where
ycr F0 yr 2/ 3
yr
Y
=
= 1
.
,
(
)
y0 yr/2 + 1
2
y0
yr /2 + 1
1
Lighthill and Whitham (1955) deduced the monoclinal profile length as the order of y0/S0.
Selecting the distance scale x0 = y0/S0 simplifies eq 35, which then describes dimensionless
monoclinal profiles that vary with yr and F0, and dimensionless monoclinaldiffusion pro-
files that depend only on yr.
The solution of eq 34 for the monoclinal wave and monoclinaldiffusion wave profiles
can be obtained by separation of variables and integration using partial fractions as
C1X + CI = C2 log(yf - y) + C3 log(y - y0 ) + C4 log(y - Y)
+ C5 (yf - y) + C6 (y - y0 ) + C7 (y - Y)
(36)
where
C1 = S0 (yf - y0 ) (yf - Y) (y0 - Y)
(
) (y - Y)
C2 = yf + ycr
3
3
0
= - (y - y ) (y - Y)
3
3
C3
0
cr
f
= (Y - y ) (y - y )
3
3
C4
0
cr
f
C5 = - yf (y0 - Y)
2
C6 = - y0 (yf - Y)
2
C7 = Y 2 (yf - y0 ) .
obtained CI by specifying X = 0 at y = 0.5(yf + y0). Other values of X are then obtained from
eq 36 by specifying the corresponding y. The contribution of inertia in eq 36 is eliminated by
setting ycr = 0, and the monoclinaldiffusion solution results. The solution for v is obtained
with y and eq 26, indicating that both monoclinal wave types have the same rating curve.
COMPARISON OF THE
Table 1. Case studies used to compare solutions.
LINEAR SOLUTIONS
η
v0
y0
D
c0
ck
c+
Case (m/s)
(m)
S0
(s)
(m2/s)
(m/s)
(m/s) (m/s)
F0
The five cases used to com-
pare the linear dynamic wave
I
0.5
1.0 0.0005
51
500
3.13 0.75 3.63 0.16
II
0.5
1.0 0.0001 255
2500
3.13 0.75 3.63 0.16
and diffusion wave solutions
III
3.0
1.0 0.0015 102
1000
3.13 4.5
6.13 0.96
are listed in Table 1. These cases IV 3.0 3.0 0.0005 306 9000 5.42 4.5 8.42 0.55
represent a wide range of con-
V
1.0
3.0 0.0001 510 15000
5.42 1.5
6.42 0.18
ditions, with initial velocities,
initial depths, and channel slopes that vary by factors of 6, 3 and 15, respectively. The corre-
sponding variations in the parameters η, D, and F0 by factors of 10, 30, and 6, are also large
and include most of the subcritical flow range. Cases I and II represent shallow, low-veloc-
ity flows. Parameters η, D and F0 each have relatively small values in case I. The channel
9
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