Geoscience Reference
In-Depth Information
where
t
0
=
[(
t
−
x
/
c
01
)(
t
−
x
/
c
02
)]
1
/
2
, and where
d
1
,
d
2
and
d
3
are constants already
defined below Equation (5.70). The symbol I
1
( ) denotes the modified Bessel function
of the first kind of order one, and H( ) is the Heaviside step function (see Appendix).
The solution (5.72) consists of two wave like motions that modify the steady uniform
flow
q
0
. The first part, given by Equation (5.73), is identical to the solution of the analogous
dynamic case given in (5.60), except that it also contains an exponential decay term in
x
.
This part retains the form of a delta function and the argument of this delta function shows
that it has the celerity of a dynamic wave, namely the same as given by Equation (5.60) or
the first of (5.64). The amplitude of
u
1
decreases exponentially, provided
d
4
>
0. However,
when
d
4
<
0, that is when Fr
0
>
a
−
1
, the wave grows exponentially; this is the well-known
criterion for bore formation, namely Fr
0
>
2 (in the case of Chezy) or Fr
0
>
3
/
2 (in the case
of GM). For small Froude numbers the amplitude of
u
1
decays as exp[
−
S
0
x
/
(Fr
0
h
0
)]
=
exp[
−
g
1
/
2
S
0
x
/
(
h
1
/
0
V
0
)] . This means that the dynamic part of the disturbance decays rapidly
and becomes unimportant over short distances
x
, whenever the bed slope is large and the
flow velocity small. Interestingly, a very similar result was obtained by an analysis of the
nonlinear equations (5.13) and (5.22) by Lighthill and Whitham (1955; see also Stoker,
1957, p. 505), who showed that the dynamic front of any surface disturbance decays as
exp(
−
gS
0
t
/
V
0
), for small Froude numbers.
The second part,
u
2
given by Equation (5.74), constitutes the main body of the wave.
Mathematically, the Heaviside step function eliminates singularities from this part of the
solution. Physically, the unit step function guarantees that this part will never forge ahead
faster or further than the position of the dynamic front
x
=
(
c
01
t
), given by Equation (5.73),
and it ensures that the first part given by (5.73) represents indeed the leading edge of the
disturbance caused by the unit impulse at
x
=
0. The celerity of the main body of the wave
can be determined from the mean travel time of the wave. This mean time of occurrence
of a wave is the first moment about the origin, denoted as
m
1
or
μ
; this is also called its
centroid or center of area. Therefore, the travel time is the difference between the mean
time of occurrence of the wave at the point of observation and that at the origin
x
=
0, that
is the time for its center of gravity to reach
x
. The upstream inflow at
x
0 is given by a
delta function, whose first moment about the origin is zero. Therefore, mathematically the
mean travel time is the first moment of the outflow rate
q
p
(
x
=
,
t
) about
t
=
0foragiven
x
,or
+∞
m
1
=
tq
(
x
,
t
)
dt
(5.75)
−∞
The Laplace transform of (5.72) can be used as a moment generating function (Dooge,
1973). Accordingly, the first moment is the first derivative of the Laplace transform at the
origin in the transform domain, and it can be readily shown that it is given by
x
m
1
=
(5.76)
(1
+
a
)
V
0
This yields a celerity
x
/
m
1
for the main body of the disturbance
c
k
0
=
(1
+
a
)
V
0
(5.77)
For reasons which will become clear in Section 5.4.3, a wave with a celerity [(1
+
a
)
V
]is
referred to as a kinematic wave; Equation (5.77) represents its linearized form, as indicated
by the 0 subscript.
