and 



= C + iD , (8) 



ti n n 



where the coefficients A^ , B n , C n , and D n are real quantities. 

 However, the symmetry condition about x = (or E, = 0) requires that 

 all Aj^ and D n be zero. The resulting relations for x and y in 

 terms of E, and n are: 



N 

 x(£,n) = C ? + I (B sinh nkn + C cosh nkn) sin nkE, , (9) 



n=l 



and 



N 



I 

 n=l 



y(5>n) =B + C n + I (B cosh nkn + C sinh nkn) cos nk£ . (10) 

 j >-•,> '^oo^n n 



The condition that the range of x and E, be the same requires that 

 C = 1. The remaining N values of C n and N+l values of B^ are 

 determined by matching the coastal and seaward boundary curves at 

 n = ± 8, respectively, g also being a parameter to be determined. 



The coordinates X,Y of the given coast or seaward boundary 

 curves are specified parametrically in terms of arc length measured 

 along each curve from some fixed point. The functions X s , Y s , 

 X c , and Y c are single-valued functions of this parameter where the 

 superscript s or c represents the seaward boundary or coastline 

 curve, respectively. This property is essential since a Fourier 

 series-type representation is employed in determining the coefficients. 

 The problem is to determine 8 , B , and the set of coefficients, 

 B n and C n , for a given N such that equations (9) and (10) give a 

 best fit to the given curves, in the sense of minimizing an appropriate 

 mean square error function. Since the specified curves are not known 

 directly in terms of E, , but, rather in terms of arc length, the 

 bicurve fitting equations require an iterative process starting from 

 some initial estimate of arc length in terms of E, for each curve. 



It is found that a convergent iterative procedure results if one 

 chooses the following as the i+1 approximation of the coefficients: 



A . A 



8 = ttC/ Y C (A X ) d? - / Y^A 1 ) d? ] , (11) 



ZA o o 



A . X 



B = TrU yC ( a1 ) d ^ + / Y S (A X ) d5 ] , (12) 



° ZA o o 



15 



