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equation
x
)
2
(1
2
Q
∂
∂
Q
∂
x
(1
−
c
0
)
∂
c
2
)
Q
(
−
c
2
)
∂
t
+
x
+
t
(1
−
∂
2
t
(1
−
c
0
)
x
2
2
Q
∂
2
Q
t
∂
x
∂
+
+
t
=
0
(7.47)
2
t
2
(1
−
c
0
)
∂
x
∂
This result is obtained from Equation (7.37). In principle, (7.37) is merely a finite differ-
ence form of the kinematic wave equation (5.111) (in the absence of lateral inflow
Q
l
).
The kinematic wave equation contains only first-order derivatives. Hence, this suggests
that the first two terms of (7.47) must represent the corresponding derivative terms in
the kinematic wave equation (5.111); it also suggests that the additional three terms in
(7.47), which contain second-order derivatives, were somehow introduced spuriously by
the finite difference approximation. This can be verified as follows. Upon substitution of
(7.38) for the constants, the coefficient of the second term of (7.47) reduces to (
x
/
K
);
since
x
is the length of the reach, and
K
the time of travel of this flood wave, their ratio
is the Muskingum celerity
c
m
, already defined in (7.20). The corresponding coefficient in
the kinematic wave equation (5.111) is
c
k
. This means that the Muskingum wave celerity
is in fact the kinematic wave celerity, or
c
m
=
c
k
(7.48)
The remaining three terms in (7.47) can be combined into one term by means of the
following kinematic wave identities (obtained by taking derivatives of (5.111) (for
Q
l
=
0)
and making use of (7.48)),
2
Q
2
Q
∂
∂
K
)
∂
t
=−
(
x
/
∂
x
∂
x
2
(7.49)
2
Q
∂
2
Q
∂
∂
K
)
2
∂
=
(
x
/
t
2
x
2
Finally, with some algebra (7.47) becomes
x
)
2
(1
2
Q
∂
∂
Q
∂
t
+
x
K
∂
Q
(
−
2
X
)
∂
x
−
=
0
(7.50)
∂
2
K
x
2
This is the standard advective diffusion equation with an advectivity, given by the kine-
matic wave celerity,
c
ko
=
x
/
K
(7.51)
and with a diffusivity
D
0
=
c
k0
(1
−
2
X
)
x
/
2
(7.52)
in which, as before, the subscript 0 indicates linearity. Equation (7.52) illustrates how the
discretization
x
is responsible for the diffusion effect inherent in the storage equation.
In the limit, when
x
is made to approach zero to obtain a derivative, the diffusivity
