Environmental Engineering Reference
In-Depth Information
Meaning of Riemann Function
Consider a source term
f
(
x
,
t
)=
δ
(
x
−
x
0
,
t
−
t
0
)
in Eq. (5.32). We have
u
(
x
,
t
)=
I
0
b
A
2
2
t
x
+
At
1
2
A
t
−
τ
2
e
−
2
τ
0
d
τ
(
t
−
τ
)
−
(
x
−
ξ
)
δ
(
ξ
−
x
0
,
τ
−
t
0
)
d
ξ
τ
0
0
x
−
At
⎛
2
⎞
a
2
τ
0
(
1
2
a
√
τ
0
I
0
1
2
a
√
τ
0
t
−
τ
2
⎠
e
−
2
⎝
=
t
−
t
0
)
−
(
x
−
x
0
)
τ
0
.
(5.34)
Therefore, the temperature distribution due to a unit impulsive source
δ
(
x
−
x
0
,
t
−
t
0
at
x
0
and
t
0
can be represented by using the modified Bessel function of order
zero of the first kind. This is also the meaning of the Riemann function.
Note that
I
)
k
ρ
1and
a
2
(
0
)=
=
c
(
k
-thermal conductivity,
ρ
-density,
c
-specific
heat). By Eq. (5.34), we obtain
ρ
1
2
c
u
(
x
0
,
t
0
)=
lim
x
u
(
x
,
t
)=
τ
0
.
(5.35)
k
→
x
0
t
→
t
0
is proportional to
√
ρ
Therefore,
the
temperature
u
(
x
0
,
t
0
)
c
and inversely propor-
tional to
√
k
τ
0
.
L
LT
.
T
−
1
and
Remark 4.
In Eq. (5.34),
[
f
(
x
,
t
)] =
Θ
[
δ
(
ξ
−
x
0
,
τ
−
t
0
)] =
5.2.2 Method of Laplace Transformation
PDS (5.18) can easily be solved by the method of Laplace transformation due to
the absence of first derivative terms in its equation. The readers are referred to Ap-
pendix B.2 for a discussion of Laplace transformations. By taking a Laplace trans-
formation of (5.18), we obtain
s
2
c
2
U
s
2
U
c
2
U
U
ξξ
−
+
ψ
+
=
0 r
U
ξξ
−
−
=
−
ψ
,
(5.36)
where
U
(
ξ
,
s
)=
L
[
u
]
. The general solution of the associated homogeneous equation
is
c
1
e
ξ
√
s
2
−
c
2
c
2
e
−
ξ
√
s
2
−
c
2
U
(
ξ
,
s
)=
+
,
Search WWH ::
Custom Search