Geoscience Reference
In-Depth Information
Significance of the approximation
Within the validity range of hydraulic theory, Equations (5.13) and (5.22) (or (5.24) and
(5.25)) describe the phenomenon of free surface flow; therefore, as the solutions of (5.44)
and (5.67) indicate, dynamic waves always occur, but as shown in (5.73), gravity modifies
their amplitude. Thus in general, small forerunners of a disturbance move with veloci-
ties given approximately by (5.64). However, as a result of gravity and friction, the main
part of the disturbance usually moves with a much smaller velocity, namely as given by
(5.107), (5.108) and (5.113) (or (5.77) and (5.91) for the linearized case). When (
gS
0
t
/
V
)
is large, the dynamic waves are damped sufficiently that the kinematic waves, which usu-
ally move at a slower speed, assume the dominant role. It is under such conditions that
Equations (5.105) and (5.106) describe the flow. As will be shown in Chapters 6 and 7,
the kinematic wave approach is useful in the solution of several problems of practical
interest.
Solution of the linearized equation
As before, when the disturbances around a steady uniform reference flow are not excessive,
one can decompose the variables into an undisturbed part and a perturbation, and the
continuity equation is the first of (5.66), rewritten here for convenience
∂
h
p
∂
t
+
∂
q
p
x
=
i
(5.115)
∂
According to (5.43) with (5.100), the rate of flow can be written in terms of the decomposed
variables as
=
q
0
1
+
a
+
1
h
p
h
0
q
0
+
q
p
=
C
r
S
0
(
h
0
+
h
p
)
a
+
1
Hence, because
q
0
=
V
0
h
0
, and presumably
h
p
<<
h
0
, one can write
q
p
=
(
a
+
1)
V
0
h
p
(5.116)
This shows that, since
q
p
is a function of
h
p
only (
V
0
is constant), just like (5.103) and
(5.104), (5.115) can be written as a total time derivative of
h
p
, that is
∂
h
p
∂
t
+
dq
p
dh
p
∂
h
p
∂
x
=
i
(5.117)
or, alternatively as a total time derivative of
q
p
,as
∂
q
p
∂
t
+
dh
p
∂
q
p
dq
p
dq
p
dh
p
i
∂
x
=
(5.118)
From (5.117) and (5.118) one can define a wave celerity (
dq
p
/
dh
p
); however, by virtue of
(5.116) and (5.43), this is equal to (
dq
0
/
dh
0
). Hence one has
dq
p
dh
p
=
dq
0
dh
0
=
c
k0
(5.119)
or
c
k0
=
(
a
+
1)
V
0
(5.120)
As anticipated, this result is the same as the celerities given by (5.77) and (5.91).
