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k = 1000
k = 100
k = 10
k = 1
k = 0, 1
k = 0, 01
k = 0, 001
t
D
< t
D0
t
D
> t
D0
t
D
< t
D0
t
D
> t
D0
t
D
< t
D0
t
D
> t
D0
h
z
= 25
h
z
= 50
h
z
= 75
h
z
= 25
h
z
= 50
h
z
= 75
h
x
= 2500
h
x
= 5000
h
x
= 7500
h
x
= 2500
h
x
= 5000
h
x
= 7500
k = 1000
k = 1000
k = 100
k = 10
k = 10
k = 1
k = 1
k = 0, 1
k = 0, 01
k = 0, 001
A
2
c
−1
C
d
3
D
e
h
j
k
2
l
L
m
M
n
p
q
3
r
R
s
S
t
T
v
V
3
X
z
0
1
a
D
e
f
final
g
i
inicial
L
m
p
r r
s
z
t
x x
xd
x
xl
x
y y
w
z z
za z
zb z
θ θ
∞
¯
ˆ
x
D
ˆ
ˆ
z
D
ˆ
ˆ
ˆ
α
µ
ρ
φ
γ
∂
L
τ
AOF
C
1
C
A
E
i
F (s)
F AF
F
fx
D
x
D
F
fz
D
z
D
H (t)
LI
LT
m (p)
P R
q
AOF
RI
1 × 10
3
1 × 10
−3
µ(p) = µ = µ(p) z(p) = z =
z(p) p =
p
2
i
+ p
2
L
(t)
2
1
2
p
i
p
L
µ z µ z
µz
µz =
p
2
i
− p
2
L
(t)
2
p
i
p
L
p
µ(p
)z(p
)
dp
m(p) = 2
p
p
m
p
µ(p
)z(p
)
dp
p
m
m(p)
(µ)
(z)
t = 229680
φµc
g
r
2
w
C
k
0.0002637
α
t
C
C =
V
∞
2πφhr
2
w
V
∞
(kh)
kh
µc
t
t
a
(t) =
t
t
0
dt
µ(t
)c
t
(t
)
t
0
t
final
= 1800
d
2
z
φµc
t
k
z
0.0002637
α
t
t
final
= 125
L
2
φµc
t
k
x
0.0002637
α
t
d
z
z L
t
inicial
= 1800
D
2
z
φµc
t
k
z
0.0002637
α
t
D
z
z
t
final
= 160
L
2
φµc
t
k
x
0.0002637
α
t
t
inicial
= 1480
L
2
φµc
t
k
x
0.0002637
α
t
t
final
= 2000
φµc
t
(d
x
+ L/4)
2
k
x
0.0002637
α
t
t
final
= 1650
φµc
t
d
2
y
k
y
0.0002637
α
t
d
x
d
y
x y
x
t
inicial
= 4800
φµc
t
(D
x
+ L/4)
2
k
x
0.0002637
α
t
t
inicial
= 1800
D
2
z
φµc
t
k
z
0.0002637
α
t
D
x
x
t
final
= 1650
d
2
y
φµc
t
k
y
0.0002637
α
t
1
r
∂
∂r
k
µ(p)
∂p
∂r
rρ
=
∂ (φρ)
∂t
(µ)
1
r
∂
∂r
r
∂p
2
∂r
=
µφc
g
k
∂p
2
∂t
(r
w
→ 0)
p(r, t = 0) = p
i
⇒ p
2
(r, t = 0) = p
2
i
lim
r→∞
p(r, t) = p
i
⇒ lim
r→∞
p
2
(r, t) = p
2
i
lim
r→0
r
∂p
∂r
=
q
w
µ
2πkh
⇒ lim
r→0
r
∂p
2
∂r
=
q
w
µ
πkh
p
w
p
2
(r, t) = p
2
i
+
µ
2πkh
p
0
q
0
T
w
T
0
E
i
−
φµc
g
r
2
4kt
p
0
q
0
T
0
∂p
2
∂t
= 0
r = r
e
⇒ p(r = r
e
) = p
e
⇒ p
2
(r = r
e
) = p
2
e
r = r
w
⇒ p(r = r
w
) = p
w
⇒ p
2
(r = r
w
) = p
2
w
p
2
(r) = p
2
w
+
p
2
e
− p
2
w
ln
r
e
r
w
ln
r
r
w
∂p
∂r
r=r
e
= 0 ⇒
∂p
2
∂r
r=r
e
= 0
r
∂p
∂r
r=r
w
=
µq
w
2πkh
⇒
r
∂p
2
∂r
r=r
w
=
p
w
q
w
µ
πkh
p (r, t)
2
− p
2
w
= −
q
0
p
0
T
w
µ
T
0
πkh
1
2
r
r
e
2
− ln
r
r
w
1
r
∂
∂r
r
∂m(p)
∂r
=
φµ(p)c
t
(p)
k
∂m(p)
∂t
m(p)
m(p)
m(p)
p(r, t = 0) = p
i
⇒ m(p) = m(p
i
)
lim
r→∞
p(r, t) = p
i
⇒ lim
r→∞
m(p) = m(p
i
)
lim
r→0
r
∂p
∂r
=
q
w
µ
2πkh
⇒ lim
r→0
r
∂m(p)
∂r
=
q
w
p
w
z
w
πkh
m(p) = m(p
i
) +
q
0
p
0
T
w
2T
0
πkh
E
i
−
φ (µc
t
)
i
r
2
4kt
∂m(p)
∂t
=
2p
µ(p)z(p)
∂p
∂t
= 0
r = r
e
⇒ p(r = r
e
, t) = p
e
⇒ m(p) = m(p
e
)
r = r
w
⇒ p(r = r
w
, t) = p
w
⇒ m(p) = m(p
w
)
m(p) − m(p
w
) =
m(p
e
) − m(p
w
)
ln
r
e
r
w
ln
r
r
w
r = r
e
⇒
∂p
∂r
r=r
e
= 0 ⇒
∂m(p)
∂r
r=r
e
=
2p
µz
∂p
∂r
r=r
e
= 0
r
∂p
∂r
r=r
w
=
q
w
µ
2πkh
⇒
r
∂m(p)
∂r
r=r
w
=
q
w
p
w
z
w
πkh
∂m(p)
∂t
= C
1
(= )
m(p) − m(p
w
) = −
q
0
p
0
T
w
T
0
πkhφ
1
2
r
r
e
2
− ln
r
r
w
p
f
q = C [m(p) − m(p
f
)]
n
p p
f
log q = log C + n log [m(p) − m(p
f
)]
log log
1/n
p
f
= p
b
, p
b
C
C =
q
i
[m(p
i
)]
n
p
b
q
AOF
= C [m(p) − m(p
b
)]
n
C n
m(p) − m(p
f
) = α
p
p
0
qT
T
0
kh
1, 151 log
10, 06A
C
A
r
2
w
−
3
4
+ s + Dq
s D A
C
A
m(p) = m(p) − m(p
f
) = aq + bq
2
a = α
p
p
0
qT
T
0
kh
1, 151 log
10, 06A
C
A
r
2
w
−
3
4
+ s
b = α
p
p
0
qT D
T
0
kh
q
m(p)
q
=
m(p) − m(p
f
)
q
= a + bq
b a
m(p)
q
q p
f
a b
q
AOF
=
−a +
a
2
+ 4b [m(p) − m(p
b
)]
2b
C
[C(t)] log [m(p) − m(p
f
)]
log q log [m(p) − m(p
f,s
)] log q
s
q
s
p
f,s
1/n C(t)
[C
s
]
C
s
=
q
s
[m(p) − m(p
f,s
)]
n
q
AOF
= C
s
[m(p) − m(p
b
)]
n
m(p)
q
=
m(p) − m(p
f
)
q
= a
t
+ bq
a
t
= α
p
p
0
qT
T
0
kh
ln
r
d
r
w
−
3
4
+ s
b = α
p
p
0
qT D
T
0
kh
r
d
=
17, 9098α
t
kt
φ (µc
t
)
i
m(p)
q
q a b b
a
a
t,s
=
m(p) − m(p
f,s
)
q
s
− bq
s
q
AOF
=
−a
t,s
+
a
2
t,s
+ 4b (m(p) − m(p
b
))
2b
m(p) = m(p
s
) − m(p
f
) = a
t
q + bq
2
a
t
= α
p
p
0
T
T
0
kh
ln
4α
t
kt
e
γ
φ (µc
t
)
i
r
2
w
+ s
b = α
p
p
0
T D
T
0
kh
p
s
γ (0.57722)
b
m(p)
q
q a
a
t,s
=
m(p
s
) − m(p
f,s
)
q
s
− bq
s
q
AOF
=
−a
t,s
+
a
2
t,s
+ 4b (m(p
s
) − m(p
b
))
2b
(kh)
(s)
m(p
i
) − m(p
f
) = 2, 302
α
p
p
0
qT
T
0
kh
log(t) + log
4α
t
k
e
γ
φ (µc
t
)
i
r
2
w
+ 0.869s
m(p
i
) − m(p
f
) = m
q [log(t) + s]
m
= 2, 302
α
p
p
0
T
T
0
kh
s = log
4α
t
k
e
γ
φ (µc
t
)
i
r
2
w
+ 0.869s
m(p
i
) − m(p
f
)
q
n
= m
n
j=1
q
j
− q
j−1
q
n
log (t − t
j−1
)
+ m
s
q
n
= 0
m(p
i
) − m(p
f
)
q
n
n
j=1
q
j
− q
j−1
q
n
log (t − t
j−1
)
k = 2, 302
α
p
p
0
T
T
0
m
h
p
q
s = 1.151
p
q
i
m
− log
4α
t
k
e
γ
φ (µc
t
)
i
r
2
w
m(p
s
) = m(p
i
) − 2, 302
α
p
p
0
qT
T
0
kh
log
t
p
+ t
t
t
p
t
(m) m(p
s
) log
t
p
+ t
t
k = 2, 302
α
p
p
0
qT
T
0
mh
s
= 1.151
m(p
1hr
) − m(p
f
)
m
− log
4α
t
k
e
γ
φ (µc
t
)
i
r
2
w
m(p
1hr
) t = 1hr m(p
f
)
(µc
g
)
k
x
k
y
∂
2
m (p)
∂x
2
+
∂
2
m (p)
∂y
2
+
k
z
k
y
∂
2
m (p)
∂z
2
=
φ (µc
g
)
i
α
t
k
y
∂m (p)
∂t
m (p (x, y, z, t = 0)) = m (p
i
)
lim
y→∞
m (p (x, y, z, t)) = m (p
i
)
lim
y→0
(L
zb
− L
za
) (L
xl
− L
xd
)
∂m (p)
∂y
=
2πα
p
q
0
p
0
T
w
T
0
k
y
L
xd
≤ x ≤ L
xl
L
za
≤ z ≤ L
zb
t < t
0
0 L
xd
≤ x ≤ L
xl
L
za
≤ z ≤ L
zb
t > t
0
∂m (p)
∂z
z=0
=
∂m (p)
∂z
z=h
z
= 0
∂m (p)
∂x
x=0
=
∂m (p)
∂x
x=h
x
= 0
p (x, y, z, t = 0) = p
i
lim
y→∞
p (x, y, z, t) = p
i
lim
y→0
(L
zb
− L
za
) (L
xl
− L
xd
)
∂p
∂y
=
2πα
p
q
w
µ (p)
2k
y
L
xd
≤ x ≤ L
xl
L
za
≤ z ≤ L
zb
t < t
0
0 L
xd
≤ x ≤ L
xl
L
za
≤ z ≤ L
zb
t > t
0
∂p
∂z
z=0
=
∂p
∂z
z=h
z
= 0
∂p
∂x
x=0
=
∂p
∂x
x=h
x
= 0
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
√
πt
D
+
1
ν
z
h
z
4r
w
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
xl
− L
xd
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
+
+2π
h
x
L
xl
− L
xd
2
h
z
4r
w
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
×
×Z
m
Z
2
n
cos (mπz
D
)]
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
√
π
√
t
D
−
√
t
D
− t
D0
+
+
1
ν
z
h
z
4r
w
∞
m=1
1
m
mπν
z
√
t
D
−
mπν
z
√
t
D
− t
D0
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
xl
− L
xd
2
∞
n=1
1
n
nπν
x
√
t
D
−
nπν
x
√
t
D
− t
D0
Z
2
n
+
+2π
h
x
L
xl
− L
xd
2
h
z
4r
w
∞
n=1
∞
m=1
1
(nπν
x
)
2
+ (mπν
z
)
2
×
×
(nπν
x
)
2
+ (mπν
z
)
2
t
D
−
(nπν
x
)
2
+ (mπν
z
)
2
(t
D
− t
D0
)
Z
m
Z
2
n
cos (mπz
D
)
t
D
=
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
ν
z
=
4r
w
h
z
k
z
k
y
ν
x
=
4r
w
h
x
k
x
k
y
Z
m
=
sin
mπ (h
s
+ 2r
w
)
h
z
− sin
mπ (h
s
− 2r
w
)
h
z
mπ
Z
n
=
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
nπ
r
w
= r
w
k
z
k
y
1
4
z
D
=
1
h
z
(h
s
+ 1, 47r
w
)
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
√
πt
D
+
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
√
π
√
t
D
−
√
t
D
− t
D0
+
+
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
−
mπν
z
√
t
D
− t
D0
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
−
nπν
x
√
t
D
− t
D0
n
Z
2
n
m
D
=
2 × 1, 8089
π
2
k
y
k
z
log
64k
z
γ
k
y
t
D
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
4α
t
k
z
t
γ
φ (µc
g
)
i
(r
w
)
2
m
D
=
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
√
πt
D
+
h
x
h
z
2L
w
ν
z
S
z
m (p) =
16α
p
q
0
p
0
T
w
r
w
T
0
k
y
h
z
L
w
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
