Environmental Engineering Reference
In-Depth Information
radius, an integration in a spherical coordinate system yields
+
∞
2π
π
1
2sin
ω
at
e
iω
r
cos θ
ω
2
sin
I
=
d
ω
d
ϕ
θ
d
θ
2
4
π
ω
0
0
0
+
∞
1
r
=
2sin
ω
at
sin
ω
r
d
ω
π
0
+
∞
1
r
=
[
cos
ω
(
r
−
at
)
−
cos
ω
(
r
+
at
)]
d
ω
,
π
0
1
r
[
δ
(
=
r
+
at
)
−
δ
(
r
+
at
)]
,
+
∞
+
∞
1
2
1
π
e
iω
x
d
where
δ
(
x
)=
ω
=
cos
ω
x
d
ω
is the integral representation of
π
−
∞
0
the
-function (see Appendix B).
Applying a Fourier transformation to
u
δ
(
r
,
t
)
, the solution of PDS (2.77) for the
case
ϕ
=
f
=
0 yields
1
e
iω
·
r
d
(
ω
,
u
(
r
,
t
)=
u
t
)
ω
1
d
ω
2
d
ω
3
,
(2.79)
(
2
π
)
3
Ω
where
u
=
F
[
u
]
. Substituting it into PDS (2.77) with
ϕ
=
f
=
0 leads to
2
u
u
tt
(
ω
,
t
)+(
ω
a
)
(
ω
,
t
)=
0
,
u
(
ω
,
0
)=
0
,
u
t
(
ω
,
0
)=
ψ
(
ω
)
,
¯
where ¯
ψ
(
ω
)=
F
[
ψ
(
M
)]
. Its solution reads
e
iω
at
e
−
iω
at
ψ
(
ω
)
ω
¯
ψ
(
ω
)
2
a
i
¯
u
=
sin
ω
at
=
−
,
(2.80)
a
ω
Ω
:
x
<
+
∞
,−
∞
<
y
<
+
∞
,−
∞
<
z
<
+
∞
;
r
=
x
i
y
j
z
k
;
x
,
y
where
−
∞
<
+
+
and
z
are integral variables; and
e
−
iω
·
r
d
x
d
y
d
z
.
r
)
ψ
(
ω
)=
¯
ψ
(
(2.81)
Ω
Equations (2.78)-(2.81) lead to
at
d
x
d
y
d
z
.
δ
r
r
−
at
−
δ
r
r
+
r
)
1
ψ
(
u
(
r
,
t
)=
−
−
r
|
4
π
a
|
r
−
Ω
(2.82)
Ω
can be regarded as a sphere of center
r
and infinite radius; and, for
Note that
a
>
0and
t
>
0,
δ
(
|
r
−
r
|
+
at
)
≡
0. Also
=
r
∈
S
at
,
δ
r
r
−
0
,
at
=
−
r
∈
S
at
,
=
0
,
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