Geoscience Reference
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t + =0.001
1
0.01
0.05
0.8
η +
0.1
0.2
0.6
0.4
0.5
0.2
1.0
0
0
0.2
0.4
0.6
0.8
1
x +
Fig. 10.22 Successive scaled positions of the water table η + = ( η − D c ) / ( D D c ) in an unconfined
riparian aquifer, as calculated with the linear Boussinesq solution (10.105), for the indicated
values of the scaled time t + = [ k 0 η 0 / ( n e B 2 )] t . The variable x + = x / B is the scaled distance
from the stream; for t + > 0 . 2, the solution is essentially reduced to the fundamental harmonic,
(10.107), that is the first term in the series expansion.
Equation (10.105) is displayed for several values of the time t + in Figure 10.22. Reverting
back to the original (dimensional) variables, by means of (10.94), one obtains from (10.105)
1) sin (2 n 1) π
x
4( D D c )
π
1
(2 n
η =
D c +
2 B
n
=
1
,
2
,...
× exp
t
(2 n 1) 2
2 k 0 η 0
π
(10.106)
4 n e B 2
This solution was already implicit in the work of Boussinesq (1903, 1904), who compared
the problem to the “cooling of a prismatic homogeneous rod, laterally impermeable, of
length L , having its extremity x = 0 immersed in melting ice and its other extremity,
x = L , impermeable to the heat just like the sides.” But he felt that the solution would
“reduce more or less rapidly to the simple fundamental solution of Fourier,” that is, the
first term in the series, so that the higher-order terms would be negligible. It was probably
not until the work of Dumm (1954) and Kraijenhoff (1958) that the full series was used in
hydrology.
The arguments of the exponential functions in the series in (10.105) (and in (10.106))
increase rapidly as 1, 9, 25,...Moreover, the amplitudes of the sine functions in the series
decrease as 1, 1 / 3, 1 / 5,...These are the two features which made Boussinesq observe,
that with time only the first term in the series survives. Thus the water table (10.105)
gradually assumes the shape of the first sine function, and the long-time linear solution is
the fundamental harmonic
sin π x 2 exp π
4
π
2 t +
4
η + =
(10.107)
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