Environmental Engineering Reference
In-Depth Information
L
u
3
r
=
R
=
t
L
W
F
τ
,
∂
r
=
R
,
∂
u
3
∂
W
F
τ
∂
τ
=
,
d
0
r
r
0
u
3
(
r
,
θ
,
0
)=
0
,
t
=
0
=
W
F
τ
|
τ
=
t
t
=
0
=
t
t
∂
∂
)=
∂
∂
∂
W
F
τ
∂
t
u
3
(
r
,
θ
,
0
W
F
τ
d
τ
d
τ
+
0
.
t
t
0
0
t
Thus
u
3
=
W
F
τ
(
r
,
θ
,
t
−
τ
)
d
τ
is the solution for the case of
Φ
=
Ψ
=
0.
0
3. Since PDS (2.51) is linear, the principle of superposition is valid, so
t
=
∂
∂
u
t
W
Φ
+
W
Ψ
(
r
,
θ
,
t
)+
W
F
τ
(
r
,
θ
,
t
−
τ
)
d
τ
0
is the solution of PDS (2.51).
Therefore, the solution structure theorem is also valid in a polar coordinate system.
2.6 Three-Dimensional Mixed Problems
The Fourier method and the method of separation of variables work for mixed prob-
lems only in some regular domains. In this section, we apply them to solve mixed
problems in cuboid and spherical domains.
2.6.1 Cuboid Domain
Consider
⎧
⎨
a
2
u
tt
=
Δ
u
+
f
(
x
,
y
,
z
,
t
)
,
Ω
×
(
0
,
+
∞
)
u
z
)
∂Ω
=
L
(
u
,
u
x
,
u
y
,
0
,
(2.52)
⎩
u
(
x
,
y
,
z
,
0
)=
ϕ
(
x
,
y
,
z
)
,
u
t
(
x
,
y
,
z
,
0
)=
ψ
(
x
,
y
,
z
)
,
where
Ω
stands for a cuboid domain: 0
<
x
<
a
1
,
0
<
y
<
b
1
,
0
<
z
<
c
1
,
∂Ω
is
the boundary of
. If all combinations of the boundary conditions of
the first, the second and the third kinds are considered, for a finite cuboid domain
D
,
there exist 729 combinations of linear boundary conditions
L
Ω
,and
t
∈
(
0
,
∞
)
u
z
)
∂Ω
=
(
u
,
u
x
,
u
y
,
0.
We solve PDS (2.52) for the case of
⎧
⎨
u
(
0
,
y
,
z
,
t
)=
u
(
a
1
,
y
,
z
,
t
)=
0
,
u
y
(
x
,
0
,
z
,
t
)=
u
y
(
x
,
b
1
,
z
,
t
)+
h
2
u
(
x
,
b
1
,
z
,
t
)=
0
,
(2.53)
⎩
u
z
(
x
,
y
,
0
,
t
)
−
h
1
u
(
x
,
y
,
0
,
t
)=
u
(
x
,
y
,
c
1
,
t
)=
0
.
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