Environmental Engineering Reference
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Poisson formula of two-dimensional wave equations, the solution of
u
tt
=
a
2
Δ
u
+
f
(
x
,
y
,
t
)
, −
∞
<
x
,
y
<
+
∞
,
0
<
t
,
(2.92)
u
(
x
,
y
,
0
)=
ϕ
(
x
,
y
)
,
u
t
(
x
,
y
,
0
)=
ψ
(
x
,
y
)
.
To find the solution of PDS (2.92) from Eqs. (2.84) or (2.91), consider a spherical
surface
S
at
for a sphere of center
M
(
x
,
y
)
and radius
at
. Its projection onto plane-
2
2
and radius
at
denoted by
D
at
:
Oxy
is a circle of center
M
(
x
,
y
)
(
ξ
−
x
)
+(
η
−
y
)
≤
2
.Let
be the angle between positive
z
-axis and normal of
S
at
. Thus
(
at
)
γ
2
2
2
(
at
)
−
(
ξ
−
x
)
−
(
η
−
y
)
cos
γ
=
±
,
at
is corresponding to points on the upper half
S
1
or the lower half
S
2
of
S
at
.
where
±
Since
ϕ
(
x
,
y
)
and
ψ
(
x
,
y
)
are symmetric with respect to
z
=
0,
M
)
at
M
)
at
M
)
at
ϕ
(
ϕ
(
ϕ
(
d
S
=
d
S
+
d
S
S
at
S
1
S
2
2
S
1
2
D
at
M
)
at
ϕ
(
ϕ
(
ξ
,
η
)
at
1
cos
=
d
S
=
d
ξ
d
η
γ
2
D
at
ϕ
(
ξ
,
η
)
=
η
,
d
ξ
d
2
2
2
(
at
)
−
(
ξ
−
x
)
−
(
η
−
y
)
which reduces the spherical surface integral into a surface integral on plane-
Oxy
.
Similarly, we can also express
S
at
M
)
d
S
by using a double integral over
D
at
.
ψ
(
Finally, we obtain the
Poisson formula
of two-dimensional wave equations
⎡
⎣
∂
∂
1
ϕ
(
ξ
,
η
)
u
(
M
,
t
)=
d
ξ
d
η
2
π
a
t
2
2
2
(
at
)
−
(
ξ
−
x
)
−
(
η
−
y
)
D
at
⎤
ψ
(
ξ
,
η
)
⎦
.
+
d
ξ
d
η
(2.93)
2
2
2
(
)
−
(
ξ
−
)
−
(
η
−
)
at
x
y
D
at
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