Environmental Engineering Reference
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and thus comes from the source term of
. Take the vibration of a string
with two fixed ends as the example. The source term of
δ
(
x
−
ξ
,
t
−
τ
)
δ
(
x
−
ξ
,
t
−
τ
)
stands for a
localized force action at spatial point
ξ
and time instant
τ
.The
G
(
x
,
ξ
,
t
−
τ
)
is thus
the displacement distribution due to such a force action.
Once the Green function is available, the solution of PDS (2.5) for the case of
ϕ
=
ψ
=
0 can be expressed explicitly as a function of
f
(
x
,
t
)
, simply by performing
the integration in Eq. (2.11).
2.2.2 Boundary Condition of the Second Kind
For the boundary condition
u
x
(
0
,
t
)=
u
x
(
l
,
t
)=
0, the solution of PDS (2.6) has the
form
∞
k
=
0
T
k
(
t
)
cos
k
π
x
u
(
x
,
t
)=
,
l
where
T
k
(
t
)
is the undetermined function, and
k
=
0
,
1
,
2
, ···
. By following the same
procedure in Section 2.2.1, we can obtain
W
ψ
(
x
,
t
)
, the solution of PDS (2.6) under
the boundary condition of the second kind,
⎧
⎨
+
∞
k
=
1
b
k
sin
k
π
at
cos
k
π
x
l
u
=
W
ψ
(
x
,
t
)=
b
0
t
+
,
l
(2.14)
⎩
l
0
ψ
(
a
0
ψ
(
1
2
cos
k
π
x
l
b
0
=
x
)
d
x
,
b
k
=
x
)
d
x
.
k
π
For any nontrivial
b
0
,
b
0
t
. There appears, therefore, an unbounded
term
b
0
t
in the solution. Take the string vibration again as the example. The bound-
ary condition
u
x
(
→
∞
as
t
→
∞
0 describes two free ends that can move freely in
the vertical direction. This free movement leads, in combination with the effect of
the initial velocity
0
,
t
)=
u
x
(
l
,
t
)=
of the string, to a uniformmoment of velocity
b
0
in addition
to the vibration due to the initial velocity. An unit analysis yields
ψ
(
x
)
1
l
1
a
[
b
0
]=
[
ψ
d
x
]=
L
/
T
,
[
b
k
]=
[
ψ
d
x
]=
L
.
Therefore,
b
0
has the units of a velocity and
b
k
(
k
=
1
,
2
, ···
)
represent the displace-
ment.
By the solution structure theorem, the solution of PDS (2.5) with
u
x
(
0
,
t
)=
u
x
(
l
,
t
)=
0is
t
=
∂
∂
u
t
W
ϕ
+
W
ψ
(
x
,
t
)+
W
f
τ
(
x
,
t
−
τ
)
d
τ
,
0
where
f
τ
=
f
(
x
,
τ
)
.
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