Environmental Engineering Reference
In-Depth Information
Solution.
We first find the solution for the case
f
=
0. It can be readily written as,
nl
t
,
ω
nl
t
in Eq. (2.59) by e
−
ω
simply by replacing sin
⎧
⎨
+
∞
∑
P
n
(
u
=
W
Φ
(
r
,
θ
,
ϕ
,
t
)=
1
(
b
mnl
cos
m
ϕ
+
d
mnl
sin
m
ϕ
)
cos
θ
)
m
,
n
=
0
,
l
=
⎛
⎝
μ
(
⎞
)
2
n
+
nl
t
l
⎠
e
−
ω
×
j
n
r
,
a
0
⎛
⎝
μ
(
⎞
)
2
n
+
1
l
⎠
r
2
sin
b
0
nl
=
Φ
(
r
,
θ
,
ϕ
)
P
n
(
cos
θ
)
j
n
r
θ
d
θ
d
r
d
ϕ
,
2
π
M
0
nl
a
0
r
≤
a
0
⎩
⎛
⎝
μ
(
⎞
2
)
n
+
1
P
n
(
l
⎠
r
2
sin
b
mnl
=
Φ
(
r
,
θ
,
ϕ
)
cos
m
ϕ
cos
θ
)
j
n
r
θ
d
θ
d
r
d
ϕ
,
π
M
mnl
a
0
r
≤
a
0
⎛
⎝
μ
(
⎞
1
2
)
n
+
1
P
n
(
l
⎠
r
2
sin
d
mnl
=
Φ
(
r
,
θ
,
ϕ
)
sin
m
ϕ
cos
θ
)
j
n
r
θ
d
θ
d
r
d
ϕ
.
π
M
mnl
a
0
r
≤
a
0
Thus the solution of PDS (3.16) is, by the solution structure theorem,
t
u
=
W
Φ
(
r
,
θ
,
ϕ
,
t
)+
W
F
τ
(
r
,
θ
,
ϕ
,
t
−
τ
)
d
τ
,
0
where
F
τ
=
,andthe
meanings of all parameters, functions and constants are the same as those for wave
equations in Section 2.6.
F
(
r
,
θ
,
ϕ
,
τ
)
,
f
(
x
,
y
,
z
,
t
)=
F
(
r
,
θ
,
ϕ
,
t
)
,
ϕ
(
x
,
y
,
z
)=
Φ
(
r
,
θ
,
ϕ
)
3.3 Well-Posedness of PDS
We examine the well-posedness of PDS using the example
⎧
⎨
a
2
u
xx
,
u
t
=
0
<
x
<
l
,
0
<
t
,
u
(
0
,
t
)=
u
(
l
,
t
)=
0
,
(3.17)
⎩
u
(
x
,
0
)=
ϕ
(
x
)
.
Search WWH ::
Custom Search