ERDC/CHL CHETN-IV-35
June 2001
general situation with time-dependent variables. An analytic solution approach for rapid desk
study is given next.
ANALYTICAL SOLUTION FOR CHANNEL INFILLING: Operation and maintenance of
channels will not allow the depth to become less than project depth or allow the width of the
channel to be greatly reduced. These conditions are equivalent to stating that interest concerns a
relatively short time interval after dredging as compared to the total time it would require to fill
the channel completely. For this case, the equations can be linearized under the assumptions
z/z0 << 1 and x/W0 << 1. By expansion of denominators, Equations 8 and 9 become
x
dz adR
z
=
qR 1 - εrd
+
z(0) = 0
(10)
za W0
dt W0
and
z
dx abR
=
q 1 +
x(0) = 0
(11)
za R
za
dt
which are now simultaneous linear equations for z and x.
Differentiating Equation 10 with respect to time and substituting Equation 11 into the resultant
equation to replace the dx/dt gives
d2z
a
dz
z(0) = 0, z ′(0) = dR qR
+ 2b - cz = d ,
(12)
dt 2
dt
W0
where the quantities b, c, and d are
ε a
ε aa
2
b = rd dR qR ,
c = rd bR dR qR ,
d = cz0
(13)
22
2 W0 z0
W0 z0
A second initial condition for z was introduced through the first derivative as determined from
Equations 8 evaluated with the initial conditions on x and z. The solution of (12) is found to be
z = C1 exp(r t ) + C2 exp(r2t)- z0
(14)
1
where
r1 = -b + b2 + c ,
r2 = -b - b2 + c
(15)
and
z ′(0) - r2 z0
C1 =
C2 = -C1 + z0
,
(16)
r1 - r2
6
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