^ = t -^ +1-3 (2) 



3x ^ ay ^ ' 



t = t k^ + 1 ky (3) 



where i and j are the unit vectors in the x and y directions respectively. 

 Equation (1) can be expressed as 



3(ksine) _ aCkcose) ,.^ 

 3x ay 



in which e is the direction of the vector wave number relative to the x-axis 



n 



and k denotes \k\. Noda expanded Equation (4) to the following form 



k cose - + sine ^ '= -k sine ^ + cose - (5) 

 ax ax ay ay 



f^ k f^ k 

 Since t— and ^^ are known from the angular frequency a, the water depth 

 dX dy 



h, and the dispersion equation 



a^ = g k tanh kh (6) 



Equation (5) can be solved numerically, although there are problems of 

 directional stability. The primary advantage of Equation (5) is that it 

 allows the wave direction e to be determined on a specified grid, compared to 

 unspecified locations that would be obtained by, for example, wave ray 

 tracing. 



2. Crenulate Bays (LeBlond, 1972) . 



LeBlond attempted to model the evaluation of an initially straight 

 shoreline between two headlands into a crenulate bay. The model constitutes 

 a one-line (shoreline) representation. The transport equation employed 

 related the total sediment transport to total water transport in the surf 

 zone as predicted by the formulation provided by Longuet-Higgins (1970). The- 

 initial shoreline patterns resemble crenulate bays in nature; however, the 

 predictions were found to be unstable for reasonably long periods of computa- 

 tional time and did not approach a realistic planform. 



3. Crenulate Bays (Rea and Komar, 1975) . 



Rea and Komar employed a rather ingenious system of orthogonal grid 

 cells to provide a cell which locally is displaced perpendicular to the 

 general shoreline orientation. A one-line representation was employed. A 

 simple and approximate representation of wave diffraction was employed. 

 Although the model yielded reasonable results for the examples presented, the 

 unique coordinate system would not be suitable for a general model as the 

 coordinate system must be "tailored" to some degree to conform to the 

 expected shoreline configurations. 



