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u
(
x, y
) =
sin
πy
2
a
4
a
2
Z
R
3
ϕ
(
t
)
2 + cos
πy
a
+ ch
π
|
x
−
t
|
a
sh
π
|
x
−
t
|
2
a
|
x
−
t
|
cos
πy
a
−
ch
π
|
x
−
t
|
a
2
dt
+
+
sin
πy
2
a
2
πa
Z
R
3
ψ
(
t
)
sh
π
|
x
−
t
|
2
a
|
x
−
t
|
cos
πy
a
+ ch
π
|
x
−
t
|
a
dt.
Application in the underground hydrodynamics.
We consider the
solution to the problem of filtration theory describing the flow under the
spot dam with an aquitard as an example of the formula for the two variables
application. Filtering the liquid (water) is caused by the pressure difference
at the upstream wall (
P
1
=
−
ϕ
1
)
and downstream wall (
P
2
=
−
ϕ
2
)
(Figure). The velocity field of the filtered fluid is described by the vector
−→
v
=
k
−−−→
grad
u
, where the coefficient
k
characterizes the permeability of
the medium (soil) [10, 11]:
Δ
u
(
x, y
) = 0
,
−∞
< x <
∞
,
0
< y < a
;
u
(
x,
0) =
ϕ
1
, x <
0
, u
(
x,
0) =
ϕ
2
, x >
0;
u
y
(
x, a
) = 0
,
−∞
< x <
∞
.
The solution to this problem is a function
u
(
x, y
) =
ϕ
1
a
sin
πy
2
a
0
Z
−∞
ch (
π
(
x
−
t
)
/
2
a
)
ch (
π
(
x
−
t
)
/a
)
−
cos (
πy/a
)
dt
+
+
ϕ
2
a
sin
πy
2
a
∞
Z
0
ch (
π
(
x
−
t
)
/
2
a
)
ch (
π
(
x
−
t
)
/a
)
−
cos (
πy/a
)
dt
=
=
ϕ
2
−
ϕ
1
π
arctg
sh (
πx/
2
a
)
sin (
πy/
2
a
)
+
ϕ
1
+
ϕ
2
2
.
Filtration flow diagram for the
point dam with an aquitard
10
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2015. No. 1