Environmental Engineering Reference
In-Depth Information
For classical heat-conduction equations, the counterpart of Eq. (5.106) is
t
+
∞
e
−
(
r
−
ρ
)
2
1
2
a
√
π
1
4
a
2
u
(
r
,
t
)=
√
t
d
τ
ρδ
(
ρ
−
r
0
,
τ
−
t
0
)
d
ρ
(
t
−
τ
)
r
−
τ
0
−
∞
r
e
−
(
r
−
r
0
)
2
r
0
2
a
π
(
4
a
2
=
.
(5.108)
(
t
−
t
0
)
t
−
t
0
)
It shows that: (1) lim
r
→
0
u
(
r
,
t
)=
∞
and lim
r
u
(
r
,
0
)=
0,thesameasthosein
→
∞
Eq. (5.107), but (2) lim
r
u
(
r
,
t
)=
∞
. Hence Eq. (5.107) appears more reasonable
→
r
0
t
→
t
0
than Eq. (5.108) because of its finite limit of
u
(
r
,
t
)
as
t
→
t
0
and
r
→
r
0
.
5.8 Methods of Fourier Transformation and Spherical Means for
Three-Dimensional Cauchy Problems
In this section we apply the Fourier transformation and the method of spherical
means to solve three-dimensional Cauchy problems. We also make a comparison
with three-dimensional wave equations to demonstrate some features of thermal
waves.
5.8.1 An Integral Formula of Bessel Function
Let
π
1
2
e
i
r
cos ϕ cos θ
sin
I
(
r
,
ϕ
)=
J
0
(
r
sin
ϕ
sin
θ
)
θ
d
θ
,
(5.109)
0
where
(
r
,
θ
,
ϕ
)
are the coordinates of a spherical coordinate system. We attempt to
prove
sin
r
r
I
(
r
,
ϕ
)=
.
Note that
J
0
(
x
)=
−
J
1
(
x
)
.Wehave
d
I
d
ϕ
=
I
1
+
I
2
,
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