Sóng Rayleigh trong mô hình hai lớp thuần nhất

• u v w 0x 0y 0z
• ˙u ˙v ˙w 0x 0y 0z
•
• α
m
• β
m
• ρ
m
• d
m
• ν
m
• C = c/β
1
• r
s
=
β
1
β
2
• c
v
= r
2

s
• r
d
= ρ
1
/ρ
2
• r
t
= d
1
/d
2
• p
• k = p/c = 2π/λ
• γ
1
= β
2
1
/α
2
1
= (1 −2ν
1
)/2(1 −ν
1
)
• γ
2

= β
2
2
/α
2
2
= (1 −2ν
2
)/2(1 −ν
2
)
• g
α
1
=
√
γ
1
C
2
− 1
• g
β
1
=
√
C
2
− 1
• g

α
2
=

r
2
s
γ
2
C
2
− 1
• g
β
2
=

r
2
s
C
2
− 1
• g
α
m
=

(c/α
m

)
2
− 1
• g
β
m
=

(c/β
m
)
2
− 1
• p
m
= kg
α
m
• q
m
= kg
β
m
• G
m
=
2
C
2
, m = 1, 2

• σ
• τ
•
¯
f
•
¯
β



x
1
c
d
1
ν
1
β
1
ρ
1
d
2
ν
2
β
2
ρ

2
u
1
u
3
u
2
= 0
u = ∇ϕ + ∇∧ ψ
ϕ ψ
ϕ = Φ(x
3
) exp[i(kx
1
− ωt)],
ψ = Ψ(x
3
) exp[i(kx
1
− ωt)].
i =
√
−1 k = 2π/λ ω = 2π/T
T λ
ω = k.c Φ(x
3
) Ψ(x
3
)
Φ(x

3
) = A sinh(px
3
) + B cosh(px
3
),
Ψ(x
3
) = C sinh(qx
3
) + D cosh(qx
3
),
p
2
= −
ω
2
α
2
+ k
2
= k
2

−
c
2
α
2

+ 1

= k
2

−γC
2
+ 1

,
q
2
= −
ω
2
β
2
+ k
2
= k
2

−
c
2
β
2
+ 1

= k

2

−C
2
+ 1

.
C = c/β γ = β
2
/α
2
e
exp[i(kx
1
−ωt)]
U
1
(x
3
) = ikΦ −
dΨ
dx
3
,
U
3
(x
3
) =
dΦ

dx
3
+ ikΨ.
σ
31
= ρβ
2

dU
1
dx
3
+ ikU
3

,
σ
33
= ρα
2

dU
3
dx
3
+ ik(1 −2γ)U
1

.
Φ

1
(x
3
) = A
1
sinh(p
1
x
3
) + B
1
cosh(p
1
x
3
),
Ψ
1
(x
3
) = C
1
sinh(q
1
x
3
) + D
1
cosh(q
1

x
3
),
γ
1
= β
2
1
/α
2
1
= (1 −2ν
1
)/2(1−ν
1
) p
1
= k
√
−C
2
γ
1
+ 1 q
1
= k
√
−C
2
+ 1

x
1
x
3
U
(1)
1
(x
3
) = ikΦ
1
−
dΨ
1
dx
3
,
U
(1)
3
(x
3
) =
dΦ
1
dx
3
+ ikΨ
1
.

σ
(1)
13
= ρ
1
β
2
1

dU
(1)
1
dx
3
+ ikU
(1)
3

,
σ
(1)
33
= ρ
1
α
2
1

dU
(1)

3
dx
3
+ ik(1 −2γ
1
)U
(1)
1

.
Φ
2
(x
3
) = A
2
sinh(p
2
x
3
) + B
2
cosh(p
2
x
3
),
Ψ
2
(x

3
) = C
2
sinh(q
2
x
3
) + D
2
cosh(q
2
x
3
).
x
1
x
3
U
(2)
1
(x
3
) = ikΦ
2
−
dΨ
2
dx
3

,
U
(2)
3
(x
3
) =
dΦ
2
dx
3
+ ikΨ
2
.
σ
(2)
13
= ρ
2
β
2
2

dU
(2)
1
dx
3
+ ikU
(2)

3

,
σ
(2)
33
= ρ
2
α
2
2

dU
(2)
2
dx
3
+ ik(1 −2γ
2
)U
(2)
1

.
γ
2
= β
2
2
/α

2
2
= (1 −2ν
2
)/2(1 −ν
2
), r
s
=
β
1
β
2
,
p
2
= k

−C
2
γ
2
r
2
s
+ 1, q
2
= k

−C

2
r
2
s
+ 1.
σ
(1)
13
(−d
1
) = 0,
σ
(1)
33
(−d
1
) = 0.
U
(1)
1
(0) = U
(2)
1
(0),
U
(1)
3
(0) = U
(2)
3

