Geoscience Reference
In-Depth Information
becomes
c
s
dh
d
d
(
Vh
)
d
−
+
=
0
(7.4)
ξ
ξ
After integration across the wave, this becomes in turn
(
c
s
−
V
)
h
=
constant
or
(7.5)
V
2
)
h
2
Equation (7.5) can now be substituted into (7.3) to integrate the latter. According to (7.5)
part of the first term in (7.3) is independent of
(
c
s
−
V
1
)
h
1
=
(
c
s
−
ξ
. Therefore after taking that part outside
the integral sign, (7.3) becomes
2
h
2
−
h
1
=
g
(
c
s
−
V
1
)
h
1
(
V
1
−
V
2
)
+
0
(7.6)
Equation (7.5) can also be used to eliminate
V
2
from (7.6); after substitution of
V
2
=
[
c
s
−
(
c
s
−
V
1
)
h
1
/
h
2
] into (7.6) and some algebra, one obtains finally
gh
2
(
h
2
+
1
/
2
h
1
)
c
s
=
V
1
±
(7.7)
2
h
1
The square root term in Equation (7.7) represents the celerity of the bore relative to the
velocity at cross section 1. Because of symmetry, the subscripts 1 and 2 in Equation (7.7)
can be interchanged, to yield the celerity relative to the velocity at section 2. Whenever
sections 1 and 2 are defined respectively as the section upstream and the downstream
from the abrupt wave, the plus sign in Equation (7.7) describes the downstream motion
of surges of type A and D, and the minus sign upstream moving surges of type B and C.
Thus flood waves, as encountered in hydrology require the plus sign in (7.7).
It can be verified that, if the analysis had been carried out for a channel with arbitrarily
shaped cross sections at points 1 and 2, the result would have been
g
(
A
2
h
2
−
1
/
2
A
1
h
1
)
c
s
=
V
1
±
(7.8)
−
A
1
/
2
A
1
(1
A
2
)
When
h
is large, or when the disturbance is small,
h
1
≈
h
2
, and (7.7) relative to
V
1
(
V
2
), reduces to Lagrange's celerity equation (5.50), as was to be expected. When
c
s
=
≈
0, Equations (7.7) and (7.8) describe a stationary hydraulic jump.
The analysis presented here is simplified considerably, in that the effects of bed slope
and resistance have been omitted. When the wave travels over large distances in a natural
river these factors can play an important role; nevertheless Equation (7.8) can sometimes
provide worthwhile first order information on the main features of such waves.
Disastrous floods
Abrupt waves of type A have been associated with some extreme flooding events in the
past. For instance, in the United States, the Johnstown flood is still among the largest
natural disasters on record (see McCullough, 1968; Degen and Degen, 1984). The flood
