DIFFHACTIOX OF R.VniO WAVFS BY A l'AHAT?OMC CYUNDFU If)!) 



2, • • • .It must not pass close to a = since the work leading to (7.61) 

 assumes (/a to be a small number. Changing the variable of integration 

 to /' = a + (/a, approximating i/a, (/a by (/i\ i/v, and evaluating 

 the integral by considering the residues of the poles at y = al gives 



// ^ ,-'M/. E (l - -4)"" ""' '"''-^rl'^f "•" • (7.63) 



This shows, to a first approximation, how the expression (7.52) is 

 modified when the cyhnder is a very good, but not perfect, conductor. 

 Of course g must be negative in (7.03). Since ( in (7.62) varies as li'"^ 

 while k in (7.59) varies as IrT^^^ it appears that the field for \'ertical 

 polarization is much more sensitive to changes in the conductivity 

 than it is for horizontal polarization. 



It may be verified that the change in the exponential terms in the 

 series (7.30) and (7.52) produced by finite conductivity, namely 



as changes to a^ — k 



(7-64) 

 tts changes to a^ — S/us, 



agrees, to a first approximation, with the change produced in the cor- 

 responding series (given, for example, by the series (27) and (28) on 

 page 45 of Reference 7) for the propagation of radio waves over the 

 earth's surface. 



8. FIELD AT A GREAT DISTANCE BEHIND THE PARABOLIC CYLINDER WHEN 

 6 = x/2 AND h IS LARGE 



In the work of Section 7 the angle of incidence 6 may lie anywhere 

 between and tt. Here vve take d = t/2, w^hich corresponds to the case 

 shown in Fig. 2.3 and described in Section 2. Some simplification is ob- 

 tained thereby. For example, the incident wave is now simj^ly exp ( — ix). 

 We shall write the expressions for the horizontal and vertical polariza- 

 tion cases as 



E = (e-''^ + S{)r + Sn + (^22 + ^23), (8.1) 



H = (e-'' + S,)r + Sn + (Sz2 + ^33), (8.2) 



respectively. Here >S2i, • • • are defined by (7.16) and (7.46) in which 



e = 7r/2, w = 1, 



(8 3) 

 /3 = t/r, = cot (<p/2 + 7r/4) = 1 - ^ + ^72 - <p'/3 + • • • 



Throughout this section jS will be defined by (8.3), i.e., by (7.4) with 