(0).
σ
(1)
13
(0) = σ
(2)
13
(0),
σ
(1)
33
(0) = σ
(2)
33
(0).
U
(2)
1
(d
2
) = 0,
U
(2)
3
(d
2
) = 0.
A
1
, B

1
, C
1
, D
1
, A
2
, B
2
, C
2
, D
2
F.[A
1
, B
1
, C
1
, D
1
, A
2
, B
2
, C
2
, D
2
]

T
= 0
F =




M
1
M
2
0 0
M
3
M
4
M
5
M
6
M
7
M
8
M
9
M
10
0 0 M
11

M
12




F
M
1
=

2πig
α
1
cosh(kd
1
g
α
1
) −2πig
α
1
sinh(kd
1
g
α
1
)
−(1 + g
2

β
1
) sinh(kd
1
g
α
1
) (1 + g
2
β
1
) cosh(kd
1
g
α
1
)

M
2
=

(1 + g
2
β
1
) sinh(kd
1
g
β

1
) −(1 + g
2
β
1
) cosh(kd
1
g
β
1
)
2πig
β
1
cosh(kd
1
g
β
1
) −2πig
β
1
sinh(kd
1
g
β
1
)

M

3
=

0 i
g
α
1
0

M
4
=

−g
β
1
0
0 i

M
5
=

0 −i
−g
α
2
0

M

6
=

g
β
2
0
0 −i

M
7
=

2ic
r
c
v
g
α
1
0
0 c
r
c
v
(1 + g
2
β
1
)


M
8
=

0 −c
r
c
v
(1 + g
2
β
1
)
2ic
r
c
v
g
β
1
0

M
9
=

−2ig
α
2

0
0 −(1 + g
2
β
2
)

M
10
=

0 1 + g
2
β
2
−2ig
β
2
0

M
11
=

ik sinh(kd
2
g
α
2
) ik cosh(kd

2
g
α
2
)
g
α
2
k cosh(kd
2
g
α
2
) g
α
2
k sinh(kd
2
g
α
2
)

M
12
=

−g
β
2

k cosh(kd
2
g
β
2
) −g
β
2
k sinh(kd
2
g
β
2
)
ik sinh(kd
2
g
β
2
) ik cosh(kd
2
g
β
2
)

det(F ) = 0
F
0
+ F

1
cosh(kd
1
g
α
1
) cosh(kd
1
g
β
1
)
+ F
2
cosh(kd
2
g
α
2
) cosh(kd
2
g
β
2
)
+ F
3
sinh(kd
1
g

α
1
) sinh(kd
1
g
β
1
)
+ F
4
sinh(kd
2
g
α
2
) sinh(kd
2
g
β
2
)
+ F
5
cosh(kd
1
g
β
1
) cosh(kd
2

g
β
2
) sinh(kd
1
g
α
1
) sinh(kd
2
g
α
2
)
+ F
6
cosh(kd
2
g
α
2
) cosh(kd
2
g
β
2
) sinh(kd
1
g
α

1
) sinh(kd
2
g
β
2
)
+ F
7
cosh(kd
2
g
α
2
) cosh(kd
1
g
β
1
) sinh(kd
1
g
α
1
) sinh(kd
2
g
β
2
)

+ F
8
cosh(kd
1
g
α
1
) cosh(kd
2
g
β
2
) sinh(kd
2
g
α
2
) sinh(kd
1
g
β
1
)
+ F
9
cosh(kd
1
g
α
1

) cosh(kd
2
g
α
2
) sinh(kd
1
g
β
1
) sinh(kd
2
g
β
2
)
+ F
10
cosh(kd
1
g
α
1
) cosh(kd
2
g
α
2
) sinh(kd
2

g
α
2
) sinh(kd
2
g
β
2
)
+ F
11
cosh(kd
1
g
α
1
) cosh(kd
2
g
α
2
) cosh(kd
1
g
β
1
) cosh(kd
2
g
β

2
)
+ F
12
sinh(kd
1
g
α
1
) sinh(kd
2
g
α
2
) sinh(kd
1
g
β
1
) sinh(kd
2
g
β
2
) = 0
F
i
, (i =
0, 12)
F

0
= 4g
α
1
g
α
2
g
β
1
g
β
2
(1 + g
2
β
1
)(4r
2
d
c
2
v
(1 + g
2
β
1
) + 4(1 + g
2
β

