelength in free space, the superposition gives the 

 following expression for the reflection coefficient of the electric vector: 



R = Rh — Re 



(8-1) 

 - '- [(2a/\of - \r(ocH/[{2a/\,f - 1] - aa^/&) 



where liran and Itvue are the total horn angles in the planes of H and E, 

 respectively. Of course this approximation can be expected to hold only 

 when uh and a^ are small. 



9. Numerical Calculations — Rh for 60° Horn 



The value of Rh , the reflection coefficient when the magnetic vector lies 

 in the plane of the flare, was computed on the assumption that only the 

 dominant mode need be considered.* Thus, instead of the system of 

 equations (2-4) and (2-5), only their simplified version, namely the single 

 second order differential equation (7-2), was used. This equation may be 

 written as 



^^ + TiTFi = (9-1) 



where, according to (5-6), 



K - -//2 = ,-2(1 j^ a^_ a.,/2) - \ (9-2) 



1 + ao - 02/2 = (e-" + l)-«F(-a, 1/2; 2; sech- i')- 



The problem was to obtain the Rh appearing in that solution Fi of (9-1) 

 which satisfies the boundary conditions (2-6). 



No computations for Re were made. 



In the first method of calculation the integrals in the approximation 

 (3-14), namely 



R„= 1 f\-^A-^'^f,A'-''^J. (9-3) 



2/ J-oo dv- 



2^ = 2icv + 2i / {K^ - c) dv, 

 * I am indebted to Miss M. Darvillc for carrying out the computations of this section. 



