Geoscience Reference
In-Depth Information
or, neglecting higher-order terms,
S
f
−
S
0
=
b
−
1
S
0
[(
V
p
/
V
0
)
−
(
ah
p
/
h
0
)]
(5.65)
for which values of
a
and
b
can be taken from Table 5.2. In many applications in hydrology,
the discharge rate
q
is of greater interest than the water depth
h
or the velocity
V
. Hence,
replacing
V
by
q
/
h
and making use of
q
=
q
0
+
q
p
, one can rewrite (5.44) for turbulent
flow with
b
=
1
/
2 as follows
∂
h
p
∂
t
+
∂
q
p
x
−
i
=
0
∂
and
(5.66)
∂
t
+
2
gh
0
S
0
q
p
gh
0
−
q
0
∂
h
p
∂
x
+
2
q
0
h
0
∂
q
p
∂
x
+
h
0
∂
q
p
(1
+
a
)
h
p
h
0
q
0
−
=
0
These may be combined into one equation by operating with (
∂/∂
t
) on the second of (5.66)
and then substituting (
∂
h
p
/∂
t
) from the first, or
2
q
p
∂
t
2
∂
t
∂
x
+
q
0
−
gh
0
∂
2
q
p
2
q
p
∂
x
2
+
2
gh
0
S
0
/
q
0
∂
q
p
∂
t
h
0
∂
+
2
q
0
h
0
∂
(5.67)
∂
x
=
q
0
−
gh
0
∂
i
+
2 (1
+
a
)
gS
0
h
0
∂
q
p
∂
x
+
2 (1
+
a
)
gh
0
S
0
i
This equation reduces to the one first derived by Deymie (1938) for the special case
without lateral inflow, i.e. for
i
=
0, and with Chezy's formula, i.e. for
a
=
1
/
2. The solution
of Deymie's equation for the propagation of a disturbance resulting from a known
q
p
=
q
u
(
t
)
at
x
=
0, has been obtained by different methods (see Deymie, 1939; Masse, 1939; Lighthill
and Whitham, 1955; Dooge and Harley, 1967).
The more general solution, for an inflow
q
p
=
q
u
(
t
)at
x
=
0 and an arbitrary non-
zero lateral inflow
i
=
i
(
x
,
t
), has been presented by Brutsaert (1973) and the reader is
referred to the journal article for the mathematical details. The conditions, that must be
satisfied by (5.67) to describe this situation, remain the same as in (5.59), and can be applied
to
q
p
,or
q
p
(0
,
t
)
=
q
u
(
t
);
x
=
0
,
t
>
0
q
p
(
∞
,
t
) finite;
x
→∞
,
t
>
0
(5.68)
q
p
(
x
,
0)
=
0;
x
>
0
,
t
=
0
∂
q
p
(
x
,
0)
=
0;
x
>
0;
t
=
0
∂
t
The solution of this problem is
t
∞
q
p
(
x
,
t
)
=
G
(
ξ,τ
;
x
,
t
)
i
(
ξ,τ
)
d
ξ
d
τ
0
0
)
∂
G
(
ξ,τ
;
x
,
t
)]
∂ξ
t
−
gh
0
−
q
0
q
u
(
τ
d
τ
(5.69)
ξ
=
0
0
