berm and beach face are assumed to move in conjunction with the shoreline 

 position. The required sediment transport is then computed by the change in 

 position of the shoreline. The two equations are 



n+1 n ^ r n+1 n -, ,->o , 



^-,0 =^i,0 ^ ^^i.l -^ij^ (38a) 



Qn.l _ tBerm_^]fyn|l_^n^^3 ^33^^ 



The offshore boundary is treated by keeping y""*"^ (i ,jmax) (the contour 

 beyond the last simulated contour) fixed, until the angle of repose is exceeded. 

 Then, the y""*"^ (i,jmax+l ) is reset (at the conclusion of the n + 1 time-step) 

 to a position such that the slope equals the angle of repose. Note that 

 y"''"^(i,0) is represented in the program by YZEROj . 



There are also no-flow boundary conditions required at each of the 

 structures being modeled. These are imposed on the adjacent y-grid points which 

 are located downdrift (i.e., in the shadow zone) of the structure and shoreward 

 of the structures' seaward extremities. They are imposed by setting S3i j of 

 equation (33) and DISTRi^j (the term in square brackets in equation (29J 

 equal to zero, thereby causing Qx(i,j) to be zero (i.e., the no-sediment flow 

 condition). This boundary condition is imposed automatically for every 

 shore-perpendicular structure. 



It was found that even with the implicit formulation, high frequency 

 oscillations occurred in the y values immediately updrift and downdrift of the 

 structure. The solution did not "blow up"; however, on larger time-steps 

 "sloshing" (oscillating) did occur. Part of this problem was due to the 

 boundary condition at the structure which had been such that either no sand was 

 allowed along a contour line or the sand determined by the equations was allowed 

 to be transported. Because of the very large angle which existed around the tip 

 of the structure when a contour first exceeded the length of the structure, \/ery 

 large amounts of sediment transport were predicted. In the nature where analog 

 sand transport rather than digitized transport occurs, this does not happen. 

 Therefore, the boundary condition was altered to constantly allow sand transport 

 around the end of the structure in proportion to that part of the contour 

 representation which exceeded the structure (i.e., the transport was calculated 

 for the location at tip of the structure as if the structure was not there and 

 then a proportion of this value was allowed to bypass). Although the transport 

 around the tip of the structure is based on the values from the past time-step, 

 it more closely simulated the natural phenomenon. 



Additionally, a dissipative interface is used on the y values as follows: 



y. . = (x) y. . . + (1 - 2t) y. . + (x) y. . . (39) 

 •^1,0 1-1, J i,J 1+1, J 



where t was again taken as 0.25. It is noted that only high frequency 

 oscillations in y are affected by the use of equation (39); the total sum 

 of y values is not affected. Also, in all the dissipative interface 



27 