2α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
S
z
S
z
=
2
π
h
z
Ω
z
Ω
z
=
1
8r
w
[ψ (η
1
) + ψ (η
2
) − ψ (η
3
) − ψ (η
4
)]
ψ (η) =
∞
n=1
sin (nη)
n
2
η
1
=
0, 52πr
w
h
z
η
2
=
π
h
z
(2h
s
+ 3, 48r
w
)
η
3
= −
3, 48πr
w
h
z
η
4
=
π
h
z
(2h
s
− 0, 52r
w
)
m
D
=
2
π
4r
w
h
z
L
w
h
x
h
x
h
z
L
w
ν
z
π
Ω
z
+
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
k
y
k
x
h
z
log
64α
t
k
x
t
γ
φ (µc
g
)
i
L
2
w
+
2α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
S
z
x
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
8r
w
L
w
k
y
k
z
(S
z
+ S
x
)
m (p) =
16α
p
q
0
p
0
T
w
r
w
T
0
k
y
h
z
h
x
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
2α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
(S
z
+ S
x
)
S
x
=
2h
2
x
h
z
L
w
k
x
k
z
∞
n=1
Z
2
n
n
m (p) = α
p
2, 303q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
t
t − t
0
− log
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
+
16r
w
L
w
√
k
z
2, 303
k
y
h
z
h
x
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+ log
γ
k
y
64k
z
+
2
2, 303
(S
x
+ S
z
)
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
π
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
√
t −
h
x
L
w
√
t − t
0
+ α
p
2q
0
p
0
T
w
T
0
L
w
k
y
k
z
S
x
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
k
y
k
x
h
z
log
t
t−t
0
− log
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+ log
L
2
w
γ
k
y
(4r
w
)
2
64k
x
+
+
4r
w
π
√
k
x
1, 8089h
x
k
y
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
h
z
π
√
k
x
2 × 1, 8089L
w
√
k
z
S
x
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
π
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
√
t −
√
t − t
0
m (p) = α
p
2, 303q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
t
t − t
0
− log
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
+
L
w
√
k
z
π
√
k
x
h
z
log
64α
t
k
x
t
γ
φ (µc
g
)
i
L
2
w
+ log
γ
k
y
64k
z
+
2
2, 303
S
z
m (p) =
16α
p
q
0
p
0
T
w
r
w
T
0
k
y
h
z
L
w
−
π
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
(t − t
0
) +
2, 303L
w
k
y
16r
w
√
k
x
log
64α
t
k
x
t
γ
φ (µc
g
)
i
L
2
w
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
k
y
k
x
h
z
log
t
t − t
0
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
t
t − t
0
− log
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
+
16r
w
√
k
z
2, 303h
z
k
y
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+ log
γ
k
y
64k
z
+
2
2, 303
S
z
m (p) =
16α
p
q
0
p
0
T
w
r
w
T
0
k
y
h
z
L
w
π
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
√
t −
√
t − t
0
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
t
t − t
0
1/2 1
∂m (p)
∂ ln t
= t
∂m (p)
∂t
= α
p
8r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
√
πt
D
×
×
1 +
h
x
h
z
2r
w
L
w
∞
m=1
exp
−
mπν
z
√
t
D
2
Z
m
cos (mπz
D
) +
+
1
2
h
x
L
w
2
∞
n=1
exp
−
nπν
x
√
t
D
2
Z
2
n
m (p) = α
p
16r
w
p
0
T
w
T
0
k
y
h
z
h
x
j
i=0
(q
0,i+1
− q
0,i
) ×
×
π (t
D
− t
Di
) +
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
(t
D
− t
Di
)
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
(t
D
− t
Di
)
n
Z
2
n
t > t
i
q
0,0
= 0 t
0
= 0 j
m (p) =
4πα
p
q
0
p
0
T
w
T
0
h
x
h
y
h
z
L
w
α
t
0
S
x
(τ) S
y
(τ) S
z
(τ) dτ
α =
φ (µc
g
)
i
α
t
S
x
(τ) = L
w
+
2h
x
π
∞
m=1
exp
−
m
2
π
2
k
x
τ
αh
2
x
cos
mπx
h
x
sin
mπL
xl
h
x
− sin
mπL
xd
h
x
S
y
(τ) = 1 + 2
∞
l=1
exp
−
l
2
π
2
k
y
τ
αh
2
y
cos
mπy
h
y
cos
mπy
0
h
y
S
z
(τ) = 1 + 2
∞
n=1
exp
−
n
2
π
2
k
z
τ
αh
2
z
cos
mπz
h
z
cos
mπz
0
h
z
m (p) =
q
0
p
0
T
w
α
p
2T
0
L
w
k
y
k
z
t
0
(x, x
0
, τ) (y, y
0
, τ) (z, z
0
, τ)
dτ
τ
(x, x
0
, τ) =
∞
m=−∞
L
xl
− x − 2mh
x
4k
x
τ/α
+
L
xl
+ x + 2mh
x
4k
x
τ/α
+
−
L
xd
− x − 2mh
x
4k
x
τ/α
−
L
xd
+ x + 2mh
x
4k
x
τ/α
(y, y
0
, τ) =
∞
l=−∞
exp
−α (y − y
0
+ 2lh
y
)
2
4k
y
τ
+ exp
−α (y + y
0
+ 2lh
y
)
2
4k
y
τ
(z, z
0
, τ) =
∞
n=−∞
exp
−α (z −z
0
+ 2nh
y
)
2
4k
z
τ
+ exp
−α (z + z
0
+ 2nh
y
)
2
4k
z
τ
m (p)
∂ ln t
= t
m (p)
∂t
=
4πα
p
q
0
p
0
T
w
T
0
h
x
h
y
h
z
L
w
α
S
x
(t) S
y
(t) S
z
(t)
m (p) =
4πα
p
p
0
T
w
T
0
h
x
h
y
h
z
L
w
α
j
i=0
(q
0,i+1
− q
0,i
)
t−t
i
0
S
x
(τ) S
y
(τ) S
z
(τ) dτ
t > t
i
q
0,0
= 0 t
0
= 0 j
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
k
y
k
z
L
w
log
4α
t
k
y
k
z
t
γ
φ (µc
g
)
i
r
2
w
m (p) =
2, 303α
p
q
0
p
0
T
w
2T
0
L
w
h
z
α
t
t
0.0002637φ (µc
g
)
i
k
y
+
+
17, 37h
z
k
y
k
z
ln
h
z
r
w
+ 0, 25 ln
k
y
k
z
− ln
sin
πz
0
h
z
− 1, 838
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
k
y
k
z
h
z
log
α
t
k
y
t
0.0002637φ (µc
g
)
i
L
2
w
− 1.76+
+0.87
k
x
k
z
h
z
L
w
ln
h
z
r
w
+ 0, 25 ln
k
y
k
z
− ln
sin
πz
0
h
z
− 1, 838
m (p) =
2, 303α
p
q
0
p
0
T
w
2T
0
h
x
h
z
α
t
t
0.0002637φ (µc
g
)
i
k
y
+
+
17, 37h
z
k
y
k
z
ln
h
z
r
w
+ 0, 25 ln
k
y
k
z
− ln
sin
πz
0
h
z
− 1, 838 + S
R
S
R
= P
xyz
+ P
xy
h
y
k
y
≥
0, 75h
x
√
k
x
0, 75h
z
√
k
z
S
R
= P
xyz
+ P
x
+ P
xy
h
x
√
k
x
>
1, 33h
y
k
y
0, 75h
z
√
k
z
P
xyz
=
h
x
L
w
− 1
ln
h
z
r
w
+ 0, 25 ln
k
y
k
z
− ln
sin
πz
0
h
z
− 1, 84
P
xy
=
2h
2
x
L
w
h
z
k
z
k
x
F
L
w
2h
x
+ 0, 5
F
4x
0
+ L
w
2h
x
− F
4x
0
− L
w
2h
x
P
x
=
6, 28h
2
x
h
z
h
y
k
y
k
z
k
x
1
3
−
x
0
h
x
+
x
2
0
h
2
x
+
L
w
24h
x
L
w
h
x
− 3
P
xy
=
h
x
L
w
− 1
6, 28h
y
h
z
k
z
k
y
1
3
−
y
0
h
y
+
y
2
0
h
2
y
(y
0
, h
y
− y
0
) ≥ 0, 25h
y
F (x) = −x
0, 0145 + ln (x) − 0, 137x
2
x ≤ 1
F (x) = (2 − x)
0, 0145 + ln (2 − x) − 0, 137 (2 − x)
2
x > 1
t
final
= 190
h
2,095
s
r
−0,095
w
φ (µc
g
)
i
k
z
0, 0002637
α
t
t
final
= 20, 8
φ (µc
g
)
i
L
2
w
k
x
0, 0002637
α
t
xy
t
inicial
= 1230
φ (µc
g
)
i
L
2
w
k
x
0, 0002637
α
t
t
final
= 297
(L
xl
+ L
xd
)
2,095
L
−0,095
w
φ (µc
g
)
i
k
x
0, 0002637
α
t
L
w
t
final1
= 1800
d
2
z
φ (µc
g
)
i
k
z
0, 0002637
α
t
t
final2
= 125
L
2
w
φ (µc
g
)
i
k
x
0, 0002637
α
t
x
t
inicial
= 1800
D
2
z
φ (µc
g
)
i
k
z
0, 0002637
α
t
t
final
= 160
L
2
w
φ (µc
g
)
i
k
x
0, 0002637
α
t
L
w
h
x
< 0, 45 L
w
h
x
t
inicial
= 1480
L
2
w
φ (µc
g
)
i
k
x
0, 0002637
α
t
t
final1
= 2000
φ (µc
g
)
i
(d
x
+ L
w
/4)
2
k
x
0, 0002637
α
t
t
final2
= 1650
d
2
y
φ (µc
g
)
i
k
y
0, 0002637
α
t
x y
z x
t
inicial1
= 4800
φ (µc
g
)
i
(D
x
+ L
w
/4)
2
k
x
0, 0002637
α
t
t
inicial2
= 1800
φ (µc
g
)
i
D
2
z
k
z
0, 0002637
α
t
t
final
= 1650
φ (µc
g
)
i
d
2
y
k
y
0, 0002637
α
t
x
t
final
=
e
γ
φ (µc
g
)
i
(d
z
− r
w
)
2
4α
t
k
z
t
inicial
=
e
γ
φ (µc
g
)
i
(D
z
− r
w
)
2
4α
t
k
z
t
final
= 0, 1
φ (µc
g
)
i
L
2
w
4α
t
k
x
t
inicial1
= 10
φ (µc
g
)
i
L
w
4α
t
k
x
1 × 10
3
1 × 10
−3
t
inicial1
=
φ (µc
g
)
i
L
w
4α
t
k
x
t
final1
=
e
γ
φ (µc
g
)
i
d
x
−
L
w
4
k
y
k
x
1/4
2
4α
t
k
x
t
final2
=
e
γ
φ (µc
g
)
i
[d
y
− r
w
]
2
4α
t
k
y
x y
t
inicial
=
e
γ
φ (µc
g
)
i
D
x
−
L
w
4
k
y
k
x
1/4
2
4α
t
k
x
t
inicial
=
e
γ
φ (µc
g
)
i
[D
y
− r
w
]
2
4α
t
k
y
h
x
2500 7500
h
y
10000
h
z
50
6000
r
w
0, 1
φ 0, 2
T
w
150
P
i
650
2
µ
c
g
L
w
600
k
x
10
k
y
10
k
z
10
q
0
2 × 10
6 3
P
0
1, 03323
2
T
0
15, 5556
h
x
= 2500
h
z
h
z
= 25
h
z
= 50
h
z
= 75
h
z
= 25
h
z
= 50
h
z
= 75
h
x
= 2500
h
z
h
z
= 25
h
z
= 50
h
z
= 75
h
z
= 50
x h
x
h
x
= 2500
h
x
= 5000
h
x
= 7500
h
z
= 25
h
