RRCTANGILAK WAVEGriDES 151 



3C X ^ 00 



S S — 7 — ir^N = 22 w"^' 

 „=i „,=i mii(m -\- II) „=i 



turiKMl up ill tlu' in\-csti,i,fati()n of the orders of magnitude of the various 

 terms. 



7. Approximation to Rcjicction Coefficient of Horn, Magiielic Vector in (x, y) 

 Plane 



The work of this section is quite similar to that in vSection 6 except that 

 here we enter into more of the details. We shall show that when a is small 

 the reflection coefficient aj)pearing in equation (2-6) is 



Rh = ^^3 + O(a^). U-1) 



From (2-4) the analogue of the differential equation (3-1) is 



F'l -^ [k--(1 + a, - aJ2) - \]b\ = (7-2) 



and the A' appearing in the second of equations (3-14) is now 



A' = -h- = K--(l + an - a./l) - 1 (7-3) 



The largest terms in the expression (5-8) for 1 + (7o — 0-2/2 yield, to within 

 terms of 0(a), 



A' = K-{e^"' + ae--') - 1 , v> 



k = K-(4a e^" - 2ae--'') (7-4) 



A = K.-{\()a-e^'- + Aae--') 



K = /c-(l + ae-') - 1 = c- + K-ae-' , z' < 



A - laK-e-" (7-5) 



A = iaK-e-" 



where the dots denote differentiation with respect to v and c- = k- — 1. 

 We have retained the a'- in A as given by (7-4) because at this stage we do not 

 know whether it may be neglected or not. 



When V < 0, the dehnition (3-5) of ^ and (7-5) yield 



^ = icv -\- i i (K' — c) dv 



(7-6) 

 = icv + ic I [(1 + K c"' ae'")' — IJ dv = icv + 0(a) 



J— 00 



