The general equation is thus transformed to 
A ag Dy) a 
B+ Di. Me ee yh dd, 1 Dy, dAy nis KpKe sin 2a, 
Mo eo 
n~ 
The hats will be dropped from all variables at this point in all discus- 
Sions except those relating to observed results. Also, the subscript s for 
y and Q will be omitted. Hence, the general equation used in the following 
is 
DO AS 
B + De. dy 1 De day 9 Knkp sin 20, 
* (We - ¥p) = + 5 eS sO) 
Bo + De at dt 2 Bo Dende dX 2 
Recall 
> _ COS A 
Rumcosmap 
where a5 = a + tan-! dy/ax and ap, = function of ap, as f(a.) (the function 
f depends only on H,/Lg as previously shown). Therefore, 
Ne 
22 . 12 9 
Kikp sin 202, = Ks cos a, Sin ap, 
Note 
ee cr ec) Bets 
ax COS % Sin ap = (cos A cos ap Jes = Sill Cy) Sala «p) aa (a5) aa 
The general equation (37) then becomes (after some rearrangements) 
BceaeD) 2 
dy O C 1 a“y 
ae 2 SS PCy), Se oe eee Rei) (38) 
ot Be De Tae ( Lax) axe 
where 
By ap 1D) 
O Cc Ja, dD 
RES yait) = B+ Ds F (a,) yx Ye 5 Yp) dt 
D oK 
1 C dAy D : 
Ss =— 9 
7 By © A ce + 2Kp 5, COS Op Sin ap (39) 
and 
a(x) in the diffraction zone 
a 
The above equation is the general dimensionless form which gives the time- 
dependent sand budget. 
The general equation is nonlinear and appears to be impossible to solve ana- 
lytically. As it is also difficult to solve computationally in the most univer- 
sal case (where lake level, beach slope, and bluff-berm height vary as functions 
of x, y, t), several simple cases are described in detail which indicate the 
behavior of the equations. Although numerical results are presented, detailed 
descriptions of the numerical methods used are given in a later discussion. 
23 
