Geoscience Reference
In-Depth Information
rate disturbance
q
p
, the solution resulting from an inflow
q
u
(
t
)at
x
=
0is
t
q
=
q
(
x
,
t
)
=
q
0
+
q
u
(
τ
)
δ
(
x
−
(
V
0
+
c
0
)(
t
−
τ
))
d
τ
(5.61)
0
or, upon integration,
q
=
q
(
x
,
t
)
=
q
0
+
q
u
(
t
−
x
/
(
V
0
+
c
0
))
(5.62)
To repeat briefly the results of this section, general solutions of the dynamic part of
the shallow water equations result in two waves; their speed of propagation
c
0
=
(
gh
0
)
1
/
2
relative to the mean motion, is also known as Lagrange's celerity equation (cf. Equation
(5.50)). One of these “dynamic” waves is moving in the direction of the current and the
other against the current. Thus two observers, one moving downstream with a velocity
c
01
≡
(
V
0
+
c
0
) and the other with a velocity against the current
c
02
≡
(
V
0
−
c
0
), would
see the small disturbance as a stationary, i.e. non-moving, displacement of the surface from
equilibrium. Recall in this context, that the ratio
V
0
/
c
0
defines the Froude number for steady
uniform flow, that is
V
0
(
gh
0
)
1
/
2
Fr
0
=
(5.63)
Therefore, when
c
02
<
0, or Fr
0
<
1, the flow is subcritical and this disturbance (or the
observer) actually moves upstream; when
c
02
>
0, the flow is supercritical, and while this
disturbance still moves against the current, it is smaller than
V
0
and is thus swept down-
stream. The paths of these two observers traced on the
x-t
plane are called characteristics.
From the above analysis it follows that these characteristics can be defined by the ordinary
differential equations
dx
dt
=
V
0
+
(
gh
0
)
1
/
2
=
c
01
and
(5.64)
dx
dt
=
(
gh
0
)
1
/
2
V
0
−
=
c
02
The concept of characteristics, which is being introduced here in an offhand way, arises
formally in the theory of partial differential equations. However, this is beyond the scope
of the present discussion. For an introduction to the mathematics, the reader is referred to
such topics as Sommerfeld (1949), and on the application of characteristics to free surface
flow to Stoker (1957) or Abbott (1975).
Solutions of complete system: two types of wave
If the last term in the second of Equations (5.44) is not neglected, it must also be expressed
in terms of the initial steady uniform flow variables and their perturbations; making use of
(5.39) for flow in a wide channel, with
R
h
=
h
(or (5.43)), one obtains
(
V
0
+
V
p
)
2
C
r
(
h
0
+
h
p
)
2
a
V
0
C
r
h
0
S
f
=
=
[(1
+
2
V
p
/
V
0
+···
)(1
−
2
ah
p
/
h
0
+···
)]
