a
dx
= b q,
x(0) = 0
(9)
dt z0 - z
Equations 8 and 9 are simultaneous non-linear equations for channel depth z and width x as
a function of the input rate (which can be time dependent) and time. Equation 8 indicates
that z will increase more rapidly as the width decreases, and Eq. 9 indicates that the width
W(x) will decrease more rapidly as the channel fills. These equations can be solved
numerically for a general situation with time-dependent variables. An analytic solution
approach for rapid desk study is given next. The analytic solution reveals physical
dependencies and yields several simple expressions governing channel performance.
ANALYTICAL SOLUTION FOR CHANNEL INFILLING
Ongoing 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 mathematically that practical applications concern a relatively short
time interval after dredging as compared to the total time required to fill a channel
completely. For this situation, the equations can be linearized under the reasonable
assumptions z/z0 << 1 and x/W0 << 1. By expansion of denominators, Eqs. 8 and 9 become,
dz ad
x
z
=
q 1 - +
z(0) = 0
(10)
dt W0 z0 W0
and
z
dx ab
=
q 1 +
x(0) = 0
(11)
dt z0
z0
which are now simultaneous linear equations for z and x.
Differentiating Eq. 10 with respect to time and substituting Eq. 11 into the resultant
equation to replace dx/dt gives,
d 2z
ad
dz
z(0) = 0, z′(0) =
+ 2b - cz = d ,
q
(12)
dt 2
dt
W0
where the quantities b, c, and d are defined as:
ad
ab ad 2
b=
c=
d = cz0
q,
q,
(13)
W02 z0
2
2 W0 z0
A second initial condition for z was introduced through the first derivative as determined
from Eq. 8 evaluated with the initial conditions on x and z. The solution of Eq. 12 is found
to be,
z = C1 exp(r1t ) + C2 exp(r2t )- z0
(14)
Kraus and Larson
6
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