130 



BELL SYSTEM TECHNICAL JOURNAL 



where 0(a-) denotes correction terms of the order a-. This result is based 

 upon the fact that when terms of order a' are neglected the set of dif- 

 ferential equations (1-4) and (1-5) reduce to the single equation 



Fo + (1 + a,)k''Fo = 



where, from (5-1, 4, 5), 



1 + a. 



V > 



V <Q 

 1 



(6-2) 



y (1 + (7o) 4ae^«" 

 av 







dv' 



(1 + a„) 16aV«"(l - e-^'^y^-' 16aV"(l - e^")2«-i 



The reflection coefiicient (6-1) is the one corresponding to the differential 

 equation (6-2) and may be computed by setting 



(1 + ao)k-' = -Ji' = K (6-3) 



in the integrals (3-14). ' • 



The expression (6-1) for Re may be obtained quickly (but the procedure 

 is not trustworthy) by assuming that the principal contribution to the first 

 .integral in (3-14) comes from the region close to v = 0, say in — e < r < e, 

 where the second derivative of Ir^ '- is infinite but integrable. When the 

 integration is performed approximately by replacing the second derivative 

 by the first, (3-14) gives 



Re 



1 



2 



m 



dv 



If 



(6-4) 



dt 



(1 + aoP 



la 



where e is assumed to be so small that 1 + (Zo is effectively unity and 

 d{l -\- ao)/dv changes from at — e to 4a at + e. 



A more careful investigation based on the second integral in (3-14) also 

 leads to the value (6-1) for Re . It further suggests that possibly most of 

 the correction term, denoted by 0(a-) in (6-1), is given by 



2 »o 



— / 



2lk Jn 



2i-2av 



dv = -^ + J. [Si(x) 

 4 IX 4? 



7r/2 + iCi(x)] (6-5) 



with X = k/a and 2^ = /.v[exp (lav) — 1]. Si(x) and C'/(.v) denote the 

 integral sine and cosine functions. Incidentally, the rather curious result 



