Environmental Engineering Reference
In-Depth Information
where the former is called the
Fourier sine transformation
and the latter is called
the
inverse Fourier sine transformation
. Similarly, we also have the
Fourier cosine
transformation
and the
inverse Fourier cosine transformation
⎧
⎨
π
f
(
k
)=
f
(
x
)
cos
kx
d
x
,
(7.34a)
0
⎩
f
f
∞
k
=
1
(
0
)
2
π
f
(
)=
+
(
)
.
x
k
cos
kx
(7.34b)
π
We can apply the Fourier sine/cosine transformation to solve some boundary-value
problems of potential equations. If the boundary conditions are all of the first kind
at two ends along one spatial direction, we should use the Fourier sine transforma-
tion. When the boundary conditions are all of the second kind at two ends along
one spatial direction, the Fourier cosine transformation becomes appropriate. After
a Fourier sine/cosine transformation, some terms involving second derivatives dis-
appear and the problem reduces to a problem of ordinary differential equations. To
illustrate this, consider
u
=
u
(
x
,
t
)
. The Fourier sine transformation of
u
xx
is
x
sin
kx
π
k
π
0
π
0
−
2
u
∂
=
∂
u
∂
u
x
2
sin
kx
d
x
x
cos
kx
d
x
∂
∂
∂
0
k
2
π
0
)
|
0
−
=
−
(
ku
cos
kx
u
sin
kx
d
x
k
u
k
u
k
2
u
=
(
0
,
t
)
−
(
−
1
)
(
π
,
t
)
−
(
k
,
t
)
,
π
where
u
(
k
,
t
)=
u
(
x
,
t
)
sin
kx
d
x
. The Fourier cosine transformation of
u
xx
is
0
π
2
u
∂
k
u
x
k
2
u
=(
−
)
(
π
,
)
−
(
,
)
−
(
,
)
,
x
2
cos
kx
d
x
1
t
u
x
0
t
k
t
∂
0
π
where
u
(
k
,
t
)=
u
(
x
,
t
)
cos
kx
d
x
.
0
Note that the Fourier sine/cosine transformation of
u
xx
contains boundary values
of the first kind and the second kind.
Example 5
. Find the solution of
⎧
⎨
u
xx
+
u
yy
=
0
,<
x
,
y
<
π
,
u
(
0
,
y
)=
u
(
π
,
y
)=
0
,
(7.35)
⎩
u
(
x
,
0
)=
0
,
u
(
x
,
π
)=
u
0
=
0
.
It governs the steady temperature distribution in a square plate of side length
, with
three sides having a temperature of zero and the fourth side a temperature of
u
0
.
π
Search WWH ::
Custom Search