Environmental Engineering Reference
In-Depth Information
where
F
[
u
tt
]=
u
tt
(
ω
,
t
)
has been used. The solution of the above equation is
u
(
ω
,
t
)=
A
(
ω
)
cos
ω
at
+
B
(
ω
)
sin
ω
at
,
A
(
ω
)
and
B
(
ω
)
can be determined by initial conditions
u
(
ω
,
0
)=
0and
u
t
(
ω
,
0
)=
¯
ψ
(
ω
)
ω
ψ
(
ω
)
¯
(
ω
)=
(
ω
)=
as
A
0,
B
. Thus
a
¯
e
−
iω
at
ψ
(
ω
)
ω
¯
1
2
a
ψ
(
ω
)
i
ψ
(
ω
)
i
¯
e
iω
at
u
(
ω
,
t
)=
sin
ω
at
=
−
.
a
ω
ω
Using the integral property and the shifting property of Fourier transformations and
taking an inverse transformation leads to
x
+
at
1
2
a
u
(
x
,
t
)=
W
ψ
(
x
,
t
)=
ψ
(
ξ
)
d
ξ
.
x
−
at
The solution of PDS (2.60) thus follows from the solution structure theorem
t
=
∂
∂
u
t
W
ϕ
+
W
ψ
(
x
,
t
)+
W
f
τ
(
x
,
t
−
τ
)
d
τ
,
(2.61)
0
where
f
τ
=
f
(
x
,
τ
)
.Since
x
+
at
∂
W
ϕ
∂
2
a
∂
1
ξ
=
ϕ
(
x
+
at
)+
ϕ
(
x
−
at
)
=
ϕ
(
ξ
)
d
,
t
∂
t
2
x
−
at
the solution of
u
tt
a
2
u
xx
=
, −
∞
<
x
<
+
∞
,
0
<
t
,
(
,
)=
ϕ
(
)
,
(
,
)=
ψ
(
)
u
x
0
x
u
t
x
0
x
reads
x
+
at
)=
ϕ
(
x
+
at
)+
ϕ
(
x
−
at
)
1
2
a
u
(
x
,
t
+
ψ
(
ξ
)
d
ξ
.
(2.62)
2
x
−
at
This is called the
D'Alembert formula of one-dimensional wave equation
.
The solution of
u
tt
=
a
2
u
xx
+
f
(
x
,
t
)
, −
∞
<
x
<
+
∞
,
0
<
t
,
u
(
x
,
0
)=
0
,
u
t
(
x
,
0
)=
0
is
t
x
+
a
(
t
−
τ
)
1
2
a
u
(
x
,
t
)=
f
(
ξ
,
τ
)
d
ξ
d
τ
,
(2.63)
0
x
−
a
(
t
−
τ
)
which is called the
Kirchhoff formula of one-dimensional wave equation
.
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