Environmental Engineering Reference
In-Depth Information
Finally, we have
W
ψ
(
x
,
t
)
⎧
⎨
∞
m
=
1
b
m
cos
μ
m
x
sin
μ
m
at
l
u
=
W
ψ
(
x
,
t
)=
,
l
l
0
ψ
(
⎩
l
μ
m
M
m
a
cos
μ
m
x
l
b
m
=
x
)
d
x
.
The solution of the original PDS follows from the solution structure theorem
t
=
∂
∂
u
t
W
ϕ
+
W
ψ
(
x
,
t
)+
W
f
τ
(
x
,
t
−
τ
)
d
τ
.
0
Remark.
The various mixed problems of one-dimensional wave equations can be
solved very efficiently and concisely by using Table 2.1, the solution structure the-
orem and the structure function
W
.
2.3.3 Important Properties of Eigenvalue Problems (2.19)
The method of separation of variables relies on the properties of eigenvalue prob-
lems (2.19). We list four important properties here and refer to Appendix D for
a discussion of the theory of eigenvalue problems.
1. All eigenvalues are non-negative and real-valued for all combinations of bound-
ary conditions. A vanished eigenvalue appears only when
X
(
X
(
0.
2. Eigenvalues form a sequence of numbers which is monotonically increasing to-
wards infinity, whatever the boundary conditions, i.e.
0
)=
l
)=
≤
λ
≤
λ
≤···≤
λ
k
≤··· ,
k
→
∞
λ
k
=
∞
.
0
lim
1
2
3. All eigenfunction sets
{
X
k
(
x
)
}
are orthogonal in
[
0
,
l
]
,i.e.
l
(
X
k
,
X
m
)=
X
k
(
x
)
X
m
(
x
)
d
x
=
0
,
k
=
m
.
0
L
2
4. Any function
f
(
x
)
∈
[
a
,
b
]
can be expanded into a generalized Fourier series
by an eigenfunction set, i.e.
⎨
∞
k
=
1
c
k
X
k
(
x
)
,
f
(
x
)=
⎩
b
b
1
M
k
2
L
2
X
k
(
c
k
=
X
k
(
x
)
f
(
x
)
d
x
,
M
k
=
X
k
]
=
x
)
d
x
,
[
a
,
b
a
a
where
√
M
k
is called the normal of
{
X
k
(
x
)
}
and serves as the measure of function
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