246 BELL SYSTEM TECHNICAL JOURNAL 



Thus, we can rewrite the expression for Ez as 



77 .-r^ // •/ N Y^ (!"" + 0o)7tn(x, y)Jn 



E. = e ^( -jM Z _ ^, 



(6.50) 

 + (i^f) S 7r„(.v, y)Jn) 



The second term in the brackets is just j/coc times the impressed current, 

 as we can see from (6.35). The first term can be rearranged 



i-jMivl + 0l)Jn 



{-j/o:e)ivl + /3o) // Tnix, y)J(x, y) dx dy {(),^\) 



11 [Ttnix, y)f dx dy 



Referring back to (6.29), let ^„ be twice the power P„ carried by the 

 unforced mode when the field strength is 



I £^no I = 1 (6.52) 



Further, let us choose the7r,j's so that, at some specified position, x = y = 0, 



„(0, 0) = 1 (6.53) 



Then 



Using this in connection with (6.51), we obtain 



TnTTnix, y) 1 1 TTnix, y)j{x, y) dx dy 



E^ = e-'\ - E 



^«(r?. - n) 



+ (i/we)y(x, y) 



(6.55) 



An expression for the forced field in terms of the parameters of the nor- 

 mal modes was given earlier '". In deriving this expression, the existence of 

 a set of modes was assumed, and the field at a point was found as an in- 

 tegral over the disturbances induced in the circuit to the right and to the i 

 left and propagated to the point in question. Such a derivation applies for 

 lossy and mixed waves, while that given here applies for lossless transverse- 

 magnetic waves only. 



