CIRCUIT DESCRIBED IN TERMS OP NORMAL MODES 245 



So 



E. = e E ^^(p. _ r^) (6.43) 



£. = Zi[i:±^ ,- Z !^|^ C6.44) 



coe 1 ;, — 1 -^ 



6.2 Comparison with Results of Chapter II 



Let us consider a case in which there is only one mode of propagation, 

 characterized by 7ri(:K, 3^), Fi, and a case in which the current flows over a 

 region in which 7ri(x, y) has a constant value, say, 7ri(0, 0). This corre- 

 sponds to the case of the transmission line which was discussed in Chapter 

 II. 



We take only the term with the subscript 1 in (6.44) and (6.30). Combin- 

 ing these equations, we obtain for the field at 0, 



E, = 



{E^/P'P)(T' + ^l) ^^-^^ // f"^^^' ^^^' ^^ ^^ 



(VI + ^i) 2f^i(o, 0) 



We have from (6.36) 



7ri(0, 0) 



Ji = 



If IHx, y)]^ dx dy 



(6.45) 



(6.46) 



From (6.45) and (6.46) we obtain 



2(r? + /3^)(r? - r') 



Let us compare this with (6.4), which came from the transmission line 

 analogy of Chapter II, identifying Ez and / with E and i. We see that, 

 for slow waves for which 



iSo « I r'x I (6.48) 



j8o « I r' I (6.49) 



(6.47) becomes the same as (6.4). It was, of course, under the assumption 

 that the waves are slow that we obtained (2.10), which led to (6.4). 



6.3 Expansion Rewritten in Another Form 



Expression (6.44) can be rewritten so as to appear quite different. We 

 can write 



r' + /3'o = r' - tI + r'„ + ^l 



