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η
and
x
with their respective maximal values
D
and
B
, as suggested by the boundary
conditions (10.65). Insertion of these normalized variables in the governing differential
equation (10.30) yields then the appropriate scaling of the time variable. Thus one ends
up with the following scaled variables
x
+
=
x
/
B
(
n
e
B
2
)]
t
t
+
=
[
k
0
D
/
(10.66)
η
+
=
η/
D
These dimensionless variables allow the Boussinesq equation (10.30) to be simplified as
∂η
+
∂
t
+
∂
∂
x
+
η
+
∂η
+
∂
x
+
=
(10.67)
and the boundary conditions (10.65) as
η
+
=
0
x
+
=
0
t
+
≥
0
∂η
+
∂
x
+
(10.68)
=
0
x
+
=
1
t
+
≥
0
η
+
=
1
x
+
=
1
t
+
=
0
This problem can be solved by separation of variables, that is, by assuming that the
solution
η
+
=
η
+
(
x
+
,
t
+
) is a product of two functions, one dependent only on
x
+
and one
dependent only on
t
+
,or
η
+
=
F
1
(
x
+
)
F
2
(
t
+
)
(10.69)
Substitution of (10.69) into (10.67) produces
F
1
dF
1
dx
+
1
F
2
dF
2
dt
+
1
F
1
d
dx
+
=
=
C
1
(10.70)
in which
C
1
must be constant; since
x
and
t
are independent of each other, the
F
1
and
F
2
dependent parts of Equation (10.70) can only be equal to each other if they are constant.
The solution of the differential equation for
F
1
cannot be expressed in terms of common
functions, but for the purpose of the present discussion it is not needed, so let it be assumed
that it is known; it will be derived below. However, the solution of the differential equation
for
F
2
yields immediately
−
F
−
1
2
=
C
1
t
+
+
C
2
(10.71)
where
C
2
is a second constant. Thus, putting
−
F
1
(
x
+
)
/
C
2
=
F
(
x
+
) and
a
=
(
C
1
/
C
2
), one
can write the solution (10.69) in the following form
F
(
x
+
)
1
+
at
+
η
+
=
(10.72)
where
a
is a dimensionless constant, and
F
(
x
+
) is a function of
x
+
that satisfies the same
differential equation as
F
1
(
x
+
) and the conditions
F
=
0 for
x
+
=
0, and
F
=
0 and
F
=
1
for
x
+
=
1, according to the boundary conditions (10.68). This solution indicates similarity,
in that the initial shape of the water table is preserved throughout. In other words, once the
