Sharma 



C*(s,y) + iS*(s,y) = ^s^- 1 ^(x,y) exp(isx) dx . (!'') 



Now, if the asymptotic behavior of wave height 



-X -»co: ^x,y) ^ (Cj cos x + Cj sin x)/\/c ^ - x (18) 



is postulated in analogy to the result obtained by the usual stationary phase 

 analysis of any theoretical wave system, then the function Ux,y) can indeed be 

 extrapolated to -x^co, once the constants Cj , Cj, and C3 have been determined 

 by, say, a least-squares fit to the known function ^ in some finite region 

 Xj < X < Xj, where -Xj, -x^ >> !• 



The contribution of the integration - 00 < x < x^, where x^ is some large 

 negative value of x , can then be derived by closed evaluation of the following 

 integrals: 



^ r e C, COS X + C, Sin X /-«v 



AC + iAS* = ys2- 1 — exp(isx)dx. (19) 



J- 00 



c , COS X + Co sin X 



7^ 



After some rewriting and simplification one obtains the following results: 



AC* = y^ [diCpCz^) + d2Sp(z*) + d3Cp(z-) + d^SpCz')] 



(20a) 



AS' „ 2 



= /^[-diSp(z*) + d^CpCz^) - d3Sp(z-) + d^CpCz-)] (20b) 



with the abbreviations 



dj - 7s - 1 [cj cos C3(s + 1) + C2 sin C3(s + 1)] , (21a) 



dj = ys-l [cj sin C3(s+ 1) - Cj cos C3(s+ 1)] , (21b) 



dj = V s + 1 [cj cos C3(s- 1) - Cj sin C3(s- 1)] , (21c) 



d^ = V s + 1 [cj sin C3(s- 1) + Cj cos C3(s- 1)] . (21d) 



Herein Cp and Sp denote the easily computable Fresnel integrals 



Cp(z) + iSp(z) = I exp^i^ tA dt , (22) 



e.g., see Ref. 12, and the argument z takes the values 



738 



