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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