Geoscience Reference
In-Depth Information
6.2
KINEMATIC WAVE APPROACH
With the kinematic approximation, only the two slope terms in Equation (6.2) (or the sec-
ond of (6.4)) are of any importance (cf. Equation (5.100)); thus the momentum equation
becomes
S
f
=
S
0
(6.7)
Physically, Equation (6.7) states that the water surface is assumed to be parallel to the
bed. Substitution of (6.7) into (5.43) yields immediately the kinematic relationship
K
r
h
a
+
1
q
=
(6.8)
(
Vh
) is the rate of flow per unit width of plane [L
2
T
−
1
], and
K
r
can be
defined with (5.43) as
K
r
=
where
q
=
C
r
S
0
, in which the values
C
r
,
a
and
b
are listed in Table 5.2
for different flow conditions.
Equation (6.8) implies a unique relationship between
q
(or
V
) and
h
. Hence, as already
indicated in Section 5.4.3, the continuity equation (6.1) is suggestive of the mathematical
form of a total derivative
∂
h
∂
dx
dt
∂
h
dh
dt
t
+
x
=
(6.9)
∂
with the following equalities
dx
dt
=
dq
dh
dh
dt
=
and
i
(6.10)
The quantity
dx
/
dt
defines a kinematic wave celerity, which by virtue of (6.8) is
dx
dt
=
1)
K
r
h
a
1)
K
1/(
a
+
1)
r
q
a
/(
a
+
1)
c
k
=
(
a
+
=
(
a
+
(6.11)
To an observer moving forward at this rate
dx
dt
, both equalities in (6.10) will appear
to hold. Recall from Chapter 5, that the path of such an imaginary observer traced on the
x-t
plane is called a
characteristic
of the wave motion.
/
6.2.1
Unsteady lateral inflow
Consider first the case of an arbitrary unsteady, but uniform, lateral inflow
i
i
(
t
), and
consider an imaginary observer moving with a velocity
dx
/
dt
given by Equation (6.11).
To this observer, that is along the characteristics, on account of (6.10), it will appear
=
(i)
that the water depth
h
changes at a rate
i
=
i
(
t
), so that at any time this depth is
the integral of
i
, namely
t
h
=
idt
(6.12)
t
0
