RECTANGULAR WAVE GUIDES 



149 



/ I6a'e'""(l - e-'y^'Ha, a; 1; e"'^), v > 



. ., (1 + oo) = „ , , ., , , (5-5) 



^^''- []6a'e"'(\ - e'T F(a, a; 1; e'"), r < 



where in obtaining (5-5) use was made of Euler's transformation 



F(a, b;c; x) = (1 - xy-''-''F(c - a, c - b;c;x) 



It is seen that </(l + ao)/dv is continuous at z; = but tlie second deriva- 

 tive becomes infinite as v°'~ . 



When 1 + do and Ci are expressed as the customary integrals dehning the 

 Fourier coefficients it is seen that one of the coefficients occurring in equation 

 (2-4) for 7^1 is given by 



1 + «o - 02/2 = - r (l - li" cos 2Q + e'^'T sin' Q eld 



^ -^0 (5-6) 



= (e'" i- \f"F(-a,^; 2; sech'v) 



At z' = 0, 1 + ao — 0-2/2 and its first and second derivatives are continu- 

 ous, their values being 



r(2 -f 2a) 2aT(2 + 2a) 



r(i -f a)r(2 -f a) ' r(i + a)r(2 + a) ' 



4ai2a + 2a + l)r(l -f 2a) 

 r(i + a)r(2 -f a) 



(5- 



respectively. These may be obtained by differentiating the integral in 

 (5-6) and setting z; = 0. 

 A second expression for 1 -}- ao — O2/2 follows from (5-1) and (5-2): 



1 + Co — ^2/2 = < 



e^'lFi-a, -a; 1; g"^") + ae^-'Fi-a, 1 - a; 2; e-")J, 



V > 

 F{-a, -a; l;e'-) + ae'''F{-a, 1 - a; 2; e*''), 



V < 0, 

 (5-8) 



6. Approximation to Reflection Coefficient of Horn, FJcctric Vector in (.v, y) 

 Plane 



When the flare angle 2a:7r of the horn is very small the reflection coefficient 

 may be shown to be 



R, 



la 

 2k 



+ 0(a2) 



(6-1; 



