niFFI^ACTIOX OF KADK) WAVFS HV A PARABOLIC CYLINDKR K).") 



obtain useful results by this method because the computation of the 

 coefficients becomes more and more difficult. 



When g is lai-ge and positi\'e it may be shown that 



^(O) f/"' + ii^yf" oxp (-hf/12). (7.35) 



The procedure used to establish (7.35) is much the same as that used 

 to establish the more general result 



S-2-1 + *S23 ~ -iMog 



-1 



'h(l - /3)' 



1/2 



exp [-W- + 2ih{l - /3)/(l +/3)] 



Lr(l + /3)J sin h(^ + d + 7r/2) ' (7.36) 



1 - /3 _ sin U^ - d + x/2) 

 1 + i3 sin h{<p + d -\- 7r/2) ' 



which holds when h^^^(l — /3) is large and positive. 



When (f is near — 7r/2 + 6, (7.36) gives the same result as (7.26) 

 plus (7.35). For <p near —ir/2 -\- 6, 



g ^ h"\l - ^)^ [2/i^'7sin 9] sin U^ - 6 + l^, (7.37) 



which shows that g is proportional to the cube root of the radius of 

 curvature 2Vsin ^d at the point where the incident ray is tangent to the 

 cylinder. 



When j3 < I, (7.36) may be obtained from 



S,, + Sn -l/i -^-^^ dn - M, / z^t-j-t. dn. 7.38 



Jc sm irn J ACE sm TrnW„{zo) 



The second integral in (7.38) represents, asymptotically, the wave 

 reflected by the cylinder. This interpretation is suggested by the fact 

 that, when the expression (7.1) for r(/i + \)Un(z)Wniz') is substituted 

 in expression (5.1) for E, the resulting integral may be written as the 

 second integral in (7.38). 



The first term in (7.36) is obtained when z'"/sin irn in the first integral 

 of (7.38) is approximated by 2i and the result integrated. When the in- 

 tegrand of the second integral is examined with the help of Table 12.3, 

 it is found to have a saddle point* at m = mi on the imaginary axis 

 between m — and rn = —ih. Near mi the integrand is approximately 



2(3~\h/hy" exp [F(m)] (7.39) 



* It is interesting to note that a saddle point also appears in the study of re- 

 flection from a sphere. See page 86 of reference.^ 



