RECTANGULAR WAVE GUIDES 153 



[ ^'^K-^'^K dv = [ e-^'e'"" - ly^ k\16(x' e'"" + 4ae"'"'] dv 

 Jo Jo 



1 /: 2 2 /* -2J -3 iav , 



= loK a I e X e dv 

 Jo 



I / -2tc»-2t> -3 2, , (7-11) 



-f- e c K ia dv 



Jo 



^_,(,_tan l.-o+tan-lc]/a_^._,^^. _^ 2aK'c~'/il + ic) 



= Oia) + 2aK-'rV(l + ir). 



That the integral having x as the variable of integration is 0(pr) may be shown 

 as in (7-10). 



When we combine our results in accordance with (3-14) we obtain 



2l J-oo 



dv 



_OCK C ^ r 1 1 



4i \_1 — ic 1 + ic 

 ia/(2c') + Q(a) 



+ O(a^) 



(7-12) 



which is (7-1). 



If, instead of discarding (7-10) because it is 0(a-), we retain it and the 

 corresponding integral in (7-11) (in the hope that they represent most of the 

 difference between the approximate value (7-1) for Rh and the true value) 

 we obtain the approximation 



i?/? = 2~3 "" T / ^ (:>x + .T ) dx (/-13) 



in which the integral may be evaluated by numerical integration. 



The approximations (6-1) and (7-1) for the reflection coefficients may also 

 be obtained from an equation given by N. H. Frank. ''^ However, care 

 must be taken to suitably define the wave guide characteristic impedance 

 which appears in his expression. 



8. Speculation on the Reflection Obtained from Horn blared in Both Directions 



All the work from Section 4 onward applies only to a horn tlared in one 

 plane. Nevertheless, it is interesting to speculate on how close an estimate 

 of the reflection from a three-dimensional horn may be obtained by super- 

 posing the two reflection coefficients (6-1) and (7-1). It must be kept in 

 mind that the flare angles (the a's) may be different in the two directions, 