2
) − r
d
c
v
(3 + g
2
β
1
)(3 + g
2
β
2
))
F
1
= −4g
α
1
g
α
2
g
β
1
g
β
2
(4r
2

d
c
2
v
(1 + g
2
β
1
)
2
+ (5 + 2g
2
β
1
+ g
4
β
1
)(1 + g
2
β
2
) − r
d
c
v
(3 + 4g
2
β
1

+ g
4
β
1
)(3 + g
2
β
2
))
F
2
= −4g
α
1
g
α
2
g
β
1
(1 + g
2
β
1
)g
β
2
(5 + 4r
2
d

c
2
v
(1 + g
2
β
1
) + 2g
2
β
2
+ g
4
β
2
− r
d
c
v
(3 + g
2
β
1
)(3 + g
2
β
2
))
F
3

= −2g
α
2
g
β
2
(−r
2
d
c
2
v
(1 + 4(1 + 4g
2
α
1
)g
2
β
1
+ 6g
4
β
1
+ 4g
6
β
1
+ g
8

β
1
− 2(1 + (2 + 4g
2
α
1
)g
2
β
1
+ g
4
β
1
)(1 + g
2
β
2
) + r
d
c
v
(1 + (3 + 8g
2
α
1
)g
2
β
1

+ 3g
4
β
1
+ g
6
β
1
)(3 + g
2
β
2
))
F
4
= 4g
α
1
g
β
1
(1 + g
2
β
1
)(1 + (2 + 4g
α
2
)g
2

β
2
+ g
4
β
2
+ 2r
2
d
c
2
v
(1 + g
2
β
1
)(1 + g
2
α
2
g
2
β
2
)
− r
d
c
v
(3 + g

2
β
1
)(1 + (1 + 2g
2
α
2
)g
2
β
2
))
F
5
= r
d
c
v
g
β
1
(−1 + g
2
β
1
)(4g
2
α
1
+ g

2
α
2
(1 + g
2
β
1
)
2
)g
β
2
(−1 + g
2
β
2
)
F
6
= −g
α
2
g
β
2
(2r
2
d
c
2

v
(1 + 4(1 + 4g
2
α
1
)g
2
β
1
+ 6g
4
β
1
+ 4g
6
β
1
+ g
8
β
1
)
− 2r
d
c
v
(1 + (3 + 8g
2
α
1

)g
2
β
1
+ 3g
4
β
1
+ g
6
β
1
)(3 + g
2
β
2
) + (1 + (2 + 4g
2
α
1
)g
2
β
1
+ g
4
β
1
)(5 + 2g
2

β
2
+ g
4
β
2
)))
F
7
= −r
d
c
v
g
α
2
g
β
2
(−1 + g
2
β
1
)(−1 + g
2
β
2
)(1 + 2g
2
β

1
+ g
4
β
1
+ 4g
2
α
1
g
2
β
2
)
F
8
= −r
d
c
v
g
α
1
(−1 + g
2
β
1
)(1 + (2 + 4g
2
α

2
)g
2
β
1
+ g
4
β
1
)g
β
2
(−1 + g
2
β
2
)
F
9
= r
d
c
v
g
α
1
g
α
2
(−1 + g

2
β
1
)(−1 + g
2
β
2
)(g
2
β
2
+ g
4
β
1
g
2
β
2
+ 2g
2
β
1
(2 + g
2
β
2
))
F
10

= −g
α
1
g
β
1
(8r
2
d
c
2
v
(1 + g
2
β
1
)
2
(1 + g
2
α
2
g
2
β
2
) − 4r
d
c
v

(3 + 4g
2
β
1
+ g
4
β
1
)(1 + (1 + 2g
2
α
2
)g
2
β
2
)
+ (5 + 2g
2
β
1
+ g
4
β
1
)(1 + (2 + 4g
2
α
2
)g

2
β
2
+ g
4
β
2
))
F
11
= g
α
1
g
α
2
g
β
1
g
β
2
(16r
2
d
c
2
v
(1 + g
2

β
1
)
2
− 4r
d
c
v
(3 + 4g
2
α
1
+ g
4
β
1
)(3 + g
2
β
2
)
+ (5 + 2 + g
2
β
1
+ g
4
β
1
)(5 + 2 + g

2
β
2
+ g
4
β
2
))
F
12
= r
2
d
c
2
v
(1 + 4(1 + 4g
2
α
1
)g
2
β
1
+ 6g
4
β
1
+ g
8

β
1
)(1 + g
2
α
2
g
2
β
2
)
− 2r
d
c
v
(1 + (3 + 8g
2
α
1
)g
2
β
1
+ 3g
4
β
1
+ g
6
β

