where the alternate forms are developed by eliminating either y or v using the Chezy equa-
tion. U in eq 27 is greater than vf, and hence all velocities along the profile. With eq 27 we
rewrite B in eq 26 as
vf v0
vf - v0
2 2
= ( g C* 0 ) 1/ 2
= ( g C* 0 )
yf y0
2 S -1
2 S 1/ 2
B = y0 yf
.
(28)
1
v0 + vf
yf - y0
yf + y0/ 2
Both U and B can be expressed in dimensionless form as
U 2 yr /2 - 1
3
=
(29)
ck 3 yr - 1
yr (yr /2 - 1)
1
B
=
(30)
.
(yr - 1)
v0 y0
These parameters increase continuously with yr from minimums of 1 and 1/2 at yr = 1,
respectively, and are related to each other by a change of scale as
2 B
U
=
+ 1 .
(31)
3 v0 y0
ck
In eq 29 and 30 vr = vf / v0 can be substituted for yr by using the Chezy equation to obtain
v r = y 1/2 .
(32)
r
Returning to the monoclinal wave equation development, we divide eq 25 by gS0, elim-
inate v and v' from the inertia term using eq 24 and 26, and obtain
B2
y′
(v / v0 )2
1 -
= 1 - (y / y ) .
(33)
g y3
S0
0
Equation 26 is used again to eliminate v from eq 33, which after rearranging becomes
2
(
B - Uy)
1
S0 y 3 -
2
S0 (y - yf )(y - y0 )(y - Y)
g C* S0
y′ =
=
(34)
y - ycr
y 3 - ycr
3
3
3
where
y0 yf
Y=
< y0
(
)
2
+
yf /2
1
y0/2
1
and ycr = B2/g represents the contribution of inertia.
3
Rewriting eq 34 in terms of convenient dimensionless variables y* = y/y0 and x = x/x0
~
yields
(y* - yr )(y* - 1) y* - y
Y
S x
dy*
0
= 0 0
(35)
3
d x y0
~
3 ycr
y* -
y0
8
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