CON FORMAL TRANSFORMATION 115 



Hy letting ro — ^ — °° we ()l)taiii the exact expression 



R„=-— dv dd f;(v,e)P{v,d)e~'" sine (5-7) 



TTC J— 00 •'0 



When dealing with the electric corner we saw that Re —^ a.s k —^ 0. 

 The presence of c in the denominator of (5-7) suggests the possibility that 

 /?// ^ — 1 as c — ^ 0. For Rh must remain finite and this may perhaps come 

 about through P{v, 6) —^ in the region, say around v = 0, where g(v, 6) is 

 appreciably different from zero. This and the fact that P{v, d) must 

 contain a unit incident wave suggest that for 2; < the dominant portion of 

 P{v, 6) is 2i sin cv which gives Ru = — 1. Incidentally, it is apparent that 

 the approximations for P{v, 6) given below in (5-8) and (5-10) (and therefore 

 also the approximations (5-11) for Rh) fail when c becomes small. 



The first approximation to the solution of the integral equation (5-6) is 



P^^\v, 6) = e-^'" sin (5-8) 



When we introduce the coefficients 



2 r 



bn{i) = - / g(v, e) sin 6 sin nd dd 



(5-9) 

 sin dg{v, 6) = ^ bn{v) sin nd 



bi(v) = aoiv) — a2{v)/2, bn(v) = [on-i(u) — an+i{v)]/2, n > 1 

 we find that the second approximation is 



P^^\vo , do) = e"'"" sin do + k^2~' Y. C sin mdo 



(5-10) 



The successive approximations to the reflection coefficient are 



R\!^ = 0, R\r = -i f\vbMe~''- 



2c J-oc 



RT = R'k^ - u E (4c5„)~' dvob,,{vo) (5-11) 



m=l J— 00 



J. A /■- \ „—ic(vi-Vn)—\ii — VQ[i„ 



dvbm{vo)e " '" . 



■X 



f 



