a= INCOMING WAVE DIRECTION —— MINIMAL SHORELINE 
——-—-— COMPUTED SHORELINE 
Ax 2Ax Ax 2Ax x 
Figure 10. Initial shoreline and incoming wave direction (A); computed and 
minimal shorelines for finite-difference scheme of time 1At (B)- 
now used to derive an equivalent equation for the transport Q which, although 
subject to similar numerical problems, will satisfy the transport boundary con- 
ditions exactly. 
In a situation where only refraction is important, the general equation 
then becomes 
OF, Ow 
dt Ox 
where Q is cos J Sin op, Ap 18 f(a), and a, is a - tan! dy/ox. Differen- 
tiating by x gives 
as i eS (43) 
The transport function Q can be considered as a function of o, which may 
be solved for a,, such as a = g(Q). Thus, the transport equation (43) 
becomes 
ane SO, Dior 
a ania ic) = op an(g(Q) - a) = aD 
therefore, 
BO) ls 37Q 
st Cosi (g(a) ao 
but 
3g(Q _ dg(Q) 2Q 
at dol) 3c 
therefore, 
aQ__ cos? (g(Q) - a) 37Q 
ot dg(Q /ag ax? 
(44) 
27 