z
= 50
h
z
= 75
h
x
= 2500
h
x
= 5000
h
x
= 7500
h
x
= 2500
h
z
= 50
x h
x
h
x
= 2500
h
x
= 5000
h
x
= 7500
h
x
= 5000
h
x
= 7500
6000
r
w
0, 1
φ 0, 2
T
w
150
P
i
650
2
z
µ
c
g
L
w
600
k
x
1 × 10
3
1 × 10
−3
k
y
1 × 10
3
1 × 10
−3
k
z
1 × 10
3
1 × 10
−3
q
0
2 × 10
6
2 × 10
7 3
P
0
1.03323
2
T
0
15.5556
h
x
2500 5000 7500 10000
h
y
10000 10000 10000 2500
h
z
50 50 50 50
1 × 10
3
2 × 10
6 3
4 × 10
6 3
6 × 10
6 3
3 × 10
6 3
1 × 10
2
2 × 10
6 3
4 × 10
6 3
6 × 10
6 3
3 × 10
6 3
1 × 10
1
2 × 10
6 3
4 × 10
6 3
6 × 10
6 3
3 × 10
6 3
1 × 10
0
1 × 10
5 3
2 × 10
5 3
3 × 10
5 3
3 × 10
5 3
1 × 10
−1
9 × 10
3 3
1, 8 × 10
4 3
2, 7 × 10
4 3
1, 35 × 10
4 3
1 × 10
−2
1 × 10
3 3
2 × 10
3 3
3 × 10
3 3
1, 5 × 10
3 3
1 × 10
−3
3 × 10
2 3
6 × 10
2 3
9 × 10
2 3
4, 5 × 10
2 3
2, 92 × 10
8
9, 29 × 10
8
8, 27 × 10
8
5, 13 × 10
7
1, 06 × 10
8
5, 73 × 10
7
1, 00 × 10
7
1, 27 × 10
7
8, 01 × 10
5
1, 27 × 10
6
1, 20 × 10
6
7, 76 × 10
4
1, 28 × 10
5
8, 01 × 10
3
1, 26 × 10
4
1, 15 × 10
3
1, 36 × 10
3
3
3
2, 92 × 10
8
8, 16 × 10
8
6, 18 × 10
8
5, 13 × 10
7
1, 11 × 10
8
5, 65 × 10
7
1, 00 × 10
7
1, 26 × 10
7
1, 17 × 10
7
8, 01 × 10
5
1, 25 × 10
6
1, 13 × 10
6
7, 76 × 10
4
1, 28 × 10
8
8, 01 × 10
3
1, 27 × 10
4
1, 15 × 10
3
1, 37 × 10
3
4, 86 × 10
8
1, 36 × 10
9
7, 75 × 10
7
1, 41 × 10
8
8, 51 × 10
7
1, 36 × 10
7
1, 65 × 10
7
1, 15 × 10
6
1, 68 × 10
6
1, 58 × 10
6
1, 12 × 10
5
1, 70 × 10
5
1, 15 × 10
4
1, 67 × 10
4
1, 54 × 10
3
1, 77 × 10
3
4, 86 × 10
8
1, 30 × 10
9
1, 21 × 10
9
7, 75 × 10
7
1, 44 × 10
8
7, 98 × 10
7
1, 36 × 10
7
1, 64 × 10
7
1, 51 × 10
7
1, 15 × 10
6
1, 65 × 10
6
1, 48 × 10
6
1, 12 × 10
5
1, 71 × 10
5
1, 15 × 10
4
1, 68 × 10
4
1, 54 × 10
3
1, 78 × 10
3
7, 42 × 10
8
1, 46 × 10
9
1, 04 × 10
8
1, 52 × 10
8
9, 10 × 10
7
1, 59 × 10
7
1, 77 × 10
7
1, 37 × 10
6
1, 81 × 10
6
1, 70 × 10
6
1, 34 × 10
5
1, 84 × 10
5
1, 37 × 10
4
1, 80 × 10
4
1, 75 × 10
3
1, 90 × 10
3
7, 42 × 10
8
1, 46 × 10
9
1, 04 × 10
8
1, 56 × 10
8
8, 53 × 10
7
1, 59 × 10
7
1, 64 × 10
7
1, 71 × 10
7
1, 37 × 10
6
1, 78 × 10
6
1, 59 × 10
6
1, 34 × 10
5
1, 85 × 10
5
1, 37 × 10
4
1, 81 × 10
4
1, 75 × 10
3
1, 91 × 10
3
2, 73 × 10
8
9, 07 × 10
8
8, 01 × 10
8
4, 92 × 10
7
1, 04 × 10
8
6, 57 × 10
7
9, 73 × 10
6
1, 24 × 10
7
7, 74 × 10
5
1, 23 × 10
6
1, 17 × 10
6
7, 49 × 10
4
1, 25 × 10
5
7, 74 × 10
3
1, 23 × 10
4
1, 12 × 10
3
1, 33 × 10
3
2, 73 × 10
8
7, 98 × 10
8
8, 70 × 10
8
5, 93 × 10
8
4, 92 × 10
7
1, 08 × 10
8
1, 14 × 10
8
6, 46 × 10
7
9, 73 × 10
6
1, 23 × 10
7
1, 20 × 10
7
7, 74 × 10
5
1, 22 × 10
6
1, 10 × 10
6
7, 49 × 10
4
1, 25 × 10
5
7, 74 × 10
3
1, 24 × 10
4
1, 12 × 10
3
1, 34 × 10
3
10 =
3, 32%
1000
3
3
100
2, 92 × 10
8
2, 87 × 10
9
5, 13 × 10
7
4, 84 × 10
8
1, 00 × 10
7
1, 07 × 10
8
8, 01 × 10
5
1, 63 × 10
7
7, 76 × 10
4
2, 03 × 10
6
8, 01 × 10
3
2, 63 × 10
5
1, 15 × 10
3
3, 94 × 10
4
2, 92 × 10
8
2, 52 × 10
9
5, 13 × 10
7
4, 16 × 10
8
1, 00 × 10
7
8, 87 × 10
7
8, 01 × 10
5
1, 47 × 10
7
7, 76 × 10
4
1, 91 × 10
6
8, 01 × 10
3
2, 41 × 10
5
1, 15 × 10
3
3, 47 × 10
4
4, 86 × 10
8
2, 86 × 10
9
7, 75 × 10
7
4, 19 × 10
8
1, 36 × 10
7
1, 07 × 10
8
1, 15 × 10
6
1, 63 × 10
7
1, 12 × 10
5
2, 03 × 10
6
1, 15 × 10
4
2, 63 × 10
5
1, 54 × 10
3
3, 94 × 10
4
4, 86 × 10
8
2, 60 × 10
9
7, 75 × 10
7
4, 07 × 10
8
1, 36 × 10
7
8, 87 × 10
7
1, 15 × 10
6
1, 47 × 10
7
1, 12 × 10
5
1, 91 × 10
6
1, 15 × 10
4
2, 41 × 10
5
1, 54 × 10
3
3, 47 × 10
4
7, 42 × 10
8
2, 79 × 10
9
1, 04 × 10
8
4, 82 × 10
8
1, 59 × 10
7
1, 07 × 10
8
1, 37 × 10
6
1, 63 × 10
7
1, 34 × 10
5
2, 03 × 10
6
1, 37 × 10
4
2, 63 × 10
5
1, 75 × 10
3
3, 94 × 10
4
7, 42 × 10
8
2, 56 × 10
9
1, 04 × 10
8
4, 07 × 10
8
1, 59 × 10
7
8, 87 × 10
7
1, 37 × 10
6
1, 47 × 10
7
1, 34 × 10
5
1, 91 × 10
6
1, 37 × 10
4
2, 41 × 10
5
1, 75 × 10
3
3, 47 × 10
4
2, 73 × 10
8
2, 87 × 10
9
4, 92 × 10
7
4, 84 × 10
8
9, 73 × 10
6
1, 07 × 10
8
7, 74 × 10
5
1, 63 × 10
7
7, 49 × 10
4
2, 03 × 10
6
7, 74 × 10
3
2, 63 × 10
5
1, 12 × 10
3
3, 94 × 10
4
2, 73 × 10
8
2, 53 × 10
9
4, 92 × 10
7
4, 16 × 10
8
9, 73 × 10
6
8, 87 × 10
7
7, 74 × 10
5
1, 47 × 10
7
7, 49 × 10
4
1, 91 × 10
6
7, 74 × 10
3
2, 41 × 10
5
1, 12 × 10
3
3, 47 × 10
4
1000
(µ (p) c
g
(p))
(µ (p) c
g
(p))
i
h
x
4860
h
y
12150
h
z
90
6000
r
w
0, 1
φ 0, 2
T
w
150
P
i
650
2
d
g
0, 7
z
µ
c
g
c
f
4.97817 × 10
−5 2
L
w
540
k
x
1 × 10
3
1 × 10
−3
k
y
1 × 10
3
1 × 10
−3
k
z
1 × 10
3
1 × 10
−3
P
0
1.03323
2
T
0
15.5556
k = 1000
810 ×
810 × 90
q
0
= 2 ×10
7 3
x 6 810
y 15 810
z 90
x 6 × 3 = 18 810 ÷ 3 = 270
y 15 × 3 = 45 810 ÷ 3 = 270
z 1 90
x 18 × 3 = 54 270 ÷ 3 = 90
y 45 × 3 = 135 270 ÷ 3 = 90
z 1 90
x 54 × 3 = 162 90 ÷ 3 = 30
y 135 × 3 = 405 90 ÷ 3 = 30
z 1 × 3 = 3 90 ÷ 3 = 30
k = 1000
k = 1000
k = 100
k = 100
k = 100
k = 10
k = 10
k = 10
k = 10
k = 1
3
q
0
= 2 × 10
7 3
q
0
= 2 ×10
6
k = 1
k = 1
k = 0, 1
k = 0, 1
q
0
= 2 × 10
5 3
k = 0, 1
k = 0, 01
q
0
= 2 × 10
4 3
k = 0, 01
k = 0, 001
q
0
= 2 × 10
3
k = 0, 001
(µ (p) c (p))
l
v
l
= −
k
µ
∂p
∂l
c
g
(p) =
1
ρ
∂ρ
∂p
ρc
g
(p)
∂p
∂t
=
∂ρ
∂t
c
f
=
1
φ
∂φ
∂p
φc
f
∂p
∂t
=
∂φ
∂t
(ρ) (p) (T )
(z)
ρ =
M
RT
p
z (p)
m (p) = 2
p
p
m
p
µ (p
) z (p
)
dp
m (p)
p
∂m (p)
∂l
=
∂m (p)
∂p
∂p
∂l
=
2p
µ (p) z (p)
∂p
∂l
l x y z r t
− =
(qρ)
r
− (qρ)
r+r
t +
(qρ)
z
− (qρ)
z+z
t +
(qρ)
θ
− (qρ)
θ+θ
t =
θ
2
(r+ r)
2
− r
2
z
(φρ)
t+t
− (φρ)
t
q
r
= v
r
A
r
= v
r
r z θ
q
z
= v
z
A
z
= v
z
2r r + ( r)
2
θ
2
q
θ
= v
θ
A
θ
= v
θ
r z
(v
r
r z θρ)
r
− (v
r
r z θρ)
r+r
t+
+
v
z
2r r + ( r)
2
θ
2
ρ
z
−
v
z
2r r + ( r)
2
θ
2
ρ
z+z
t+
+
(v
θ
r zρ)
θ
− (v
θ
r zρ)
θ+θ
t =
θ
2
2r r + ( r)
2
z
(φρ)
t+t
− (φρ)
t
t z
θ
2
2r r + ( r)
2
(v
r
rρ)
r
− (v
r
rρ)
r+r
1
2
2r r + ( r)
2
+
(v
z
ρ)
z
− (v
z
ρ)
z+z
z
+
(v
θ
ρ)
θ
− (v
θ
ρ)
θ+θ
θ
2
[2r+ r]
=
(φρ)
t+t
− (φρ)
t
t
lim
r→0
(v
r
rρ)
r
− (v
r
rρ)
r+r
1
2
2r r + ( r)
2
= −
1
r
∂ (v
r
rρ)
∂r
lim
z→0
(v
z
ρ)
z
− (v
z
ρ)
z+z
z
= −
∂ (v
z
ρ)
∂z
lim
r→0,θ→0
(v
θ
ρ)
θ
− (v
θ
ρ)
θ+θ
θ
2
[2r+ r]
= −
1
r
∂ (v
θ
ρ)
∂θ
lim
t→0
(φρ)
t+t
− (φρ)
t
t
=
∂ (φρ)
∂t
1
r
∂ (v
r
rρ)
∂r
+
∂ (v
z
ρ)
∂z
+
1
r
∂ (v
θ
ρ)
∂θ
= −
∂ (φρ)
∂t
1
r
∂ (v
r
rρ)
∂r
= −
∂ (φρ)
∂t
v
r
= −
k
µ(p)
∂p
∂r
1
r
∂
∂r
k
µ(p)
∂p
∂r
rρ
=
∂ (φρ)
∂t
∂ (ρφ)
∂t
= ρ
∂φ
∂t
+ φ
∂ρ
∂t
= ρφc
f
∂p
∂t
+ ρφc
g
(p)
∂p
∂t
= ρφ (c
g
(p) + c
f
)
∂p
∂t
c
g
c
f
c
f
∂ (ρφ)
∂t
= ρφ (c
g
(p) + c
f
)
∂p
∂t
= ρφc
g
(p)
∂p
∂t
1
r
∂
∂r
k
µ(p)
∂p
∂r
rρ
= ρφc
g
(p)
∂p
∂t
(ρφxyz)
t+t
−(ρφxyz)
t
=
(ρv
x
yz)
x
−(ρv
x
yz)
x+x
t+
+
(ρv
y
xz)
y
−(ρv
y
xz)
y+y
t +
(ρv
z
xy)
z
−(ρv
z
xy)
z+z
t
ρ φ v
x
v
y
v
z
x y z
xyzt
(ρφ)
t+t
−(ρφ)
t
t
=
(ρv
x
)
x
−(ρv
x
)
x+x
x
+
+
(ρv
y
)
y
−(ρv
y
)
y+y
y
+
(ρv
z
)
z
−(ρv
z
)
z+z
z
x → 0 y → 0 z → 0 t → 0
∂ (ρv
x
)
∂x
+
∂ (ρv
y
)
∂y
+
∂ (ρv
z
)
∂z
= −
∂ (ρφ)
∂t
∂
∂x
ρ
k
x
µ (p)
∂p
∂x
+
∂
∂y
ρ
k
y
µ (p)
∂p
∂y
+
∂
∂z
ρ
k
z
µ (p)
∂p
∂z
=
∂ (ρφ)
∂t
∂
∂x
p
z (p)
k
x
µ (p)
∂p
∂x
+
∂
∂y
p
z (p)
k
y
µ (p)
∂p
∂y
+
∂
∂z
p
z (p)
k
z
µ (p)
∂p
∂z
=
p
z (p)
φc
g
(p)
∂p
∂t
µ k
1
r
∂
∂r
ρr
∂p
∂r
=
µ
k
ρφc
g
(p)
∂p
∂t
c
g
=
1
p
−
1
z(p)
dz(p)
dp
c
g
=
1
p
ρ =
Mp
RT
1
r
∂
∂r
M
RT
pr
∂p
∂r
=
µ
k
φ
M
RT
∂p
∂t
1
r
∂
∂r
r
∂p
2
∂r
=
µφ
kp
∂p
