Environmental Engineering Reference
In-Depth Information
and D M 0
At 0
Fig. 5.8 D 0
ε
Step 2. Apply a constant initial temperature K on D O
ε
and start to record time. Math-
ematically, we consider
u t
τ 0 +
A 2
R 2
u tt =
Δ
u
,
× (
0
+ ) ,
K
D O
, (
,
)
ε ,
x
y
u
(
x
,
y
,
0
)= ϕ (
x
,
y
)=
u t (
x
,
y
,
0
)=
0
.
D O
0
, (
x
,
y
)
ε ,
Its solution reads
1
1
2
0 +
t
e
u
(
x
,
y
,
t
)=
2
τ 0
A (
t )
K
( ϕ )
d
ξ
d
η .
2
π
τ
D At
Step 3. Observe the variation of temperature at point M 0 and record the time instant
t 0 when there is a reading of temperature at point M 0 ( u
(
0
,
y 0 ,
t 0 ) =
0, no matter how
small).
) t 0 by using known y 0 , y 1 and t 0 Th is
comes from the fact that the characteristic cone surface
=(
Step 4. Calculate the wave speed A
y 0
y 1
(
x
x 0
)
2
+(
y
y 0
)
2
=
A
(
t 0
must cross the y -axis at point M 1 (Fig. 5.8).
Once A is available, we can obtain
t
)
a 2 A 2 .
τ 0 by the definition of A as
τ 0 =
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