Environmental Engineering Reference
In-Depth Information
Hankel Transformation
An axially-
symme
tric function
f
(
x
,
y
)
can be written as
f
(
x
,
y
)=
f
(
r
)=
f
(
r
)
x
2
with
r
=
+
y
2
. Its do
uble Four
ier transformation is also axially symmetric,
f
f
2
2
(
ω
)=
(
ω
)
with
ω
=
ω
1
+
ω
2
,0
≤
ω
<
+
∞
. Therefore, the double Fourier
(
,
)
transformation of an axially symmetric function
f
x
y
and its inverse transforma-
tion have a special form
+
∞
r
d
r
2π
0
1
2
1
2
f
e
−
iω
·
r
d
x
d
y
e
−
iω
r
cos θ
d
(
ω
)=
f
(
x
,
y
)
=
f
(
r
)
θ
π
π
0
R
2
+
∞
=
rf
(
r
)
J
0
(
ω
r
)
d
r
,
(5.81)
0
+
∞
1
2
f
e
iω
·
r
d
f
f
(
r
)=
(
ω
1
,
ω
2
)
ω
1
d
ω
2
=
ω
(
ω
)
J
0
(
ω
r
)
d
ω
,
(5.82)
π
0
R
2
in which we have applied Eq. (5.80). Denote Eqs. (5.81) and (5.82) by
f
H
−
1
0
f
(
ω
)=
H
0
[
f
(
r
)]
,
f
(
r
)=
[
(
ω
)]
,
respectively. They are called the
Hankel transformation
and the
inverse Hankel
transformation
. As a special case of Fourier transformation, the Hankel transfor-
mation shares some similar properties with the Fourier transformation.
Axially Symmetric Cauchy Problems
The axially symmetric Laplace operator reads, in a polar coordinate system
r
∂
∂
2
2
Δ
=
∂
x
2
+
∂
1
r
∂
∂
y
2
=
.
∂
∂
r
r
We now consider the axially symmetric Cauchy problem
⎧
⎨
r
∂
u
t
τ
A
2
1
r
∂
∂
u
0
+
u
tt
=
+
f
(
r
,
t
)
,
0
<
r
<
+
∞
,
0
<
t
,
r
∂
r
(5.83)
⎩
(
,
)=
ϕ
(
)
,
(
,
)=
ψ
(
)
.
u
r
0
r
u
t
r
0
r
By the solution structure theorem, we can focus on seeking the solution of
⎧
⎨
r
∂
u
t
τ
0
+
A
2
1
r
∂
∂
u
u
tt
=
,
0
<
r
<
+
∞
,
0
<
t
,
r
∂
r
(5.84)
⎩
u
(
r
,
0
)=
0
,
u
t
(
r
,
0
)=
ψ
(
r
)
.
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