2
∂t
c
g
=
1
p
1
r
∂
∂r
r
∂p
2
∂r
=
φµc
g
k
∂p
2
∂t
p(r, t = 0) = p
i
⇒ p
2
(r, t = 0) = p
2
i
lim
r→∞
p(r, t) = p
i
⇒ lim
r→∞
p
2
(r, t) = p
2
i
lim
r→0
r
∂p
∂r
=
q
w
µ
2πkh
⇒ lim
r→0
r
∂p
2
∂r
=
q
w
µ
πkh
p
w
X =
φµc
g
r
2
4kt
1
r
∂
∂X
r
∂p
2
∂X
∂X
∂r
∂X
∂r
=
φµc
g
k
∂p
2
∂X
∂X
∂t
∂X
∂r
=
φµc
g
r
2kt
∂X
∂t
= −
φµc
g
r
2
4kt
2
1
r
∂
∂X
r
φµc
g
r
2kt
∂p
2
∂X
φµc
g
r
2kt
= −
φµc
g
r
2
4kt
2
φµc
g
k
∂p
2
∂X
∂
∂X
X
∂p
2
∂X
= −X
∂p
2
∂X
X
∂
∂X
∂p
2
∂X
+
∂p
2
∂X
= −X
∂p
2
∂X
p
2
(r, t = 0) = p
2
(X = ∞) = p
2
i
lim
r→∞
p
2
(r, t) = lim
X→∞
p
2
(X) = p
2
i
lim
r→0
r
∂p
2
∂r
= lim
X→0
2X
∂p
2
∂X
=
q
w
µ
πkh
p
w
∂p
2
∂X
= Y
X
∂Y
∂X
+ Y = −XY
dY
Y
= −
dX
X
− dX
dY
Y
= −
dX
X
−
dX
Y = C
1
exp(−X)
X
dp
2
= C
1
exp(−X)
X
dX
p
i
p
dp
2
= C
1
∞
X
exp(−X)
X
dX
p
2
i
− p
2
(r, t) = C
1
[E
i
(X)]
∞
X
exp(−X)
X
dX = −E
i
(−X) = E
i
(X)
C
1
lim
X→0
2X
∂p
2
∂X
= lim
X→0
(2C
1
exp(−X)) =
q
w
µ
πkh
p
w
C
1
=
q
w
µ
2πkh
p
w
C
1
p
2
(r, t) = p
2
i
+
q
w
µ
2πkh
p
w
E
i
(−X)
z = 1
p
0
q
0
z
0
T
0
=
p
w
q
w
z
w
T
w
p
0
q
0
T
0
=
p
w
q
w
T
w
p
2
(r, t) = p
2
i
+
µ
2πkh
p
0
q
0
T
w
T
0
E
i
−
φµc
g
r
2
4kt
p
0
q
0
T
0
∂p
∂t
= 0 ⇒
∂p
2
∂t
= 0
r = r
e
⇒ p(r = r
e
) = p
e
⇒ p
2
(r = r
e
) = p
2
e
r = r
w
⇒ p(r = r
w
) = p
w
⇒ p
2
(r = r
w
) = p
2
w
1
r
d
dr
r
dp
2
dr
= 0
r
dp
2
dr
= C
1
(= )
p(r)
p(r
w
)
dp
2
= C
1
r
r
w
dr
r
p
2
(r) − p
2
(r
w
) = C
1
ln
r
r
w
p
2
(r
e
) − p
2
(r
w
) = C
1
ln
r
e
r
w
C
1
=
p
2
e
− p
2
w
ln
r
e
r
w
C
1
p
2
(r) = p
2
w
+
p
2
e
− p
2
w
ln
r
e
r
w
ln
r
r
w
∂p
∂r
r=r
e
= 0 ⇒
∂p
2
∂r
r=r
e
= 0
r
∂p
∂r
r=r
w
=
µq
w
2πkh
⇒
r
∂p
2
∂r
r=r
w
=
p
w
q
w
µ
πkh
∂p
∂t
= C
1
(= )
∂
∂r
r
∂p
2
∂r
=
2φµ
k
C
1
r
(r = r
w
) (r = r
e
)
r
e
r
w
∂
∂r
r
∂p
2
∂r
dr =
r
e
r
w
2φµ
k
C
1
rdr
r
∂p
2
∂r
r
e
−r
∂p
2
∂r
r
w
=
2φµ
k
C
1
(r
2
e
− r
2
w
)
2
p
w
q
w
µ
πkh
= −
φµ
k
C
1
r
2
e
− r
2
w
z
0
= z
w
= 1
q
0
p
0
T
w
T
0
πh
= −φC
1
r
2
e
− r
2
w
C
1
C
1
= −
q
0
p
0
T
w
T
0
φπh
1
(r
2
e
− r
2
w
)
r
∂p
2
∂r
= Y
∂Y
∂r
=
2φµ
k
C
1
r
C
1
∂Y
∂r
= −
2q
0
p
0
T
w
µ
T
0
πkh
1
(r
2
e
− r
2
w
)
r
∂Y = −
2q
0
p
0
T
w
µ
T
0
πkh
1
(r
2
e
− r
2
w
)
r∂r
∂Y = −
2q
0
p
0
T
w
µ
T
0
πkh
1
(r
2
e
− r
2
w
)
r∂r
Y = −
q
0
p
0
T
w
µ
T
0
πkh
r
2
(r
2
e
− r
2
w
)
+ C
2
r
∂p
2
∂r
= −
q
0
p
0
T
w
µ
T
0
πkh
r
2
(r
2
e
− r
2
w
)
+ C
2
∂p
2
∂r
r
e
= 0 = −
q
0
p
0
T
w
µ
T
0
πkh
r
(r
2
e
− r
2
w
)
r
e
+
C
2
r
r
e
C
2
=
q
0
p
0
T
w
µ
T
0
πkh
r
2
e
(r
2
e
− r
2
w
)
r
∂p
2
∂r
= −
q
0
p
0
T
w
µ
T
0
πkh
r
2
(r
2
e
− r
2
w
)
+
q
0
p
0
T
w
µ
T
0
πkh
r
2
e
(r
2
e
− r
2
w
)
∂p
2
= −
q
0
p
0
T
w
µ
T
0
πkh
r
(r
2
e
− r
2
w
)
∂r −
r
2
e
(r
2
e
− r
2
w
)
∂r
r
p
p
w
∂p
2
= −
q
0
p
0
T
w
µ
T
0
πkh
r
r
w
r
(r
2
e
− r
2
w
)
∂r −
r
r
w
r
2
e
(r
2
e
− r
2
w
)
∂r
r
p
2
− p
2
w
= −
q
0
p
0
T
w
µ
T
0
πkh
(r
2
− r
2
w
)
2 (r
2
e
− r
2
w
)
−
r
2
e
)
(r
2
e
− r
2
w
)
ln
r
r
w
r
e
r
w
p
2
− p
2
w
= −
q
0
p
0
T
w
µ
T
0
πkh
1
2
r
r
e
2
− ln
r
r
w
m(p)
1
r
∂
∂r
r
k
µ(p)
p
z(p)
∂p
∂r
= φc
t
p
z(p)
∂p
∂t
l = r t
1
r
∂
∂r
r
∂m(p)
∂r
=
φµ(p)c
t
(p)
k
∂m(p)
∂t
p(r, t = 0) = p
i
⇒ m(p) = m(p
i
)
lim
r→∞
p(r, t) = p
i
⇒ lim
r→∞
m(p) = m(p
i
)
lim
r→0
r
∂p
∂r
=
q
w
µ
2πkh
⇒ lim
r→0
r
∂m(p)
∂r
=
q
w
p
w
z
w
πkh
X =
φ (µc
t
)
i
r
2
4kt
µc
t
(µc
t
)
i
1
r
∂
∂X
r
∂m(p)
∂X
∂X
∂r
∂X
∂r
=
φ (µc
t
)
i
k
∂m(p)
∂X
∂X
∂t
∂X
∂r
=
φ (µc
t
)
i
r
2kt
∂X
∂t
= −
φ (µc
t
)
i
r
2
4kt
2
1
r
∂
∂X
r
∂m(p)
∂X
φ (µc
t
)
i
r
2kt
φ (µc
t
)
i
r
2kt
=
φ (µc
t
)
i
k
−
φ (µc
t
)
i
r
2
4kt
2
∂m(p)
∂X
∂
∂X
X
∂m(p)
∂X
= −X
∂m(p)
∂X
X
d
dX
dm(p)
dX
+
dm(p)
dX
= −X
dm(p)
dX
p(r, t = 0) = p(X = ∞) = p
i
⇒ m(p) = m(p
i
)
lim
r→∞
p(r, t) = lim
X→∞
p(X) = p
i
⇒ lim
X→∞
m(p) = m(p
i
)
lim
r→0
r
∂m(p)
∂r
= lim
X→0
2X
∂m(p)
∂X
=
q
w
p
w
z
w
πkh
dm(p)
dX
= Y
X
dY
dX
+ Y = −XY
dY
Y
= −
dX
X
(1 + X)
dY
Y
= −
dX −
dX
X
Y = C
1
exp(−X)
X
Y
dm(p)
dX
= C
1
exp(−X)
X
m(p
i
)
m(p)
dm(p) = C
1
∞
X
exp(−X)
X
dX
m(p
i
) − m(p) = −C
1
E
i
(−X)
C
1
lim
X→0
2X
∂X
∂r
= lim
X→0
2XC
1
exp(−X)
X
=
q
w
p
w
z
w
πkh
C
1
=
q
w
p
w
2z
w
πkh
m(p
i
) − m(p) = −
q
w
p
w
2z
w
πkh
E
i
(−X)
m(p) = m(p
i
) +
q
0
p
0
T
w
2T
0
πkh
E
i
−
φ (µc
t
)
i
r
2
4kt
∂m(p)
∂t
= 0
1
r
∂
∂r
r
∂m(p)
∂r
= 0
r = r
e
⇒ p(r = r
e
, t) = p
e
⇒ m(p) = m(p
e
)
r = r
w
⇒ p(r = r
w
, t) = p
w
⇒ m(p) = m(p
w
)
r
∂m(p)
∂r
= C
1
(= )
(r = r
w
)
p
p
w
∂m(p) =
r
r
w
C
1
r
∂r
m(p) − m(p
w
) = C
1
ln
r
r
w
m(p
e
) − m(p
w
) = C
1
ln
r
e
r
w
C
1
=
m(p
e
) − m(p
w
)
ln
r
e
r
w
m(p) − m(p
w
) =
m(p
e
) − m(p
w
)
ln
r
e
r
w
ln
r
r
w
r = r
e
⇒
∂p
∂r
r=r
e
= 0 ⇒
∂m(p)
∂r
r=r
e
=
2p
µz
∂p
∂r
r=r
e
= 0
r
∂p
∂r
r=r
w
=
q
w
µ
2πkh
⇒
r
∂m(p)
∂r
r=r
w
=
q
w
p
w
z
w
πkh
∂m(p)
∂t
= C
1
(= )
µc
t
(µc
t
)
i
1
r
∂
∂r
r
∂m(p)
∂r
=
φ (µc
t
)
i
k
C
1
(r = r
w
) (r = r
e
)
r
e
r
w
∂
∂r
r
∂m(p)
∂r
dr =
r
e
r
w
φ (µc
t
)
i
k
C
1
rdr
r
∂m(p)
∂r
r
e
−r
∂m(p)
∂r
r
w
=
φ (µc
t
)
i
k
C
1
(r
2
e
− r
2
w
)
2
−
q
w
p
w
z
w
1
πkh
=
φ (µc
t
)
i
k
C
1
(r
2
e
− r
2
w
)
2
−
q
0
p
0
T
w
T
0
1
πkh
=
φ (µc
t
)
i
k
C
1
(r
2
e
− r
2
w
)
2
C
1
C
1
= −
q
0
p
0
T
w
T
0
2
πhφ (µc
t
)
i
(r
2
e
− r
2
w
)
r
∂m(p)
∂r
= Y
∂Y
∂r
=
φ (µc
t
)
i
k
C
1
r
∂Y =
φ (µc
t
)
i
k
C
1
r∂r
Y =
φ (µc
t
)
i
2k
C
1
r
2
+ C
2
Y
r
∂m(p)
∂r
=
φ (µc
t
)
i
2k
C
1
r
2
+ C
2
∂m(p)
∂r
r
e
= 0 =
φ (µc
t
)
i
2k
C
1
r
r
e
+
C
2
r
r
e
C
2
C
2
= −
φ (µc
t
)
i
2k
C
1
r
2
e
=
q
0
p
0
T
w
T
0
r
2
e
πkh (r
2
e
− r
2
w
)
∂m(p)
∂r
=
φ (µc
t
)
i
2k
C
1
r +
C
2
r
(r = r
w
)
(r)
p
p
w
∂m(p) =
r
r
w
φ (µc
t
)
i
2k
C
1
r∂r +
r
r
w
C
2
r
∂r
m(p) − m(p
w
) =
φ (µc
t
)
i
2k
C
1
(r
2
− r
2
w
)
2
+ C
2
ln
r
r
w
C
1
C
2
m(p) − m(p
w
) = −
q
0
p
0
T
w
2T
0
πkh
(r
2
− r
2
w
)
(r
2
e
− r
2
w
)
+
q
0
p
0
T
w
T
0
πkh
r
2
e
(r
2
e
− r
2
w
)
ln
r
r
w
m(p) − m(p
w
) = −
q
0
p
0
T
w
2T
0
πkh
(r
2
− r
2
w
)
(r
2
e
− r
2
w
)
−
2r
2
e
(r
2
e
− r
2
w
)
ln
r
r
w
r
e
r
w
m(p) − m(p
w
) = −
q
0
p
0
T
w
T
0
πkh
1
2
r
r
e
2
− ln
r
r
w
f : R → R f (t) = 0, t > 0. f
L(f (t)) = F (s) =
∞
0
f (t) e
−st
dt,
f (t) =
1
2πi
c+∞i
c−∞i
F (z) e
tz
dz
L
L
d
n
f(t)
dt
n
= s
n
F (s) − s
n−1
f (0) − s
n−2
f
(
)
(0) − ···− f
(n−1)
(0)
L
e
bt
f (t)
= F (s − b)
L(f (t − a) H (t − a)) = e
−as
F (s) , H (t) = 0 t < 0 H (t) = 1 t ≥ 0
L
−1
[F (s − a
1
)] = e
a
1
t
f (t)
L
−1
e
−a
1
s
F (s)
= f (t − a
1
) H (t − a
1
) =
f (t − a
1
) t > a
1
0 t < a
1
L
−1
e
−a
1
√
s
√
s
=
e
−
a
2
1
4t
√
πt
L
−1
F (s)
s
=
t
0
f (u) du
F (s) =
e
−y
D
√
s+a
√
s + a
=
G (s + a) G (s) =
e
−y
D
√
s
√
s
L
−1
(F (s)) = L
−1
(G (s + a)) = e
−at
e
−
y
2
D
4t
√
πt
L
−1
e
−y
D
√
s+a
s
√
s + a
=
t
D
0
e
−au
e
−
y
2
d
4u
√
πu
du
L
−1
e
−st
D0
e
−y
D
√
s+a
s
√
s + a
=
t
D
−t
D0
0
e
−au
e
−
y
2
d
4u
√
πu
du t
D
> t
D0
0 t
D
< t
D0
G
mn
(t
D
, y
D
)
G
mn
(y
D
, t
D
) =
t
D
0
e
−au
e
−
y
2
d
4u
√
πu
du
v =
√
u dv =
1
2
u
−
1
2
du du = 2vdv
G
mn
(y
D
, t
D
) =
1
√
π
√
t
D
0
e
−av
2
e
−
y
2
d
4v
2
v
2vdv =
2
√
π
√
t
D
0
e
−av
2
e
−
y
2
d
4v
2
dv
b =
y
2
d
4
G
mn
(y
D
, t
D
) =
2
√
π
√
t
D
0
e
−
au
2
+
b
u
2
du
G
mn
(y
D
, t
D
) =
1
√
a
√
π
√
t
D
0
e
−
au
2
+
b
u
2
√
a −
√
b
u
2
+
√
a +
√
b
u
2
du
G
mn
(y
D
, t
D
) =
1
√
a
√
π
√
t
D
0
e
−
au
2
+
b
u
2
√
a −
√
b
u
2
du +
√
t
D
0
e
−
au
2
+
b
u
2
√
a +
√
b
