Environmental Engineering Reference
In-Depth Information
By Eq. (5.70) in Section 5.4, we obtain the fundamental solution
ch
A
A
2
t
2
2
2
1
(
t
−
τ
)
−
(
x
−
ξ
)
−
(
y
−
η
)
t
−
τ
2
e
−
u
2
(
x
,
y
,
t
)=
τ
0
d
τ
A
2
2
π
A
τ
0
(
t
−
τ
)
2
−
(
x
−
ξ
)
2
−
(
y
−
η
)
2
0
D
A
(
t
−
τ
)
·
δ
(
ξ
−
x
0
,
η
−
y
0
,
τ
−
t
0
)
d
ξ
d
η
ch
A
A
2
(
t
−
t
0
)
2
−
(
x
−
x
0
)
2
−
(
y
−
y
0
)
2
1
t
−
t
0
e
−
A
2
=
2
τ
0
2
π
A
τ
(
t
−
t
0
)
2
−
(
x
−
x
0
)
2
−
(
y
−
y
0
)
2
0
1
−
(
x
−
x
0
)
2
+(
y
−
y
0
)
2
A
2
ch
t
−
t
0
2
1
τ
0
2
t
−
t
0
2τ
0
(
t
−
t
0
)
e
−
=
1
,
(5.78)
a
2
2
π
(
t
−
t
0
)
−
(
x
−
x
0
)
2
+(
y
−
y
0
)
2
A
2
2
(
t
−
t
0
)
where
x
,
y
and
t
satisfy
2
2
(
x
−
x
0
)
+(
y
−
y
0
)
≤
A
(
t
−
t
0
)
so that
M
(
x
,
y
,
t
)
∈
Ω
P
0
,
the characteristic cone of
P
0
(
x
0
,
y
0
,
t
0
)
. Outside
Ω
P
0
,
u
(
x
,
y
,
t
)
≡
0. Therefore,
⎧
⎨
Eq. (5.78)
(=
0
)
,
M
(
x
,
y
,
t
)
∈
Ω
P
0
,
u
2
(
x
,
y
,
t
)=
=
∞
,
M
(
x
,
y
,
t
)
on the characteristic cone surface,
⎩
=
0
,
M
(
x
,
y
,
t
)
outside
Ω
P
0
.
(5.79)
5.6.2 Common Properties
1. By using ch0
=
1, we obtain from Eqs. (5.77) and (5.78)
lim
M
u
1
(
x
,
y
,
t
)=
∞
,
lim
M
u
2
(
x
,
y
,
t
)=
∞
.
→
P
0
→
P
0
This is physically-grounded because
δ
(
x
−
x
0
,
y
−
y
0
,
t
−
t
0
)
implies a very high
temperature at point
(
x
0
,
y
0
)
and time instant
t
0
.
T
−
1
and
u
2
(
Note here that
[
δ
(
x
−
x
0
,
y
−
y
0
,
t
−
t
0
)] =
Θ
x
,
y
,
t
)
is a higher-order
P
0
.
2. By Eq. (5.77), we have, for any point
infinity than
u
1
(
x
,
y
,
t
)
as
M
→
R
2
,
(
,
)
∈
x
y
lim
u
1
(
x
,
y
,
t
)=
0
.
t
→
+
∞
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