Environmental Engineering Reference
In-Depth Information
The integrand in Eq. (5.38) contains a factor that is exponentially decreasing with
respect to
ξ
. The demand for the smoothness of
ϕ
(
x
)
is thus very weak. Eq. (5.38)
can also be written as
4
a
2
t
+
∞
−
∞
x
2
2
e
−
ξ
−
2
x
ξ
1
2
a
√
π
t
e
−
u
(
x
,
t
)=
4
a
2
t
ϕ
(
ξ
)
d
ξ
.
Therefore, the temperature decreases as time increases, regardless of the value of
point
x
, and lim
|
u
(
x
,
t
)=
0 for all time instants.
x
|→
+
∞
. The superimposed forward and
backward waves and the integral limits in Eq. (5.39) are the signature of so-called
thermal waves. The thermal waves in Eq. (5.39) differs, however, from the waves
in Eq. (5.37) mainly on the appearance of an exponentially decaying factor e
−
Equation (5.39) has two terms involving
ϕ
(
x
)
t
2τ
0
with respect to
t
. This is quite similar to Eq. (5.38) where a decaying factor 1
√
t
occurs.
Note that e
−
o
1
√
t
as
t
t
in
Eq. (5.39) decays with respect to
t
more quickly than in Eq. (5.38) . The smaller
τ
0
is, the faster the decay.
One main difference between solutions in Eqs. (5.38) and (5.39) is the propaga-
tion of singularities of
2
τ
0
=
→
+
∞
for all positive
τ
0
.The
u
(
x
,
t
)
ϕ
(
x
)
and
ψ
(
x
)
at
x
0
along the characteristic curves
x
−
At
=
x
0
x
0
in Eq. (5.39), which is a typical wave property. While the integral
in Eq. (5.39) shares the same format as that in Eq. (5.37), the integrand contains
both
and
x
+
At
=
as well as the modified Bessel function of the first kind. Hence
the shape of thermal waves is intrinsically more complicated. However, the integral
limit is still determined by the characteristic curves
ϕ
(
x
)
and
ψ
(
x
)
ξ
∈
[
−
,
+
]
x
At
x
At
,whichisa
property of the traveling waves in Eq. (5.37).
Another striking difference between solutions in Eqs. (5.38) and (5.39) becomes
visible by considering the solutions due to
ϕ
(
x
)
such that
>
0
,
x
∈
[
x
1
,
x
2
]
,
ϕ
(
x
)
=
0
,
x
∈
[
x
1
,
x
2
]
.
By Eq. (5.38),
x
2
u
(
x
,
t
)=
V
(
x
,
ξ
,
t
)
ϕ
(
ξ
)
d
ξ
>
0
,
x
∈
(
−
∞
,
+
∞
)
.
x
1
Therefore, the effect of
ϕ
(
x
)
in
[
x
1
,
x
2
]
can be sensed instantly at all points including
those towards
x
→−
∞
and
x
→
+
∞
. This is not the case, however, for the
u
(
x
,
t
)
in
Eq. (5.39). Note that the domain of dependence of
x
0
is
[
x
0
−
At
,
x
0
+
At
]
(Fig. 5.4).
For a sufficiently short time
t
1
,
[
x
0
−
At
1
,
x
0
+
At
1
]
is outside of
[
x
1
,
x
2
]
,wherethe
disturbance exists, so that
u
(
x
0
,
t
1
)=
0. For a sufficiently long time
t
2
,however,
[
x
0
−
At
2
,
0 in general.
Since the propagation of any disturbance always requires some time, hyperbolic
heat-conduction equation is thus a better representation of real heat conduction.
x
0
+
At
2
]
has some part in common with
[
x
1
,
x
2
]
so that
u
(
x
0
,
t
2
)
=
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