Environmental Engineering Reference
In-Depth Information
1
t
A e
+
2
τ 0
4
π
A ¯
ψ (
At
)+
4
π
At ¯
ψ t (
At
)
4
π
bI 1 b (
r 2
At
At
)
2
1
2
M )
d r
(
+
ψ (
d S
At
)
2
r 2
At
S M
r
b 2 I 1 b
r 2
(
At
)
2
1
2
M )
+
b
ψ (
d S
| r = At ·
A
2
r 2
(
At
)
S r
b 2 I 1 b
r 2
(
)
2
At
M )
b (
ψ (
d S
| r = At · (
A
)
.
At
)
2
r 2
S r
From the point view of continuation,
M )
M )
ψ (
d S
| r = At =
ψ (
d S
| r = At .
S r
S r
Also
M )
lim
t + 0
ψ (
d S
=
0
.
S At
Therefore, the u
(
M
,
t
)
in Eq. (5.120) also satisfies the initial condition u t
(
M
,
0
)=
ψ (
M
)
.
Comparison with Three-Dimensional Wave Equations
The counterpart of PDS (5.119) in wave equations reads
u tt =
A 2
R 3
Δ
u
,
× (
0
, + ) ,
(5.122)
u
(
M
,
0
)=
0
,
u t (
M
,
0
)= ψ (
M
) .
Its solution is, by the Poisson formula
1
M )
u
(
M
,
t
)=
ψ (
d S
,
(5.123)
4
π
A 2 t
S At
which is the first term of the right-hand side of Eq. (5.120) without the decaying
factor e
t
2
τ 0 . Since the solution is a surface integral of the first kind, there exist a
 
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