u
2
du
=
√
t
D
0
e
−
au
2
+
b
u
2
√
a −
√
b
u
2
du
=
√
t
D
0
e
−
au
2
+
b
u
2
√
a +
√
b
u
2
du
=
√
t
D
0
e
−
√
au+
√
b
u
2
+2
√
a
√
b
√
a −
√
b
u
2
du = e
2
√
a
√
b
√
t
D
0
e
−
√
au+
√
b
u
2
√
a −
√
b
u
2
du
η =
√
au +
√
b
u
e
2
√
a
√
b
√
t
D
0
e
−
√
au+
√
b
u
2
√
a −
√
b
u
2
du = e
2
√
a
√
b
√
a
√
t
D
+
√
b
√
t
D
∞
e
−η
2
dη
= e
2
√
a
√
b
−
∞
√
a
√
t
D
+
√
b
√
t
D
e
−η
2
dη
= e
2
√
a
√
b
−
∞
0
e
−η
2
dη −
−
√
a
√
t
D
+
√
b
√
t
D
0
e
−η
2
dη
erf
= e
2
√
a
√
b
√
π
2
√
a
√
t
D
+
√
b
√
t
D
−
√
π
2
=
√
t
D
0
e
−
√
au−
√
b
u
2
−2
√
a
√
b
√
a +
√
b
u
2
du
= e
−2
√
a
√
b
√
t
D
0
e
−
√
au−
√
b
u
2
√
a +
√
b
u
2
du
η =
√
au −
√
b
u
e
−2
√
a
√
b
√
t
D
0
e
−
√
au−
√
b
u
2
√
a +
√
b
u
2
du = e
−2
√
a
√
b
√
a
√
t
D
−
√
b
√
t
D
−∞
e
−η
2
dη
= e
−2
√
a
√
b
√
a
√
t
D
−
√
b
√
t
D
0
e
−η
2
dη −
−∞
0
e
−η
2
dη
= e
−2
√
a
√
b
√
π
2
√
a
√
t
D
−
√
b
√
t
D
+
√
π
2
G
mn
(y
D
, t
D
) =
1
√
a
√
π
e
2
√
a
√
b
√
π
2
√
a
√
t
D
+
√
b
√
t
D
−
√
π
2
+
+e
−2
√
a
√
b
√
π
2
√
a
√
t
D
−
√
b
√
t
D
+
√
π
2
G
mn
(y
D
, t
D
) =
1
2
√
a
e
−2
√
a
√
b
√
a
√
t
D
−
√
b
√
t
D
+ 1
− e
2
√
a
√
b
√
a
√
t
D
+
√
b
√
t
D
b
G
mn
(y
D
, t
D
) =
1
2
√
a
e
−y
D
√
a
1 +
√
at
D
−
y
D
2
√
t
D
− e
y
D
√
a
√
at
D
+
y
D
2
√
t
D
m
D
= m
D
(x
D
, y
D
, z
D
, t
D
) x
D
ˆm
D
(n, y
D
, z
D
, t
D
) =
1
0
m
D
cos (nπx
D
) dx
D
x
D
m
D
= F
−1
fx
D
[ ˆm
D
] = ˆm
D
n=0
+2
∞
n=1
ˆm
D
cos (nπx
D
)
m
D
x
D
m
D
1
0
∂
2
m
D
∂y
2
D
cos (nπx
D
) dx
D
=
∂
2
ˆm
D
∂y
2
D
1
0
∂
2
m
D
∂z
2
D
cos (nπx
D
) dx
D
=
∂
2
ˆm
D
∂z
2
D
1
0
∂m
D
∂t
D
cos (nπx
D
) dx
D
=
∂ ˆm
D
∂t
D
m
D
x
D
= 0 1
m
D
x
D
1
0
∂
2
m
D
∂x
2
D
cos (nπx
D
) dx
D
= −(nπ)
2
ˆm
D
u = cos (nπx
D
) dv =
∂
2
m
D
∂x
2
D
dx
D
du = −sin (nπx
D
) nπdx
D
v =
∂m
D
∂x
D
1
0
∂
2
m
D
∂x
2
D
cos (nπx
D
) dx
D
=
cos (nπx
D
)
∂m
D
∂x
D
1
0
+
1
0
∂m
D
∂x
D
sin (nπx
D
) nπdx
D
=
1
0
∂m
D
∂x
D
sin (nπx
D
) nπdx
D
m
D
u = sin (nπx
D
)
dv =
∂m
D
∂x
D
v = m
D
du = cos (nπx
D
) nπdx
D
1
0
∂m
D
∂x
D
sin (nπx
D
) nπdx
D
= nπ
m
D
sin(nπx
D
)
1
0
−
1
0
m
D
cos (nπx
D
) nπdx
D
= −(nπ)
2
ˆm
D
m
D
= m
D
(x
D
, y
D
, z
D
, t
D
) z
D
ˆ
ˆm
D
(x
D
, y
D
, m, t
D
) =
1
0
m
D
cos (mπz
D
) dz
D
m
D
= F
−1
fz
D
[ ˆm
D
] =
ˆ
ˆm
D
m=0
+2
∞
m=1
ˆ
ˆm
D
cos (mπz
D
)
z
D
x
D
m
D
z
D
m
D
1
0
∂
2
m
D
∂y
2
D
cos (mπz
D
) dz
D
=
∂
2
ˆ
ˆm
D
∂y
2
D
1
0
∂
2
m
D
∂x
2
D
cos (mπz
D
) dz
D
=
∂
2
ˆ
ˆm
D
∂x
2
D
1
0
∂m
D
∂t
D
cos (mπz
D
) dz
D
=
∂
ˆ
ˆm
D
∂t
D
m
D
z
D
= 0, 1
m
D
z
D
1
0
∂
2
m
D
∂z
2
D
cos (mπz
D
) dz
D
= −(mπ)
2
ˆ
ˆm
D
(µc
g
)
x z y
k
x
∂
2
m (p)
∂x
2
+ k
y
∂
2
m (p)
∂y
2
+ k
z
∂
2
m (p)
∂z
2
= φµ (p) c
g
(p)
∂m (p)
∂t
µ (p) c
g
(p)
(µc
g
)
i
k
x
∂
2
m (p)
∂x
2
+ k
y
∂
2
m (p)
∂y
2
+ k
z
∂
2
m (p)
∂z
2
= φ (µc
g
)
i
∂m (p)
∂t
k
y
k
x
k
y
∂
2
m (p)
∂x
2
+
∂
2
m (p)
∂y
2
+
k
z
k
y
∂
2
m (p)
∂z
2
=
φ (µc
g
)
i
k
y
∂m (p)
∂t
α
t
k
x
k
y
∂
2
m (p)
∂x
2
+
∂
2
m (p)
∂y
2
+
k
z
k
y
∂
2
m (p)
∂z
2
=
φ (µc
g
)
i
α
t
k
y
∂m (p)
∂t
p(x, y, z, t = 0) = p
i
p y → ∞
lim
y→∞
p(x, y, z, t) = p
i
lim
y→0
(L
zb
− L
za
) (L
xl
− L
xd
)
∂p
∂y
=
2πα
p
q
w
µ (p)
2k
y
L
xd
≤ x ≤ L
xl
L
za
≤ z ≤ L
zb
t < t
0
0 L
xd
≤ x ≤ L
xl
L
za
≤ z ≤ L
zb
t > t
0
z
∂p
∂z
z=0
=
∂p
∂z
z=h
z
= 0
x
∂p
∂x
x=0
=
∂p
∂x
x=h
x
= 0
p
m(p)
m (p) = m (p
i
)
lim
y→∞
m (p) = m (p
i
)
lim
y→0
(L
zb
− L
za
) (L
xl
− L
xd
)
∂m (p)
∂y
=
2πα
p
q
0
p
0
T
w
z
0
k
y
L
xd
≤ x ≤ L
xl
L
za
≤ z ≤ L
zb
t < t
0
0 L
xd
≤ x ≤ L
xl
L
za
≤ z ≤ L
zb
t > t
0
∂m (p)
∂z
z=0
=
∂m (p)
∂z
z=h
z
= 0
∂m (p)
∂x
x=0
=
∂m (p)
∂x
x=h
x
= 0
x
D
=
x
h
x
y
D
=
y
(L
zb
− L
za
)
z
D
=
z
h
z
t
D
=
α
t
k
y
t
φ (µc
g
)
i
(L
zb
− L
za
)
2
ν
x
=
(L
zb
− L
za
)
h
x
k
x
k
y
ν
z
=
(L
zb
− L
za
)
h
z
k
z
k
y
m
D
=
z
w
k
y
(L
xl
− L
xd
) m (p)
2πα
p
q
w
p
w
=
T
0
k
y
(L
xl
− L
xd
) m (p)
2πα
p
q
0
p
0
T
w
m (p) = m (p
i
) − m (p)
m
D
x
∂m
D
∂x
=
∂m
D
∂m (p)
∂m (p)
∂x
= −
T
0
k
y
(L
xl
− L
xd
)
2πα
p
q
0
p
0
T
w
∂m (p)
∂x
∂m
D
∂x
=
∂m
D
∂x
D
∂x
D
∂x
=
1
h
x
∂m
D
∂x
D
∂m (p)
∂x
= −
2πα
p
q
0
p
0
T
w
T
0
k
y
(L
xl
− L
xd
)
1
h
x
∂m
D
∂x
D
y
∂m
D
∂y
=
∂m
D
∂y
D
∂y
D
∂y
=
1
(L
zb
− L
za
)
∂m
D
∂y
D
∂m (p)
∂y
= −
2πα
p
q
0
p
0
T
w
T
0
k
y
(L
xl
− L
xd
)
1
(L
zb
− L
za
)
∂m
D
∂y
D
z
∂m
D
∂z
=
∂m
D
∂z
D
∂z
D
∂z
=
1
h
z
∂m
D
∂z
D
∂m (p)
∂z
= −
2πα
p
q
0
p
0
T
w
T
0
k
y
(L
xl
− L
xd
)
1
h
z
∂m
D
∂z
D
m
D
∂m
D
∂t
=
∂m
D
∂t
D
∂t
D
∂t
=
α
t
k
y
φ (µc
g
)
i
(L
zb
− L
za
)
2
∂m
D
∂t
D
∂m (p)
∂t
= −
2πα
p
q
0
p
0
T
w
T
0
k
y
(L
xl
− L
xd
)
α
t
k
y
φ (µc
g
)
i
(L
zb
− L
za
)
2
∂m
D
∂t
D
ν
2
x
∂
2
m
D
∂x
2
D
+
∂
2
m
D
∂y
2
D
+ ν
2
z
∂
2
m
D
∂z
2
D
=
∂m
D
∂t
D
m
D
m(p)
m
D
(x
D
, y
D
, z
D
, t
D
= 0) = 0
lim
y
D
→∞
m
D
= 0, t
D
> 0
(L
zb
− L
za
) (L
xl
− L
xd
)
∂m (p)
∂y
= −
2πα
p
q
0
p
0
T
w
T
0
k
y
∂m
D
∂y
D
lim
y→0
(L
zb
− L
za
) (L
xl
− L
xd
)
∂m (p)
∂y
= lim
y
D
→0
(−
2πα
p
q
0
p
0
T
w
T
0
k
y
∂m
D
∂y
D
) =
2πα
p
q
0
p
0
T
w
T
0
k
y
lim
y
D
→0
∂m
D
∂y
D
=
−1
L
xd
h
x
≤ x
D
≤
L
xl
h
x
L
za
h
z
≤ z
D
≤
L
zb
h
z
t
D
<= t
D0
0
L
xd
h
x
≤ x
D
≤
L
xl
h
x
L
za
h
z
≤ z
D
≤
L
zb
h
z
t
D
> t
D0
∂m
D
∂z
D
z
D
=0
=
∂m
D
∂z
D
z
D
=1
= 0
∂m
D
∂x
D
x
D
=0
=
∂m
D
∂x
D
x
D
=1
= 0
x
D
1
0
ν
2
x
∂
2
m
D
∂x
2
D
cos (nπx
D
) dx
D
+
1
0
∂
2
m
D
∂y
2
D
cos (nπx
D
) dx
D
+
1
0
ν
2
z
∂
2
m
D
∂z
2
D
cos (nπx
D
) dx
D
=
=
1
0
∂m
D
∂t
D
cos (nπx
D
) dx
D
∂
2
ˆm
D
∂y
2
D
+ ν
2
z
∂
2
ˆm
D
∂z
2
D
− (nπν
x
)
2
ˆm
D
=
∂ ˆm
D
∂t
D
ˆm
D
x
D
m
D
ˆm
D
= 0 t
D
= 0
lim
y
D
→∞
ˆm
D
= 0
lim
y
D
→0
∂ ˆm
D
∂y
D
= −
1
0
cos (nπx
D
) dx
D
= −
L
xl
h
x
L
xd
h
x
cos (nπx
D
) dx
D
= −
1
nπ
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
L
za
h
z
≤ z
D
≤
L
zb
h
z
t
D
< t
D0
∂ ˆm
D
∂z
D
z
D
=0
=
∂ ˆm
D
∂z
D
z
D
=1
= 0
z
D
1
0
∂
2
ˆm
D
∂y
2
D
cos (mπz
D
) dz
D
+
1
0
ν
2
z
∂
2
ˆm
D
∂z
2
D
cos (mπz
D
) dz
D
−
1
0
(nπν
x
)
2
ˆm
D
cos (mπz
D
) dz
D
=
=
1
0
∂ ˆm
D
∂t
D
cos (mπz
D
) dz
D
∂
ˆ
ˆm
D
∂t
D
=
∂
2
ˆ
ˆm
D
∂y
2
D
−
(nπν
x
)
2
+ (mπν
z
)
2
ˆ
ˆm
D
ˆ
ˆm
z
D
ˆm
D
ˆ
ˆm
D
= 0 t
D
= 0
lim
y
D
→∞
ˆ
ˆm
D
= 0
lim
y
D
→0
∂
ˆ
ˆm
D
∂y
D
= −
1
nπ
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
1
0
cos (mπz
D
) dz
D
= −
1
mnπ
2
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
×
×
sin
mπL
zb
h
z
− sin
mπL
za
h
z
∂
2
ˆ
ˆ
ˆm
D
∂y
2
D
−
s + (nπν
x
)
2
+ (mπν
z
)
2
ˆ
ˆ
ˆm
D
= 0
ˆ
ˆ
ˆm
D
,
lim
y
D
→∞
ˆ
ˆ
ˆm
D
= 0
lim
y
D
→0
∂
ˆ
ˆ
ˆm
D
∂y
D
=
1
mnπ
2
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
sin
mπL
zb
h
z
− sin
mπL
za
h
z
∞
0
e
−st
D
dt
D
= −
1
mnπ
2
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
×
×
sin
mπL
zb
h
z
− sin
mπL
za
h
z
−
e
−st
D
s
t
D0
0
= −
1
mnπ
2
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
×
×
sin
mπL
zb
h
z
− sin
mπL
za
h
z
1 − e
−st
D0
s
a
1
ˆ
ˆ
ˆm
D
+ a
0
ˆ
ˆ
ˆm
D
= 0
a
0
= −
s + (nπν
x
)
2
+ (mπν
z
)
2
a
1
= 1
a
1
λ
2
+ a
0
= 0,
λ
1
=
s + (nπν
x
)
2
+ (mπν
z
)
2
λ
2
= −
s + (nπν
x
)
2
+ (mπν
z
)
2
ˆ
ˆ
ˆm
D
= B
mn
e
λ
1
y
D
+ A
mn
e
λ
2
y
D
⇒
ˆ
ˆ
ˆm
D
= B
mn
e
√
s+(nπν
x
)
2
+(mπν
z
)
2
y
D
+ A
mn
e
−
√
s+(nπν
x
)
2
+(mπν
z
)
2
y
D
lim
y
D
→∞
ˆ
ˆ
ˆm
D
= lim
y
D
→∞
B
mn
e
√
s+(nπν
x
)
2
+(mπν
z
)
2
y
D
+ A
mn
e
−
√
s+(nπν
x
)
2
+(mπν
z
)
2
y
D
= 0
lim
y
D
→∞
B
mn
e
√
s+(nπν
x
)
2
+(mπν
z
)
2
y
D
= 0 ⇒ B
mn
