Environmental Engineering Reference
In-Depth Information
Here,
G
is called the
Green function
of a one-dimensional wave equation
subjected to the boundary condition of the first kind, and is defined by
(
x
,
ξ
,
t
−
τ
)
+
∞
k
=
1
2
a
sin
k
πξ
l
sin
k
π
x
sin
k
π
a
(
t
−
τ
)
G
(
x
,
ξ
,
t
−
τ
)=
.
(2.12)
k
π
l
l
For the case
f
=
0,
=
∂
W
ϕ
∂
u
t
+
W
ψ
(
x
,
t
)
a
k
cos
k
sin
k
∞
k
=
1
π
at
l
+
b
k
sin
k
π
at
π
x
=
l
l
k
=
1
a
k
sin
k
π
(
x
−
at
)
+
∞
1
2
sin
k
π
(
x
+
at
)
=
+
l
l
k
=
1
b
k
cos
k
π
(
x
−
at
)
+
∞
1
2
cos
k
π
(
x
+
at
)
+
−
l
l
x
+
at
=
ϕ
(
x
−
at
)+
ϕ
(
x
+
at
)
1
2
a
+
ψ
(
ξ
)
d
ξ
,
(2.13)
2
x
−
at
which depends on the initial values
explicitly and is called the
D'Alembert
formula
of wave equations. The D'Alembert formula shows that the solution
u
con-
sists of two parts: the contribution of
ϕ
and
ψ
ϕ
and of
ψ
. Note that both
ϕ
(
x
)
and
ψ
(
x
)
are defined in
[
0
,
l
]
. For a sufficiently large
t
,however,
x
±
at
would be outside of
[
should be
made, similarly to in Fourier series. In arriving at Eq. (2.13), the initial conditions
are specified at
t
0
,
l
]
. In applying Eq. (2.13), therefore, an odd continuation of
ϕ
and
ψ
=
0. If they are specified at
t
=
τ
Eq. (2.13) is still valid simply by
−
τ
replacing
t
by
t
.
Remark 1.
The solution structure theorem is valid for PDS with homogeneous
boundary conditions. Otherwise, the homogenization of boundary conditions should
first be made by some appropriate function transformations. In the homogenization,
the nonhomogeneous term (the source term) and the initial conditions are also var-
ied. The function transformation is normally not unique.
For any function
w
(
x
,
t
)
with
w
(
0
,
t
)=
f
1
(
t
)
and
w
(
l
,
t
)=
f
2
(
t
)
such as
w
(
x
,
t
)=
+
l
f
1
(
t
)
[
f
2
(
t
)
−
f
1
(
t
)]
, a function transformation of
u
(
x
,
t
)=
v
(
x
,
t
)+
w
(
x
,
t
)
always
reduces
⎧
⎨
a
2
u
xx
,
u
tt
=
(
0
,
l
)
×
(
0
,
+
∞
)
,
u
(
0
,
t
)=
f
1
(
t
)
,
u
(
l
,
t
)=
f
2
(
t
)
,
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
0
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