Environmental Engineering Reference
In-Depth Information
t
e
−
By the function transformation
u
(
r
,
t
)=
v
(
r
,
t
)
2
τ
0
for eliminating terms involving
the first derivative
u
t
, PDS (5.84) reduces to
⎧
⎨
⎩
r
∂
A
2
1
r
∂
∂
v
1
2
c
2
v
v
tt
=
+
,
c
=
τ
0
,
r
∂
r
(5.85)
v
(
r
,
0
)=
0
,
v
t
(
r
,
0
)=
ψ
(
r
)
.
Applying a Hankel transformation to (5.85) and using
H
0
1
r
r
∂
∂
∂
v
2
v
=
−
ω
(
ω
,
t
)
r
∂
r
leads to an initial-value problem of the ordinary differential equation
v
tt
A
2
2
c
2
(
ω
,
)+(
−
)
(
ω
,
)=
,
t
ω
v
t
0
v
(
ω
,
0
)=
0
,
v
t
(
ω
,
0
)=
ψ
(
ω
)
,
¯
where
¯
ψ
(
ω
)=
[
ψ
(
)]
H
0
r
. Its solution can be readily obtained as
sin
√
A
2
c
2
t
ψ
(
ω
)
¯
ω
2
−
√
A
2
v
(
ω
,
t
)=
.
2
c
2
ω
−
Its inverse Hankel transformation will yield the solution of PDS (5.85), and conse-
quently, the solution of PDS (5.84)
sin
√
A
2
c
2
t
τ
0
+
∞
0
ψ
(
ω
)
¯
ω
2
−
t
e
−
(
,
)=
√
A
2
u
r
t
W
ψ
(
r
,
t
)=
2
ω
J
0
(
ω
r
)
d
ω
.
2
c
2
ω
−
Thus the solution of PDS (5.83) is, by the solution structure theorem,
1
W
ϕ
(
t
τ
0
+
∂
u
(
r
,
t
)=
r
,
t
)+
W
ψ
(
r
,
t
)+
W
f
τ
(
r
,
t
−
τ
)
d
τ
,
∂
t
0
where
f
τ
=
f
(
r
,
τ
)
.
5.7.2 Spherical Bessel Transformation for Spherically-Symmetric
Cauchy Problems
Spherical Bessel Transformation of Order Zero
A sphericall
y-symmetri
c function
f
(
x
,
y
,
z
)
can be written as
f
(
x
,
y
,
z
)=
f
(
r
)=
f
(
r
)
x
2
with
r
=
+
y
2
+
z
2
. Its triple Fourier transformation is also spherically sym-
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