For convenience, let Q = coS dp sin ap 
Ty ce ee 
Q = K2Q (51) 
As before, referring to equations (43) and (44) 
CON “heyy apace a uals (CQ) OQ es ule as 
ot ox dt cco hie COSS Ce (O)Mena) dOMN ots icoscmcs(@) momo more 
Also, 
2Q _ 2 && oND 
0 Be OD 
from equation (51). 
The second term in each of the above two equations is negligible in physi- 
cal situations of usual interest where the distance between the shoreline and 
the tip of the breakwater is large compared to the distance the shoreline 
changes during a time At. 
Therefore, the transport equation becomes 
aq a2Q 
— = eos (BOQ) > &) == (52) 
at dE /y9 ox 
and 
oY. 2 eR ly, (53) 
ot ox mdt 
This system is solved using the same type of algorithm as used previously. 
In the present situation where only refraction is important, several approx- 
imations are possible which produce problems having analytic solutions. The 
most direct approximation, and essentially the assumption of Penard-Considere 
(1956), is to approximate 
a2y 
p) i 
ung (2 cos A cos ap - sin 9 sin of) ——7> 7-5 (54) 
We CVax) 
at 
where z = 0.25 + 5.5 H,/Lo (see eq. 16), subject to the boundary conditions 
OL 
ox 
= anino 
tad 
i 
° 
by 
