Geoscience Reference
In-Depth Information
in which the dependent variable
y
can represent
h
p
,
V
p
or
q
p
, and in which
c
0
≡
(
gh
0
)
1
/
2
(5.50)
is a constant introduced at this point for conciseness of notation. The general solution of
this wave equation can be written as
F
1
[
x
m
−
c
0
t
m
]
+
F
2
[
x
m
+
c
0
t
m
]
(5.51)
where
F
1
and
F
2
are arbitrary functions, which must be determined from the initial and
boundary conditions. It can be readily verified that Equation (5.51) is indeed a solution
by substituting it back into (5.49) and carrying out the differentiations. It can also be
shown, that the solution must have the form of (5.51), as follows. Consider for this purpose
the coordinate transformations
ξ
=
x
m
−
c
0
t
m
and
η
=
x
m
+
c
0
t
m
, which lead from
y
as a
function of
x
m
and
t
m
to (say)
Y
as a function of
ξ
and
η
,or
y
(
x
m
,
t
m
)
=
Y
(
ξ,η
)
By applying the chain rule of differentiation to (5.49) with this equality, one obtains the
differential equation for
Y
,or
2
Y
∂ξ∂η
=
∂
0
This shows that (
∂
Y
/∂ξ
) is independent of
η
, and conversely that (
∂
Y
/∂η
) is independent
of
ξ
. Hence if
Y
is to depend on both
ξ
and
η
it must have the form
Y
=
F
1
(
ξ
)
+
F
2
(
η
),
which is the same as (5.51).
The form of Equation (5.51) describes actually two waves, each with a speed of prop-
agation
c
0
(relative to a reference moving with the fluid velocity
V
0
of the undisturbed
flow), but traveling in opposite directions. To distinguish the speed of propagation of a
disturbance or of a wave, from the velocity of the fluid itself, it is common to refer to it
as
celerity
. As an illustration, consider now the case where
y
represents the water surface
elevation
h
p
, which is the easiest to visualize. In this case, initially at
t
m
=
0, the function
F
1
(
x
m
−
c
0
t
m
) defines a water surface configuration
h
p
=
F
1
(
x
m
); at a later time
t
m
=
t
m1
it
describes the configuration
h
p
=
F
1
(
x
m
−
c
0
t
m1
). However the water surface shape is still
the same, except that during the
t
m1
units of time it has moved to the right without distortion
over a distance
c
0
t
m
1
. The same can be said about the function
F
2
(
x
m
+
c
0
t
m
), which defines
a water surface configuration moving in the opposite direction, with the same celerity
c
0
.
The actual displacement of the water surface is the sum of these two waves. Equation (5.50)
is commonly referred to as Lagrange's celerity equation.
Example 5.3. Long channel with arbitrary initial conditions
To determine the arbitrary functions
F
1
and
F
2
, initial and boundary conditions are needed.
Consider a wide uniform channel extending to
x
m
=±∞
from the (moving) origin at
x
m
=
0. Assume as initial conditions that the variable
y
=
f
(
x
m
) and also its time derivative
(
∂
y
/∂
t
m
)
=
g
(
x
m
) are known for any value of
x
m
. Thus with (5.51) one has
y
(
x
m
,
0)
=
f
(
x
m
)
=
F
1
(
x
m
)
+
F
2
(
x
m
)
x
m
,
0
=
g
(
x
m
)
=
c
0
−
F
1
(
x
m
)
+
F
2
(
x
m
)
(5.52)
∂
y
∂
t
m
