Geoscience Reference
In-Depth Information
In the conceptual derivation of Equation (7.15),
X
was introduced simply to weight
the relative effects of the inflow and outflow sections. Accordingly, in cases of pure
reservoir action, that is, when the flood passes through a level pool whose stage (or level)
is controlled by a spillway at the downstream end, the storage
S
should be independent of
the inflow rate, and therefore
X
0. On the other hand, in a uniform rectangular channel
with a plane water surface the two sections should be weighted equally and ideally
X
=
0.5. Equation (7.25) allows now a fuller interpretation of
X
. Equation (7.19) already
showed that the parameter
K
is the mean residence time of the flood wave in th
e reach;
both second moments and
K
2
in (7.25) have the basic dimensions [T
2
]. Hence
√
1
=
2
X
reflects the rate of increase of the (strea
mwise)
width of the wave as it travels through
the reach; because mass is conserved,
√
1
−
2
X
must also reflect the rate of decrease of
its height, that is, the rate of subsidence of the peak discharge of the flood hydrograph.
According to (7.25), the difference between the two second moments is maximal when
X
−
0, that is under conditions of pure reservoir action. On the other hand, (7.25) indicates
that, when
X
=
2, the wave does not undergo deformation, but it retains its original
shape as it travels. Because the peak of a flood wave normally decreases along its path,
in principle
X
should be smaller than 0.5.
=
1
/
7.2.2
Analytical solution
The ordinary differential equation (7.16) can be readily solved. One common technique is
to multiply both sides by exp[
t
/
K
(1
−
X
) ]. This allows it to be written as
dt
e
t
/
K
(1
−
X
)
K
(1
−
X
)
Q
e
=−
KX
d
dt
e
t
/
K
(1
−
X
)
Q
i
+
d
K
1
−
X
e
t
/
K
(1
−
X
)
Q
i
(7.26)
Finally, the integral of (7.26) provides the outflow rate
Q
e
resulting from a given inflow
rate into the reach
Q
i
=
Q
i
(
t
), as follows
K
(1
−
X
)
2
Q
i
(
e
−
t
/
K
(1
−
X
)
X
(1
−
X
)
Q
i
(
t
)
)
e
τ/
K
(1
−
X
)
d
τ
−
Q
e
=
τ
+
constant
(7.27)
in which the value of the constant depends on the values of the flow rates at some reference
time.
Unit response function
This is the outflow from the channel reach in response to a unit impulse inflow into the
reach at the inflow section at
t
=
0 (see Appendix). Thus with a Dirac delta function inflow,
Q
i
=
δ
(
t
), Equation (7.27) immediately yields
e
−
t
/
K
(1
−
X
)
K
(1
−
X
)
2
X
δ
(0)
(1
−
X
)
u
(
t
)
=
−
(7.28)
The first two moments of the unit response
An alternative way to describe a function is by means of its moments. These can be deter-
mined for the unit response function (7.28) as follows. The first moment of the unit response
