Environmental Engineering Reference
In-Depth Information
The periodic condition in Eq. (2.40) and the bounded condition in Eq. (2.42) are
called the
natural boundary condition
, which is another kind of CDS.
The auxiliary problem defined by Eq. (2.42) is called an
eigenvalue problem of
the Bessel equations
, because by letting
x
kr
, the equation in (2.42) becomes
a Bessel equation of
n
-th order. The general solution of Eq. (2.42) is
=
R
n
(
r
)=
C
n
J
n
(
kr
)+
D
n
Y
n
(
kr
)
,
where
C
n
and
D
n
are arbitrary constants, and
J
n
and
Y
n
are the
n
-th order Bessel
functions of the first and the second kind, respectively. A discussion of Bessel func-
tions is available in Appendix A.
Applying the bounded condition in Eq. (2.42) and using lim
r
→
(
)=
∞
0
Y
n
kr
leads to
R
n
)
|
r
=
a
0
=
D
n
=
0. For the boundary condition of the first kind,
L
(
R
n
,
0 reduces to
R
n
(
0. Therefore we obtain the eigenvalues
and the eigenfunctions of problem (2.42) subject to the boundary condition of the
first kind
a
0
)=
0; to satisfy it we have
J
n
(
ka
0
)=
μ
(
n
m
a
0
2
k
mn
=
Eigenvalues
λ
m
=
,
(
m
=
1
,
2
, ···
)
(2.43)
k
mn
=
μ
(
n
m
a
0
,
Eigenfunctions
J
n
(
k
mn
r
)
,
μ
(
n
)
where
(
m
=
1
,
2
, ···
)
are the zero-points of
J
n
(
x
)
.
m
Note.
For
n
=
0,
J
0
(
0
)=
1sothat
x
=
0 is not a zero-point of
J
0
(
x
)
. Although
x
=
0 is a zero-point of
J
n
(
x
)
for
n
≥
1,
J
n
(
k
mn
r
)
is not an eigenfunction because
J
n
(
k
mn
r
)=
J
n
(
0
)=
0. When
k
=
0, Eq. (2.42) reduces to an Euler equation and
a trivial solution of
R
n
(
0. By Eq. (2.38), on the other hand, we also
arrive at a trivial solution of
v
when
k
r
)
when
k
=
0. Therefore
k
mn
=
=
0. The distribution
of zero-points of
J
n
(
is symmetric around the origin. We only account for their
positive zero-points; therefore
x
)
μ
(
n
m
are the positive zero-points of
J
n
(
x
)
in Eq. (2.43):
μ
(
n
)
1
<
μ
(
n
)
2
< ··· <
μ
(
n
)
< ···
.
m
With these eigenvalues
λ
m
, the solution of the equation for
T
(
t
)
yields
T
mn
(
t
)=
E
mn
cos
ω
mn
t
+
F
mn
sin
ω
mn
t
,
where
k
mn
a
,
E
mn
and
F
mn
are arbitrary constants. Since PDS (2.37) is linear,
by principle of superposition,
ω
mn
=
=
∑
(
)
(
)
Θ
(
θ
)
u
T
mn
t
R
n
r
n
∞
∑
=
n
=
0
,
m
=
1
(
a
mn
cos
ω
mn
t
+
b
mn
sin
ω
mn
t
)
J
n
(
k
mn
r
)
cos
n
θ
+(
c
mn
cos
ω
mn
t
+
d
mn
sin
ω
mn
t
)
J
n
(
k
mn
r
)
sin
n
θ
is also a solution of the wave equation. Here
a
mn
,
b
mn
,
c
mn
and
d
mn
are arbitrary
constants. In order that
u
satisfies the initial condition
u
(
r
,
θ
,
0
)=
0, we must choose
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