Allowing the wave direction to alternate between +a and -a at a fixed 
rate gives an interesting variant to the previous example. The boundary condi- 
tions are as before for a> 0. The results are shown in Figure 9(B) for the 
same values as the physical parameters used in the problem of Figure 9(A). Of 
interest is that the undulatory patterns of the shoreline in Figure 9(A) dis- 
appear in Figure 9(B). Hence, diffraction-induced undulations in a natural 
shoreline probably rarely appear since offshore wave climates are often 
multidirectional. 
The numerical scheme used to generate the preceding examples was based on 
the use of implicit finite differences. Such schemes, whether implicit or ex- 
plicit (or both), are commonly used to efficiently solve parabolic problems. 
However, even in the case where only refraction is considered (Fig. 8), the 
boundary condition 
numerically gives a solution which initially may not conserve mass, i.e., the 
integrated transport equation 
L 
-- I ydx = Q(L) 
may not be satisfied. 
Unfortunately, this feature is unavoidable for most such schemes (the ex- 
ceptions are discussed below) as the following demonstrates. Figure 10(A) 
shows an initially straight shoreline. In any finite-difference scheme, after 
one time increment At, the shoreline is bounded below by the solid shoreline 
in Figure 10(B). This shoreline has the least possible area, A, where 
A= as tan a (41) 
The conservation of mass equations requires 
ACQ (LS TAtcosmaksanwicn =A 
Thus, At must satisfy the inequality 
Ne > Hist D | (42) 
2 Sin) Gp Cos= -a 
Since the accuracy (and in explicit schemes, stability as well) depends on the 
ratio } = At/Ax*, the above inequality places a lower bound on the accuracy of 
the solution which may be unacceptable in practice. The finite-difference form 
of the equation for the conservation of mass may be incorporated directly into 
the numerical scheme. In this case a solution exists which is similar to the 
previous case but shows a small erosion throughout the reach. For engineering 
applications, the primary quantity of interest is the amount of sand on a given 
shoreline. Then it is more important to conserve mass than to satisfy the shore- 
line boundary condition as written in the present form. The general equation is 
26 
