Environmental Engineering Reference
In-Depth Information
The results for the remaining 728 combinationsmay be readily obtained using a sim-
ilar approach, with Table 2.1and the solution structure theorem.
By the solution structure theorem, we first develop
u
=
W
ψ
(
x
,
y
,
z
,
t
)
, the solution
for the case of
f
0. Based on the given boundary conditions (2.53), we should
use the eigenfunctions in Rows 1,6 and 7 in Table 2.1 to expand the solution
=
ϕ
=
sin
γ
l
z
c
1
+
ϕ
l
+
∞
∑
m
,
n
,
l
=
1
sin
m
π
x
cos
μ
n
y
b
1
u
(
x
,
y
,
z
,
t
)=
T
mnl
(
t
)
,
a
1
y
b
1
h
2
where
μ
n
and
γ
l
are the positive zero points of
f
(
y
)=
cot
y
−
and
g
(
z
)=
z
c
1
h
1
, respectively, and tan
ϕ
l
=
γ
l
tan
z
c
1
h
1
. Substituting this into the wave equation
in PDS (2.52) and comparing the coefficients yields
+
T
mnl
(
2
mnl
T
mnl
(
t
)+
ω
t
)=
0
,
a
2
m
2
. The general solution of this equa-
2
μ
n
b
1
2
γ
l
c
1
π
a
1
2
mnl
where
ω
=
+
+
tion reads
T
mnl
(
t
)=
a
mnl
cos
ω
mnl
t
+
b
mnl
sin
ω
mnl
t
.
sin
γ
l
z
c
1
+
ϕ
l
.
+
∞
∑
sin
m
π
x
cos
μ
n
y
b
1
Thus
u
=
m
,
n
,
l
=
1
(
a
mnl
cos
ω
mnl
t
+
b
mnl
sin
ω
mnl
t
)
a
1
Applying the initial condition
u
(
x
,
y
,
z
,
0
)=
0 yields
a
mnl
=
0. The
b
mnl
can be de-
termined by the initial condition
u
t
(
x
,
y
,
z
,
0
)=
ψ
(
x
,
y
,
z
)
. Finally, we have
⎧
⎨
⎩
sin
γ
l
z
c
1
+
ϕ
l
sin
+
∞
∑
b
mnl
sin
m
π
x
cos
μ
n
y
b
1
u
=
W
ψ
(
x
,
y
,
z
,
t
)=
ω
mnl
t
,
a
1
m
,
n
,
l
=
1
sin
γ
l
z
c
1
+
ϕ
l
d
x
d
y
d
z
1
ω
mnl
M
mnl
sin
m
π
x
cos
μ
n
y
b
1
b
mnl
=
ψ
(
x
,
y
,
z
)
,
a
1
Ω
where
M
mnl
=
M
m
M
n
M
l
,
M
m
,
M
n
and
M
l
are the normal squares of the three eigen-
function sets, respectively.
The solution of PDS (2.52) under the boundary conditions (2.53) follows straight-
forwardly from the solution structure theorem
=
∂
∂
t
u
t
W
ϕ
+
W
ψ
(
x
,
y
,
z
,
t
)+
W
f
τ
(
x
,
y
,
z
,
t
−
τ
)
d
τ
,
0
where
f
τ
=
f
(
x
,
y
,
z
,
τ
)
.
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