Environmental Engineering Reference
In-Depth Information
Fig. 5.6
Domain of dependence
As for the effect of the nonhomogeneous source term
f
(
x
,
y
,
t
)
,the
u
(
x
0
,
y
0
,
t
0
)
in
Eq. (5.70) depends
only on
f
(
x
,
y
,
t
)
in a cone
Ω
P
0
of top point
P
0
(
x
0
,
y
0
,
t
0
)
:
2
2
(
x
−
x
0
)
+(
y
−
y
0
)
≤
A
(
t
0
−
t
)
,0
≤
t
≤
t
0
.The
Ω
P
0
is thus called the
domain of
dependence
of point
P
0
on the source term. The
Ω
P
0
is also called the
characteristic
2
2
A
2
2
cone
of passing point
P
0
. The surface
(
x
−
x
0
)
+(
y
−
y
0
)
=
(
t
−
t
0
)
is called
the
characteristic cone surface
. For any point
P
1
(
x
1
,
y
1
,
t
1
)
∈
Ω
P
0
, its domain of
dependence on the initial value
D
M
1
At
1
is always in
D
M
0
At
0
,i.e.
D
M
1
D
M
0
At
1
⊂
At
0
. Hence the
initial values in
D
M
0
At
0
completely determine the solution due to the initial values at all
points in
Ω
P
0
.
For the one-dimensional case, the
D
M
0
At
0
reduces to the region
[
x
0
−
At
0
,
x
0
+
At
0
]
.
The
Ω
P
0
becomes a triangle region formed by
t
=
0 and characteristic curves
x
C
(constant) are very impor-
tant in studying one-dimensional hyperbolic heat-conduction equations. Similarly,
the characteristic surface plays an important role in studying two-dimensional hy-
perbolic heat-conduction equations.
−
x
0
=
±
A
(
t
−
t
0
)
. Characteristic curves
x
±
At
=
5.5.2 Domain of Influence
The structure of Eqs. (5.67) and (5.69) in Section 5.4 is similar to the Poisson for-
mula of two-dimensional wave equations. The propagation of thermal waves shares
those features of mechanical waves including the domain of influence of initial dis-
turbances discussed in Section 2.8.4. Here the initial v
alue of point
M
0
(
,
,
)
x
0
y
0
0
:
(
−
)
2
+(
−
)
2
≤
can affect all points in an infinite cone region
Ω
x
x
0
y
y
0
At
,
M
0
t
>
0 (Fig. 5.7). Point
M
0
is always in the domain of dependence of all points
P
1
(
x
1
,
y
1
,
t
1
)
∈
Ω
M
0
. For any point
P
2
(
x
2
,
y
2
,
t
2
)
outside
Ω
M
0
, its domain of de-
Search WWH ::
Custom Search