Geoscience Reference
In-Depth Information
The integrals in the denominators of (7.17), which are the zeroth moments, are required
to normalize
Q
; ideally, they should equal each other, if there are no lateral inflows or
outflows in the reach. Substitution of (7.16) in (7.17) produces
X
)
Q
e
]
dt
∞
∞
d
dt
[
XQ
i
+
t
t
=−
K
t
(1
−
Q
i
dt
(7.18)
0
0
Integration by parts, and imposition of the condition that both
Q
i
and
Q
e
are zero for
t
at
infinity, leads finally to the desired result
t
t
=
K
(7.19)
In words, Equation (7.19) states that the parameter
K
can be interpreted as a measure of
the lag or the time of travel
t
t
of the flood wave through the reach. Accordingly, when
the channel reach has a length
x
, the celerity of a Muskingum wave is
c
m
=
x
/
K
(7.20)
The width or average duration of a flood wave hydrograph, which is one of the more
obvious measures of its shape, can be conveniently characterized
by i
ts standard deviation
σ
, that is, the square root of its second moment about the mean,
√
m
2
. Thus, the change in
shape of a flood wave hydrograph, after passing through a channel reach, can be described
by the difference between the second moments of the outflow and inflow hydrographs,
namely (
m
e2
−
m
i2
). Since the second moment about the mean is related to the moments
about the origin as indicated in Equation (13.12), this difference can be written as
(
m
e2
−
m
i2
)
(
m
e1
)
2
(
m
i1
)
2
(
m
e2
−
m
i2
)
=
−
+
(7.21)
in which the difference between the two moments about the origin is
0
0
∞
∞
t
2
Q
e
dt
t
2
Q
i
dt
(
m
e2
−
m
i2
)
=
−
(7.22)
0
∞
∞
0
Q
e
dt
Q
i
dt
As before, in (7.22) the two terms in the denominators should be equal to each other,
when there are no additional in- or outflows in the reach. Substitution of Equation (7.16)
into (7.22) produces now
X
)
Q
e
]
dt
∞
∞
d
dt
[
XQ
i
+
(
m
e2
−
m
i2
)
t
2
=−
K
(1
−
Q
i
dt
(7.23)
0
0
Integrating (7.23) by parts, and making use of the same operations that led from (7.18)
to (7.19), one finds
(
m
e2
−
m
i2
)
2
K
(
m
e1
−
=
KX
)
(7.24)
Finally, substituting (7.24) into (7.21), and recalling that according to (7.19)
m
e1
−
m
i1
=
K
, one obtains
K
2
(1
(
m
e2
−
m
i2
)
=
−
2
X
)
(7.25)
