II. CONFORMAL MAPPING 

 1 . Development . 



The selection of appropriate conformal mapping and error functions 

 is of principal concern in achieving the first stated objective. We 

 desire to map a region R of the Z-plane into a rectangle in the ?- 

 plane (Figure 1), in which the seaward boundary and coastline curves 

 are specifically transformed into the image plane as constant values 

 of n • Furthermore, the curves in the negative x region are re- 

 quired to be the mirror image of those in the positive x region. 

 As an artifice to assure that the lateral boundaries of the mapped 

 region represent straight parallel lines normal to the shoreline and 

 the seaward boundary, symmetry about one of these boundaries (x = 0) 

 is imposed and the whole range in x {-A to A) is considered to be one 

 wavelength of a periodic function. Thus only the range < x < X 

 in the top panel of Figure 1 corresponds to the real region of the 

 shelf. 



The conformal mapping relation is taken in the form: 



Z = F(C) , (1) 



where 



Z = x + iy , (2) 



5 = K + in , (3) 



i = /T . (4) 



An appropriate form of the transformation for the mapping considered 

 in Figure 1 is : 



N 

 F(c) = P + Q £ + I (P cos nk? + sin nk?) , (5) 



n=l 



and 



where 



tt/X , (6) 



and X is half the horizontal extent of the region in the Z , or t, 

 plane. The coefficients P and Q n , n=0,l...N, are, in general, 

 complex and independent of £ or n • Let 



P = A + iB , (7) 



n n n 



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