Environmental Engineering Reference
In-Depth Information
Also, by Eq. (2.88), let
t
=
t
0
,
r
→
0, i.e
2
f
(
u
(
M
0
,
t
0
)=
u
(
0
,
t
0
)=
at
0
)
.
Therefore
∂
(
r
=
at
0
,
t
=
0
,
r u
)
1
a
∂
(
r u
)
u
(
M
0
,
t
0
)=
r
+
∂
∂
t
or, by the definition of
u
r
=
at
0
,
t
=
0
⎡
⎣
∂
∂
⎛
⎝
⎞
⎠
+
⎤
⎦
r
S
M
0
r
r
S
M
0
r
1
4
π
1
a
1
4
π
∂
u
u
(
M
0
,
t
0
)=
u
d
S
t
d
S
.
(2.90)
r
∂
Applying the initial conditions
u
(
M
,
0
)=
ϕ
(
M
)
and
u
t
(
M
,
0
)=
ψ
(
M
)
leads to
⎡
⎛
⎞
⎤
⎣
⎝
⎠
⎦
,
1
∂
∂
1
t
0
1
t
0
u
(
M
0
,
t
0
)=
ϕ
(
M
)
d
S
+
ψ
(
M
)
d
S
a
2
4
π
t
0
S
M
0
at
0
S
M
0
at
0
or, as both
M
0
and
t
0
are arbitrary in
Ω
and
(
0
,
+
∞
)
, respectively,
⎡
⎣
∂
∂
⎛
⎝
⎞
⎠
⎤
⎦
,
1
1
t
1
t
M
)
M
)
u
(
M
,
t
)=
ϕ
(
d
S
+
ψ
(
d
S
(2.91)
a
2
4
π
t
S
at
S
at
which is the same as the Poisson formula (2.84).
2.8.3 Method of Descent
Three-dimensional results such as Eqs. (2.84) and (2.85) can be used to obtain the
corresponding results of the two- and one-dimensional cases by using the method
of descent.
Two-dimensional Wave Equation
Expressing spherical surface integrals by double integrals in the
Oxy
-plane is the
key for the reduction from three spatial dimensions to two dimensions. This can be
achieved by noting that all functions
ϕ
(
x
,
y
)
,
ψ
(
x
,
y
)
,
f
(
x
,
y
,
t
)
and
u
(
x
,
y
,
t
)
of two
spatial variables are even functions with respect to plane
z
k
(constant) because
of their independency of
z
. We illustrate this by developing, from Eq. (2.84), the
=
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