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
0
1
2
1
2
f
e · r d x d y
e 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 · 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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