= 0
lim
y
D
→0
∂
ˆ
ˆ
ˆm
D
∂y
D
= lim
y
D
→0
−
s + (nπν
x
)
2
+ (mπν
z
)
2
A
mn
e
−
√
s+(nπν
x
)
2
+(mπν
z
)
2
y
D
= −
1
mnπ
2
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
×
sin
mπL
zb
h
z
− sin
mπL
za
h
z
1 − e
−st
D0
s
A
mn
=
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
sin
mπL
zb
h
z
− sin
mπL
za
h
z
1 − e
−st
D0
s
mnπ
2
s + (nπν
x
)
2
+ (mπν
z
)
2
ˆ
ˆ
ˆm
D
=
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
sin
mπL
zb
h
z
− sin
mπL
za
h
z
1 − e
−st
D0
s
mnπ
2
s + (nπν
x
)
2
+ (mπν
z
)
2
×
×e
−
√
s+(nπν
x
)
2
+(mπν
z
)
2
y
D
ˆ
ˆ
ˆm
D
= C
mn
(1 − e
−st
D0
) e
−y
D
√
s+a
s
√
s + a
C
mn
=
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
sin
mπL
zb
h
z
− sin
mπL
za
h
z
mnπ
2
a = (nπν
x
)
2
+ (mπν
z
)
2
ˆ
ˆm
D
= C
mn
L
−1
e
−y
D
√
s+a
s
√
s + a
−
e
−st
D0
e
−y
D
√
s+a
s
√
s + a
= C
mn
L
−1
e
−y
D
√
s+a
s
√
s + a
− L
−1
e
−st
D0
e
−y
D
√
s+a
s
√
s + a
ˆ
ˆm
D
=
Z
m
Z
n
G
mn
(y
D
, t
D
) t
D
< t
D0
Z
m
Z
n
[G
mn
(y
D
, t
D
) − G
mn
(y
D
, t
D
− t
D0
)] t
D
> t
D0
G
mn
(y
D
, t
D
) =
2
√
π
√
t
D
0
e
−
au
2
+
y
2
d
4u
2
du
G
mn
(y
D
, t
D
− t
D0
) =
2
√
π
√
t
D
−t
D0
0
e
−
au
2
+
y
2
d
4u
2
du
Z
m
=
sin
mπL
zb
h
z
− sin
mπL
za
h
z
mπ
Z
n
=
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
nπ
G
mn
(y
D
, t
D
) =
1
2
√
a
e
−y
D
√
a
1 +
√
at
D
−
y
D
2
√
t
D
− e
y
D
√
a
√
at
D
+
y
D
2
√
t
D
t
D
< t
D0
z
D
ˆm
D
= G
0n
(y
D
, t
D
) Z
m=0
Z
n
+ 2
∞
m=1
G
mn
(y
D
, t
D
) Z
m
Z
n
cos (mπz
D
)
x
D
m
D
= G
00
(y
D
, t
D
) Z
m=0
Z
n=0
+ 2
∞
m=1
G
m0
(y
D
, t
D
) Z
m
Z
n=0
cos (mπz
D
) +
+2
∞
n=1
G
0n
(y
D
, t
D
) Z
m=0
Z
n
+ 2
∞
m=1
G
mn
(y
D
, t
D
) Z
m
Z
n
cos (mπz
D
)
cos (nπx
D
)
m
D
= G
00
(y
D
, t
D
) Z
m=0
Z
n=0
+ 2
∞
m=1
G
m0
(y
D
, t
D
) Z
m
Z
n=0
cos (mπz
D
) +
+2
∞
n=1
G
0n
(y
D
, t
D
) Z
m=0
Z
n
cos (nπx
D
) + 4
∞
n=1
∞
m=1
G
mn
(y
D
, t
D
) Z
m
Z
n
cos (mπz
D
) cos (nπx
D
)
m = 0 n = 0 a = 0
G
00
(y
D
, t
D
) =
2
√
π
√
t
D
0
e
−
y
2
d
4u
2
du
v =
y
2
D
4u
2
=
y
D
2u
dv =
y
D
2
(−u
−2
) du du = −
2
y
D
u
2
dv = −
y
D
2v
2
dv
G
00
(y
D
, t
D
) = −
2
√
π
y
D
4t
D
∞
y
D
2v
2
e
−v
2
dv = −
y
D
√
π
y
D
4t
D
∞
e
−v
2
v
2
dt
G
00
(y
D
, t
D
) = −
y
D
√
π
−
e
−v
2
v
y
D
4t
D
∞
−
y
D
4t
D
∞
−
1
v
−2ve
−v
2
dv
G
00
(y
D
, t
D
) = −
y
D
√
π
−
e
−v
v
y
D
4t
D
∞
−2
y
D
4t
D
∞
e
−v
2
dv
G
00
(y
D
, t
D
) = −
y
D
√
π
−
e
−
y
2
D
4t
D
y
2
D
4t
D
−
√
π
y
2
D
4t
D
− (∞)
G
00
(y
D
, t
D
) = −
y
D
√
π
−
e
−
y
2
D
4t
D
y
D
2
√
t
D
+
√
π
y
2
D
4t
D
G
00
(y
D
, t
D
) =
2
√
t
D
√
π
e
−
y
2
D
4t
D
− y
D
y
D
2
√
t
D
m > 1 n = 0
G
m0
(y
D
, t
D
) =
1
2mπν
z
e
−y
D
mπν
z
1 +
mπν
z
√
t
D
−
y
D
2
√
t
D
+
−e
y
D
mπν
z
mπν
z
√
t
D
+
y
D
2
√
t
D
m = 0 n > 1
G
0n
(y
D
, t
D
) =
1
2nπν
x
e
−y
D
nπν
x
1 +
nπν
x
√
t
D
−
y
D
2
√
t
D
+
−e
y
D
nπν
x
nπν
x
√
t
D
+
y
D
2
√
t
D
m = 0
lim
m→0
Z
m
= lim
m→0
sin
mπL
zb
h
z
− sin
mπL
za
h
z
mπ
=
L
zb
− L
za
h
z
n = 0
lim
n→0
Z
n
= lim
n→0
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
nπ
=
L
xl
− L
xd
h
x
m
D
= G
00
(y
D
, t
D
)
L
zb
− L
za
h
z
L
xl
− L
xd
h
x
+ 2
∞
m=1
G
m0
(y
D
, t
D
) Z
m
L
xl
− L
xd
h
x
cos (mπz
D
) +
+2
∞
n=1
G
0n
(y
D
, t
D
)
L
zb
− L
za
h
z
Z
n
cos (nπx
D
) + 4
∞
n=1
∞
m=1
G
mn
(y
D
, t
D
) Z
m
Z
n
cos (mπz
D
) cos (nπx
D
)
G
mn
y
D
= 0
G
mn
(y
D
= 0, t
D
) =
1
2
√
a
1 +
√
at
D
−
√
at
D
a
G
mn
(y
D
= 0, t
D
) =
√
at
D
√
a
=
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
G
00
(y
D
= 0, t
D
) =
2
√
t
D
√
π
= 2
t
D
π
G
m0
(y
D
= 0, t
D
) =
1
2mπν
z
1 +
mπν
z
√
t
D
−
mπν
z
√
t
D
=
mπν
z
√
t
D
mπν
z
G
0n
(y
D
= 0, t
D
) =
1
2nπν
x
1 +
nπν
x
√
t
D
−
nπν
x
√
t
D
=
nπν
x
√
t
D
nπν
x
m
D
(y
D
= 0) = 2
t
D
π
L
zb
− L
za
h
z
L
xl
− L
xd
h
x
+ 2
∞
m=1
mπν
z
√
t
D
mπν
z
Z
m
L
xl
− L
xd
h
x
cos (mπz
D
) +
+2
∞
n=1
nπν
x
√
t
D
nπν
x
L
zb
− L
za
h
z
Z
n
cos (nπx
D
) +
4
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
Z
m
Z
n
cos (mπz
D
) cos (nπx
D
)
m
D
(y
D
= 0) = 2
t
D
π
L
zb
− L
za
h
z
L
xl
− L
xd
h
x
+ 2
L
xl
− L
xd
h
x
πν
z
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+2
L
zb
− L
za
h
z
πν
x
∞
n=1
nπν
x
√
t
D
n
Z
n
cos (nπx
D
) +
4
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
Z
m
Z
n
cos (mπz
D
) cos (nπx
D
)
m
D
m
D
=
L
xl
h
x
L
xd
h
x
m
D
dx
D
L
xl
− L
xd
h
x
m
D
L
xl
h
x
L
xd
h
x
2
t
D
π
L
zb
− L
za
h
z
L
xl
− L
xd
h
x
dx
D
= 2
t
D
π
L
zb
− L
za
h
z
L
xl
− L
xd
h
x
2
L
xl
h
x
L
xd
h
x
2
L
xl
− L
xd
h
x
πν
z
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) dx
D
=
=
2
πν
z
L
xl
− L
xd
h
x
2
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
)
L
xl
h
x
L
xd
h
x
2
L
zb
− L
za
h
z
πν
x
∞
n=1
nπν
x
√
t
D
n
Z
n
cos (nπx
D
) =
= 2
L
zb
− L
za
h
z
π
2
ν
x
∞
n=1
nπν
x
√
t
D
n
2
Z
n
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
L
xl
h
x
L
xd
h
x
4
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
Z
m
Z
n
cos (mπz
D
) cos (nπx
D
) =
= 4
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
Z
m
Z
n
cos (mπz
D
) ×
×
1
nπ
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
m
D
= 2
t
D
π
L
zb
− L
za
h
z
L
xl
− L
xd
h
x
+
2
πν
z
L
xl
− L
xd
h
x
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+2
h
x
(L
zb
− L
za
)
h
z
π
2
ν
x
(L
xl
− L
xd
)
∞
n=1
nπν
x
√
t
D
n
2
Z
n
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
+
+
4h
x
(L
xl
− L
xd
)
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
Z
m
Z
n
cos (mπz
D
) ×
×
1
nπ
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
m
D
=
2
π
L
zb
− L
za
h
z
L
xl
− L
xd
h
x
√
πt
D
+
1
ν
z
h
z
L
zb
− L
za
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+
1
πν
x
h
x
L
xl
− L
xd
2
∞
n=1
nπν
x
√
t
D
n
2
Z
n
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
+
+2π
h
x
L
xl
− L
xd
2
h
z
L
zb
− L
za
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
×
×Z
m
Z
2
n
cos (mπz
D
)]
m
D
=
2
π
L
zb
− L
za
h
z
L
xl
− L
xd
h
x
√
πt
D
+
1
ν
z
h
z
L
zb
− L
za
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
xl
− L
xd
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
+
+2π
h
x
L
xl
− L
xd
2
h
z
L
zb
− L
za
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
×
×Z
m
Z
2
n
cos (mπz
D
)]
r
w
(L
zb
− L
za
) = 4r
w
k
z
k
y
1
4
= 4r
w
r
w
= r
w
k
z
k
y
1
4
h
s
=
L
zb
+ L
za
2
L
za
= h
s
− 2r
w
L
zb
= h
s
+ 2r
w
t
D
t
D
=
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
Z
m
=
sin
mπ (h
s
+ 2r
w
)
h
z
− sin
mπ (h
s
− 2r
w
)
h
z
mπ
m
D
=
2
π
4r
w
h
z
L
xl
− L
xd
h
x
√
πt
D
+
1
ν
z
h
z
4r
w
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
xl
− L
xd
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
+
+2π
h
x
L
xl
− L
xd
2
h
z
4r
w
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
×
×Z
m
Z
2
n
cos (mπz
D
)]
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
+
1
ν
z
h
z
4r
w
∞
m=1
mπν
z
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
Z
m
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
xl
− L
xd
2
∞
n=1
nπν
x
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
Z
2
n
n
+
+2π
h
x
L
xl
− L
xd
2
h
z
4r
w
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
×
×
Z
m
Z
2
n
cos (mπz
