Environmental Engineering Reference
In-Depth Information
The latter is called the
eigenvalue problem of Bessel equations
. Its eigenvalues
λ
=
k
2
are boundary-condition dependent, nonzero and real valued, and depend
on the zero-points of Bessel functions (Sect. 2.5). Its eigenfunctions are Bessel
functions and form an orthogonal set in
[
0
,
a
0
]
with respect to the weight function
ρ
(
r
. The normal square of the eigenfunction set is also available in Sect. 2.5
and is listed in Table 4.1.
While the form of eigenfunctions is boundary-condition independent,
r
)=
μ
(
n
)
m
depends on boundary conditions. The general solution of Eq. (4.49) reads, with
μ
(
n
m
a
0
2
k
mn
=
λ
m
=
,
t
e
−
T
mn
(
t
)=
2
τ
0
(
a
mn
cos
γ
mn
t
+
b
mn
sin
γ
mn
t
)
,
4
1
2
where
a
mn
and
b
mn
are constants, and
γ
=
τ
0
(
k
mn
a
)
−
1.
mn
2
τ
0
Fourier Method of Expansion for PDS (4.48)
Consider the solution of PDS (4.48) at
Φ
=
F
=
0,
a
(
1
)
γ
mn
t
J
n
(
+
∞
∑
t
e
−
b
(
1
)
u
=
2
τ
0
mn
cos
γ
mn
t
+
mn
sin
k
mn
r
)
cos
n
θ
m
=
1
,
n
=
0
a
(
2
)
γ
mn
t
J
n
(
b
(
2
)
+
mn
cos
γ
mn
t
+
mn
sin
k
mn
r
)
sin
n
θ
.
0 yields
a
(
1
)
a
(
2
)
Applying the initial condition
u
(
r
,
θ
,
0
)=
=
=
0. We can also
mn
mn
determine
b
(
1
)
mn
and
b
(
2
)
mn
by satisfying the initial condition
u
t
(
r
,
θ
,
0
)=
Ψ
(
r
,
θ
)
.
Finally,
⎧
⎨
b
(
1
)
J
n
+
∞
∑
t
2τ
0
e
−
b
(
2
)
u
=
W
Ψ
(
r
,
θ
,
t
)=
mn
cos
n
θ
+
mn
sin
n
θ
(
k
mn
r
)
sin
γ
mn
t
,
m
=
1
,
n
=
0
π
a
0
1
b
(
1
)
m
0
=
Ψ
(
r
,
θ
)
J
0
(
k
m
0
r
)
r
d
r
d
θ
,
2
πγ
m
0
M
m
0
−
π
0
π
a
0
⎩
1
πγ
mn
M
mn
b
(
1
)
=
Ψ
(
r
,
θ
)
J
n
(
k
mn
r
)
cos
n
θ
·
r
d
r
d
θ
,
mn
−
π
0
π
a
0
1
πγ
mn
M
mn
b
(
2
)
=
Ψ
(
r
,
θ
)
J
n
(
k
mn
r
)
sin
n
θ
·
r
d
r
d
θ
.
mn
−
π
0
(4.51)
The solution of PDS (4.48) is, thus, by the solution structure theorem
1
W
Φ
(
t
τ
0
+
∂
u
=
r
,
θ
,
t
)+
W
Ψ
(
r
,
θ
,
t
)+
W
F
τ
(
r
,
θ
,
t
−
τ
)
d
τ
,
∂
t
0
,
θ
,
τ
)
τ
0
.
where
F
τ
=
F
(
r
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