Environmental Engineering Reference
In-Depth Information
Fig. 5.7
Domain of influence
pendence never contains the point
M
0
. Therefore, the
Ω
M
0
is called the
domain of
influence
of the initial disturbance at point
M
0
.For
ψ
(
x
,
y
)=
δ
(
x
−
x
0
,
y
−
y
0
)
in
Eq. (5.67), in particular, we have
⎧
⎨
ch
A
(
At
)
2
−
(
x
−
x
0
)
2
−
(
y
−
y
0
)
2
1
t
A
e
−
2
τ
0
,
(
x
,
y
,
t
)
∈
Ω
M
0
,
2
π
u
(
x
,
y
,
t
)=
(
At
)
2
−
(
x
−
x
0
)
2
−
(
y
−
y
0
)
2
⎩
0
,
(
x
,
y
,
t
)
∈
Ω
.
M
0
5.5.3 Measuring
τ
0
by Characteristic Cones
Characteristic curves and cones play an important role in studying hyperbolic heat-
conduction equations. They have a close relation with the solutions of Cauchy prob-
lems and they reflect features of thermal wave propagation. In the one-dimensional
case, for example, the singularity of
ϕ
(
x
)
at
x
0
of type
ϕ
(
x
0
−
0
)
=
ϕ
(
x
0
+
0
)
will
ϕ
(
)
=
ϕ
(
at least lead to
x
0
−
0
x
0
+
0
)
on both sides of the characteristic curves
x
x
0
so that
u
tt
and
u
xx
do not exist on them (Eq. (5.30)
in Section 5.2). The characteristic curves at point
−
At
=
x
0
and
x
+
At
=
are in fact the discontinu-
ous representation of the solution. Therefore, a singularity of initial values at a point
will propagate along the characteristic curves. Solutions of two-dimensional Cauchy
problems also possess similar features. Based on these observations, we may design
a method of measuring
(
x
0
,
0
)
τ
0
as follows.
Step 1.
Choose a circle
D
O
ε
of center
O
and radius
ε
on a sufficiently large and thin
plane (Fig. 5.8). The
D
O
ε
crosses
Oy
−
axis
at
M
1
(
0
,
y
1
,
0
)
. Choose point
M
0
(
0
,
y
0
,
0
)
sufficiently far away from
D
O
ε
as the test point.
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