D
)
(nπν
x
)
2
+ (mπν
z
)
2
z
D
=
1
h
z
(h
s
+ 1, 47r
w
)
t
D
> t
D0
t
D
> t
D0
m
D
=
2
π
4r
w
h
z
L
xl
− L
xd
h
x
√
π
√
t
D
−
√
t
D
− t
D0
+
+
1
ν
z
h
z
4r
w
∞
m=1
1
m
mπν
z
√
t
D
−
mπν
z
√
t
D
− t
D0
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
xl
− L
xd
2
∞
n=1
1
n
nπν
x
√
t
D
−
nπν
x
√
t
D
− t
D0
Z
2
n
+
+2π
h
x
L
xl
− L
xd
2
h
z
4r
w
∞
n=1
∞
m=1
1
(nπν
x
)
2
+ (mπν
z
)
2
×
×
(nπν
x
)
2
+ (mπν
z
)
2
t
D
−
(nπν
x
)
2
+ (mπν
z
)
2
(t
D
− t
D0
)
Z
m
Z
2
n
cos (mπz
D
)
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
π
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
√
t −
√
t − t
0
+
+
1
ν
z
h
z
4r
w
∞
m=1
mπν
z
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
−
mπν
z
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
(t − t
0
)
×
×
Z
m
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
xl
− L
xd
2
∞
n=1
nπν
x
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
−
nπν
x
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
(t − t
0
)
Z
2
n
n
+
+2π
h
x
L
xl
− L
xd
2
h
z
4r
w
∞
n=1
∞
m=1
1
(nπν
x
)
2
+ (mπν
z
)
2
×
×
(nπν
x
)
2
+ (mπν
z
)
2
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
−
(nπν
x
)
2
+ (mπν
z
)
2
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
(t − t
0
)
×
×Z
m
Z
2
n
cos (mπz
D
)}
t
D
< t
D0
(L
za
, h
z
− L
zb
)
L
w
k
x
k
z
< 0, 25
(ν
x
πn)
2
∞
n=1
∞
m=1
(nπν
x
)
2
+ (mπν
z
)
2
t
D
(nπν
x
)
2
+ (mπν
z
)
2
Z
m
Z
2
n
cos (mπz
D
)
∼
=
∼
=
∞
n=1
Z
2
n
∞
m=1
mπν
z
√
t
D
mπν
z
Z
m
cos (mπz
D
)
m
D
=
2
π
4r
w
h
z
L
xl
− L
xd
h
x
√
πt
D
+
1
ν
z
h
z
4r
w
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
xl
− L
xd
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
+
+2π
h
x
L
xl
− L
xd
2
h
z
4r
w
∞
n=1
Z
2
n
∞
m=1
mπν
z
√
t
D
mπν
z
Z
m
cos (mπz
D
)
Z
2
n
=
sin
nπL
xl
h
x
− sin
nπL
xd
h
x
2
(nπ)
2
=
=
sin
2
nπL
xl
h
x
− 2 sin
nπL
xl
h
x
sin
nπL
xd
h
x
+ sin
2
nπL
xd
h
x
(nπ)
2
−2 sin
nπL
xl
h
x
sin
nπL
xd
h
x
−2 sin
nπL
xl
h
x
sin
nπL
xd
h
x
=
= −
cos
nπ (L
xl
− L
xd
)
h
x
− cos
nπ (L
xl
+ L
xd
)
h
x
=
= −
1 − 2 sin
2
nπ (L
xl
− L
xd
)
2h
x
− 1 + 2 sin
2
nπ (L
xl
+ L
xd
)
2h
x
=
=
2 sin
2
nπ (L
xl
− L
xd
)
2h
x
− 2 sin
2
nπ (L
xl
+ L
xd
)
2h
x
Z
2
n
=
sin
2
nπL
xl
h
x
+ 2 sin
2
nπ (L
xl
− L
xd
)
2h
x
− 2 sin
2
nπ (L
xl
+ L
xd
)
2h
x
+ sin
2
nπL
xd
h
x
(nπ)
2
∞
m=1
sin
2
(mu)
m
2
=
1
2
u (π − u) 0 ≤ u ≤ π
∞
n=1
sin
2
nπL
xl
h
x
(nπ)
2
=
1
π
2
1
2
πL
xl
h
x
π −
πL
xl
h
x
=
1
2
L
xl
h
x
1 −
L
xl
h
x
∞
n=1
sin
2
nπL
xd
h
x
(nπ)
2
=
1
π
2
1
2
πL
xl
h
x
π −
πL
xd
h
x
=
1
2
L
xd
h
x
1 −
L
xd
h
x
∞
n=1
sin
2
nπ (L
xl
− L
xd
)
2h
x
(nπ)
2
=
1
π
2
1
2
π (L
xl
− L
xd
)
2h
x
π −
π (L
xl
− L
xd
)
2h
x
=
=
1
2
(L
xl
− L
xd
)
2h
x
1 −
(L
xl
− L
xd
)
2h
x
∞
n=1
sin
2
nπ (L
xl
+ L
xd
)
2h
x
(nπ)
2
=
1
π
2
1
2
π (L
xl
+ L
xd
)
2h
x
π −
π (L
xl
+ L
xd
)
2h
x
=
=
1
2
(L
xl
+ L
xd
)
2h
x
1 −
(L
xl
+ L
xd
)
2h
x
∞
n=1
Z
2
n
=
1
2
L
xl
h
x
1 −
L
xl
h
x
+
1
2
L
xd
h
x
1 −
L
xd
h
x
+
(L
xl
− L
xd
)
2h
x
1 −
(L
xl
− L
xd
)
2h
x
+
−
(L
xl
+ L
xd
)
2h
x
1 −
(L
xl
+ L
xd
)
2h
x
=
=
1
2
L
xl
h
x
−
1
2
L
xl
h
x
2
+
1
2
L
xd
h
x
−
1
2
L
xd
h
x
2
+
(L
xl
− L
xd
)
2h
x
−
L
xl
− L
xd
2h
x
2
+
−
L
xl
+ L
xd
2h
x
+
L
xl
+ L
xd
2h
x
2
=
= −
1
2
L
xl
h
x
2
−
1
2
L
xd
h
x
2
+
(L
xl
− L
xd
)
2h
x
−
L
xl
− L
xd
2h
x
2
+
L
xl
+ L
xd
2h
x
2
=
= −
1
2
L
xl
h
x
2
−
1
2
L
xd
h
x
2
+
(L
xl
− L
xd
)
2h
x
−
L
xl
2h
x
2
+
L
xd
2h
x
2
−
2L
xl
L
xd
(2h
x
)
2
+
+
L
xl
2h
x
2
+
L
xd
2h
x
2
+
2L
xl
L
xd
(2h
x
)
2
=
= −
1
2
L
xl
h
x
2
−
1
2
L
xd
h
x
2
+
(L
xl
− L
xd
)
2h
x
+
L
xl
L
xd
(h
x
)
2
=
= −
1
2
L
xl
h
x
2
+
L
xd
h
x
2
−
2L
xl
L
xd
(h
x
)
2
+
(L
xl
− L
xd
)
2h
x
=
= −
1
2
L
xl
− L
xd
h
x
2
+
(L
xl
− L
xd
)
2h
x
=
=
1
2
(L
xl
− L
xd
)
h
x
−
L
xl
− L
xd
h
x
2
=
=
1
2
L
w
h
x
−
L
w
h
x
2
=
=
1
2
L
w
h
x
1 −
L
w
h
x
m
1
ν
z
h
z
4r
w
+
2
ν
z
h
x
L
w
2
h
z
4r
w
1
2
L
w
h
x
1 −
L
w
h
x
=
=
h
z
4r
w
ν
z
1 +
h
x
L
w
− 1
=
h
z
4r
w
ν
z
h
x
L
w
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
+
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
Z
m
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
Z
2
n
n
t
D
> t
D0
t
D
< t
D0
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
π
√
t
D
−
√
t
D
− t
D0
+
+
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
−
mπν
z
√
t
D
− t
D0
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
−
nπν
x
√
t
D
− t
D0
n
Z
2
n
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
√
t −
√
t − t
0
+
+
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
−
mπν
z
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
(t − t
0
)
×
×
Z
m
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
−
nπν
x
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
(t − t
0
)
Z
2
n
n
t
D
< t
D0
u
(u) =
2
√
π
u
< 1% u < 0, 1
h
x
ν
x
π
π
L
w
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
∼
=
π
1
L
w
−
1
h
x
√
πt
D
π
4r
w
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
)
∼
=
∼
=
1, 8089
h
z
log
64k
z
γ
k
y
t
D
−
ν
z
π
√
πt
D
h
z
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
4r
w
L
w
ν
z
4r
w
π
1, 8089
h
z
log
64k
z
γ
k
y
t
D
−
ν
z
π
√
πt
D
h
z
+
+
1
ν
x
h
x
L
w
2
ν
x
π
h
x
L
w
π
2
π
1
L
w
−
1
h
x
√
πt
D
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
1, 8089h
x
L
w
ν
z
π
log
64k
z
γ
k
y
t
D
−
h
x
L
w
√
πt
D
+
+
h
x
L
w
− 1
√
πt
D
m
D
=
2
π
4r
w
h
z
L
w
h
x
1, 8089h
x
L
w
ν
z
π
log
64k
z
γ
k
y
t
D
m
D
=
2 × 1, 8089
π
2
k
y
k
z
log
64k
z
γ
k
y
t
D
γ
= 1, 78108
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
4α
t
k
z
t
γ
φ (µc
g
)
i
(r
w
)
2
u > 1, 8 (u)
∼
=
1
π
4r
w
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
)
∼
=
Ω
z
Ω
z
=
1
8r
w
[ψ (η
1
) + ψ (η
2
) − ψ (η
3
) − ψ (η
4
)]
ψ (η) =
∞
n=1
sin (nη)
n
2
η
1
=
0, 52πr
w
h
z
η
2
=
π
h
z
(2h
s
+ 3, 48r
w
)
η
3
= −
3, 48πr
w
h
z
η
4
=
π
h
z
(2h
s
− 0, 52r
w
)
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
4r
w
L
w
ν
z
4r
w
π
Ω
z
+
+
1
ν
x
h
x
L
w
2
ν
x
π
h
x
L
w
π
2
π
1
L
w
−
1
h
x
√
πt
D
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
πL
w
ν
z
Ω
z
+
h
x
L
w
− 1
√
πt
D
m
D
=
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
√
πt
D
+
h
x
h
z
πL
w
ν
z
Ω
z
S
z
= 0, 6366h
z
Ω
z
=
2
π
h
z
Ω
z
m
D
=
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
√
πt
D
+
h
x
2L
w
ν
z
S
z
m (p) =
2πα
p
q
0
p
0
T
w
T
0
k
y
L
w
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
√
πt
D
+
h
x
2L
w
ν
z
S
z
m (p) =
16α
p
q
0
p
0
T
w
r
w
T
0
k
y
h
z
L
w
√
πt
D
+
2α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
S
z
m (p) =
16α
p
q
0
p
0
T
w
r
w
T
0
k
y
h
z
L
w
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
2α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
S
z
π
L
w
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
∼
=
∼
=
1, 8089π
h
2
x
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
−
ν
x
π
2
√
πt
D
h
2
x
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
4r
w
L
w
ν
z
4r
w
π
Ω
z
+
+
1
ν
x
h
x
L
w
2
L
w
π
2
1, 8089π
h
2
x
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
−
ν
x
π
2
√
πt
D
h
2
x
