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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