Environmental Engineering Reference
In-Depth Information
Note that
l
l
l
l
0
w
x
w
xt
d
x
=
w
x
d
w
t
=
w
x
w
t
|
−
w
t
w
xx
d
x
0
0
0
l
=
−
w
t
w
xx
d
x
.
0
l
l
0
w
t
w
tt
+
a
2
w
x
w
xt
d
x
0
w
tt
−
a
2
w
xx
w
t
d
x
d
E
d
t
=
Therefore,
=
=
0,
so
E
(
t
)=
E
(
0
)
.As
w
(
x
,
0
)=
0, we have
w
x
(
x
,
0
)=
0. Also,
w
t
(
x
,
0
)=
0, so
l
w
t
+
a
2
w
x
t
=
0
d
x
1
2
E
(
0
)=
=
0
.
0
l
0
w
t
+
a
2
w
x
d
x
1
2
Thus
E
0.
This and the fundamental lemma of variational method lead to
w
t
(
(
t
)=
=
x
,
t
)=
w
x
(
x
,
t
)=
0
so
w
(
x
,
t
)
is independent of
x
and
t
. Thus
w
(
x
,
t
)
≡
w
(
x
,
0
)
≡
0, which establishes
the uniqueness of the solution.
Remark 2.
The auxiliary function
E
(
t
)
is the total energy of the string of unit den-
is viewed as the string displacement in the field of vibration.
d
E
sity if
w
0is
thus a mathematical expression of the conservation of energy, a fundamental physi-
cal law.
(
x
,
t
)
d
t
=
2.4.3 Stability
To establish the stability of solution (2.20) with respect to initial conditions, we first
need to develop an important inequality for PDS (2.21).
Consider an auxiliary function
l
u
2
d
x
E
0
(
t
)=
,
0
where
u
(
x
,
t
)
is the solution of PDS (2.21). Thus
2
l
0
l
l
E
0
(
u
2
d
x
u
t
d
x
t
)=
uu
t
d
x
≤
+
≤
E
0
(
t
)+
2
E
(
t
)
,
0
0
l
0
u
t
+
a
2
u
x
d
x
.
1
2
(
)=
where the energy integral
E
t
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