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
L
w
ν
z
π
Ω
z
+
+
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
−
√
πt
D
m
D
=
2
π
4r
w
h
z
L
w
h
x
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
+
h
x
2L
w
ν
z
S
z
m (p) =
4α
p
q
0
p
0
T
w
T
0
k
y
h
x
4r
w
h
z
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
+
h
x
2L
w
ν
z
S
z
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
k
y
k
x
h
z
log
64α
t
k
x
t
γ
φ (µc
g
)
i
L
2
w
+
2α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
S
z
x
h
x
ν
x
π
π
L
w
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
∼
=
∼
=
h
x
ν
x
π
π
L
w
2
∞
n=1
Z
2
n
n
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
L
w
ν
z
π
Ω
z
+
1
ν
x
h
x
L
w
2
∞
n=1
Z
2
n
n
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
L
w
ν
z
π
Ω
z
+
1
ν
x
h
x
L
w
2
∞
n=1
Z
2
n
n
S
x
=
2h
2
x
h
z
L
w
k
x
k
z
∞
n=1
Z
2
n
n
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
8r
w
L
w
k
y
k
z
(S
z
+ S
x
)
m (p) =
2πα
p
q
0
p
0
T
w
T
0
k
y
L
w
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
8r
w
L
w
k
y
k
z
(S
z
+ S
x
)
m (p) =
16α
p
q
0
p
0
T
w
r
w
T
0
k
y
h
z
h
x
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
2α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
(S
z
+ S
x
)
t
D
> t
D0
t
D0
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
+
−
2
π
4r
w
h
z
L
w
h
x
π (t
D
− t
D0
) +
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
− t
D0
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
− t
D0
n
Z
2
n
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
8r
w
L
w
k
y
k
z
(S
z
+ S
x
)
+
−
2
π
4r
w
h
z
L
w
h
x
π (t
D
− t
D0
) +
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
− t
D0
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
− t
D0
n
Z
2
n
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
8r
w
L
w
k
y
k
z
(S
z
+ S
x
)
−
2 × 1, 8089
π
2
k
y
k
z
log
64k
z
γ
k
y
(t
D
− t
D0
)
m (p) = α
p
2, 303q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
t
t − t
0
− log
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
+
16r
w
L
w
√
k
z
2, 303
k
y
h
z
h
x
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+ log
γ
k
y
64k
z
+
2
2, 303
(S
x
+ S
z
)
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
8r
w
L
w
k
y
k
z
(S
z
+ S
x
)
+
−
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
π (t
D
− t
D0
) +
h
x
2L
w
ν
z
S
z
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
π
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
√
t −
h
x
L
w
√
t − t
0
+ α
p
2q
0
p
0
T
w
T
0
L
w
k
y
k
z
S
x
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
8r
w
L
w
k
y
k
z
(S
z
+ S
x
)
+
−
2
π
4r
w
h
z
L
w
h
x
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
(t
D
− t
D0
)
+
h
x
2L
w
ν
z
S
z
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
k
y
k
x
h
z
log
t
t−t
0
− log
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+ log
L
2
w
γ
k
y
(4r
w
)
2
64k
x
+
+
4r
w
π
√
k
x
1, 8089h
x
k
y
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
h
z
π
√
k
x
2 × 1, 8089L
w
√
k
z
S
x
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
8r
w
L
w
k
y
k
z
(S
z
+ S
x
)
+
−
2
π
4r
w
h
z
L
w
h
x
π (t
D
− t
D0
) +
h
x
h
z
8r
w
L
w
k
y
k
z
(S
z
+ S
x
)
+
m (p) = α
p
16r
w
q
0
p
0
T
w
T
0
k
y
h
z
h
x
π
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
√
t −
√
t − t
0
m
D
=
2
π
4r
w
h
z
L
w
h
x
√
πt
D
+
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
n
Z
2
n
+
−
2
π
4r
w
h
z
L
w
h
x
π (t
D
− t
D0
) +
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
− t
D0
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
− t
D0
n
Z
2
n
m
D
=
2
π
4r
w
h
z
L
w
h
x
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
+
h
x
2L
w
ν
z
S
z
+
−
2
π
4r
w
h
z
L
w
h
x
π (t
D
− t
D0
) +
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
− t
D0
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
− t
D0
n
Z
2
n
m
D
=
2
π
4r
w
h
z
L
w
h
x
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
+
h
x
2L
w
ν
z
S
z
+
−
2 × 1, 8089
π
2
k
y
k
z
log
64k
z
γ
k
y
(t
D
− t
D0
)
m (p) = α
p
2, 303q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
t
t − t
0
− log
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
+
L
w
√
k
z
√
k
x
h
z
log
64α
t
k
x
t
γ
φ (µc
g
)
i
L
2
w
+ log
γ
k
y
64k
z
+
2
2, 303
S
z
m
D
=
2
π
4r
w
h
z
L
w
h
x
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
+
h
x
2L
w
ν
z
S
z
+
−
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
π (t
D
− t
D0
) +
h
x
2L
w
ν
z
S
z
m (p) =
16α
p
q
0
p
0
T
w
r
w
T
0
k
y
h
z
L
w
−
π
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
(t − t
0
) +
2, 303L
w
k
y
16r
w
√
k
x
log
64α
t
k
x
t
γ
φ (µc
g
)
i
L
2
w
m
D
=
2
π
4r
w
h
z
L
w
h
x
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
t
D
+
h
x
2L
w
ν
z
S
z
+
−
2
π
4r
w
h
z
L
w
h
x
1, 8089
ν
x
π
log
(4r
w
)
2
64k
x
L
2
w
γ
k
y
(t
D
− t
D0
)
+
h
x
2L
w
ν
z
S
z
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
k
y
k
x
h
z
log
t
t − t
0
m
D
=
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
√
πt
D
+
h
x
2L
w
ν
z
S
z
+
−
2
π
4r
w
h
z
L
w
h
x
π (t
D
− t
D0
) +
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
− t
D0
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
− t
D0
n
Z
2
n
m
D
=
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
√
πt
D
+
h
x
2L
w
ν
z
S
z
−
2 × 1, 8089
π
2
k
y
k
z
log
64k
z
γ
k
y
(t
D
− t
D0
)
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
t
t − t
0
− log
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+
+
16r
w
√
k
z
2, 303h
z
k
y
π
α
t
k
y
t
φ (µc
g
)
i
(4r
w
)
2
+ log
γ
k
y
64k
z
+
2
2, 303
S
z
m
D
=
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
√
πt
D
+
h
x
2L
w
ν
z
S
z
+
−
2
π
4r
w
h
z
L
w
h
x
h
x
L
w
π (t
D
− t
D0
) +
h
x
2L
w
ν
z
S
z
+ S
z
m (p) =
16α
p
q
0
p
0
T
w
r
w
T
0
k
y
h
z
L
w
π
α
t
k
y
φ (µc
g
)
i
(4r
w
)
2
√
t −
√
t − t
0
m
D
=
2 × 1, 8089
π
2
k
y
k
z
log
64k
z
γ
k
y
t
D
+
−
2
π
4r
w
h
z
L
w
h
x
π (t
D
− t
D0
) +
h
x
h
z
4r
w
L
w
ν
z
∞
m=1
mπν
z
√
t
D
− t
D0
m
Z
m
cos (mπz
D
) +
+
1
ν
x
h
x
L
w
2
∞
n=1
nπν
x
√
t
D
− t
D0
n
Z
2
n
m
D
=
2 × 1, 8089
π
2
k
y
k
z
log
64k
z
γ
k
y
t
D
−
2 × 1, 8089
π
2
k
y
k
z
log
64k
z
γ
k
y
(t
D
− t
D0
)
m (p) =
2, 303α
p
q
0
p
0
T
w
T
0
L
w
k
y
k
z
log
t
t − t
0
k = 1000
144
n
6000
r
w
0, 1
φ 0, 2
T
w
150
P
i
650
2
d
G
0, 7
Z
µ
c
g
c
f
4.97817 × 10
−5 2
L
w
600
k
x
1 × 10
3
1 × 10
−3
k
y
1 × 10
3
1 × 10
−3
k
z
1 × 10
3
1 × 10
−3
q
0
2 × 10
6
2 × 10
7 3
P
0
1.03323
2
T
0
15.5556
h
x
2500 5000 7500 10000
h
y
10000 10000 10000 2500
h
z
50 50 50 50
1 × 10
3
2 × 10
6 3
4 × 10
6 3
6 × 10
6 3
3 × 10
6 3
1 × 10
2
2 × 10
6 3
4 × 10
6 3
6 × 10
6 3
3 × 10
6 3
1 × 10
1
2 × 10
6 3
4 × 10
6 3
6 × 10
6 3
3 × 10
6 3
1 × 10
0
1 × 10
5 3
2 × 10
5 3
3 × 10
5 3
3 × 10
5 3
1 × 10
−1
9 × 10
3 3
1, 8 × 10
4 3
2, 7 × 10
4 3
1, 35 × 10
4 3
1 × 10
−2
1 × 10
3 3
2 × 10
3 3
3 × 10
3 3
1, 5 × 10
3 3
1 × 10
−3
3 × 10
2 3
6 × 10
2 3
9 × 10
2 3
4, 5 × 10
2 3
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