1
)(1 + (1 + 2g
2
α
2
)g
2
β
2
)
+ (1 + (2 + 4g
2
α
1
)g
2
β
1
+ g
4
β
1
)(1 + (2 + 4g
2
α
2
)g
2
β
2

+ g
4
β
2
)
Ox
1
Ox
3
x
3
= −d
1
x
1
U
(1)
1
(−d
1
) x
3
U
(1)
3
(−d
1
)
χ =
U

(1)
1
(−d
1
)
U
(1)
3
(−d
1
)
=
−B
1
cosh(d
1
g
α
1
k) −iC
1
g
β
1
cosh(d
1
g
β
1
k) + A

1
sinh(d
1
g
α
1
k) + iD
1
g
β
1
sinh(d
1
g
β
1
k)
A
1
cosh(d
1
g
α
1
k) + iD
1
g
β
1
cosh(d

1
g
β
1
k) −B
1
sinh(d
1
g
α
1
k) −iC
1
g
β
1
sinh(d
1
g
β
1
k)
.
χ A
1
, B
1
, C
1
, D

1
A
1
B
1
C
1
D
1
A
2
B
2
C
2
D
2
A
1
χ =
U
(1)
1
(−d
1
)
U
(1)
3
(−d

1
)
=
−g
β
1
T
g
α
1
M
T, M
T = T
1
cosh(kd
2
g
α
2
) cosh(kd
1
g
β
1
) + T
2
cosh(kd
1
g
β

1
) cosh(kd
2
g
β
2
)
+ T
3
cosh(kd
1
g
α
1
) cosh(kd
2
g
α
2
) + T
4
cosh(kd
1
g
α
1
) cosh(kd
2
g
β

2
)
+ T
5
sinh(kd
1
g
α
1
) sinh(kd
2
g
α
2
) + T
6
sinh(kd
2
g
α
2
) sinh(kd
1
g
β
1
)
+ T
7
sinh(kd

1
g
α
1
) sinh(kd
2
g
α
2
) + T
8
sinh(kd
1
g
β
1
) sinh(kd
2
g
β
2
)
M = M
1
cosh(kd
2
g
β
2
) sinh(kd

1
g
α
1
) + M
2
cosh(kd
2
g
β
2
) sinh(kd
1
g
β
1
)
+ M
3
cosh(kd
2
g
α
2
) sinh(kd
1
g
α
1
) + M

4
cosh(kd
2
g
α
2
) sinh(kd
1
g
β
1
)
+ M
5
cosh(kd
1
g
β
1
) sinh(kd
2
g
α
2
) + M
6
cosh(kd
1
g
β

1
) sinh(kd
2
g
β
2
)
+ M
7
cosh(kd
1
g
α
1
) sinh(kd
2
g
α
2
) + M
8
cosh(kd
1
g
α
1
) sinh(kd
2
g
β

2
)
T
i
M
i
(i = 1, 8)
T
1
= −2g
α
1
g
β
2
(1 −r
d
c
v
(1 + g
2
β
1
) + g
2
β
2
)
T
2

= 2g
α
2
(2 + r
d
c
v
(1 + g
2
β
1
) + g
β
2
T
3
= g
α
1
(1 + g
2
β
1
)g
β
2
(1 −2r
d
c
v

+ g
2
β
2
)
T
4
= 2g
α
1
g
β
2
(−1 + r
d
c
v
)
T
5
= g
α
2
g
β
2
(−2 + r
d
c
v

(1 + g
2
β
1
))(1 + g
2
β
1
)
T
6
= −4(−1 + r
d
c
v
)g
α
1
g
β
1
g
α
2
g
β
2
T
7
= −(1 + g

2
β
1
)(−1 + r
d
c
v
(1 + g
2
β
1
) −g
2
β
2
)
T
8
= 2g
α
1
g
β
1
(−1 + 2r
d
c
v
− g
2

β
2
)
M
1
= 4(−1 + r
d
c
v
)g
α
1
g
β
1
g
β
2
M
2
= −(1 + g
2
β
1
)(−2 + r
d
c
v
(1 + g
2

β
1
))g
β
2
M
3
= (1 + g
2
β
1
)g
β
2
(−1 + r
d
c
v
(1 + g
2
β
1
) −g
2
β
2
)
M
4
= (1 + g

2
β
1
)g
β
2
(−1 + r
d
c
v
(1 + g
2
β
1
) −g
2
β
2
)
M
5
= −2(−1 + r
d
c
v
)(1 + g
2
β
1
)g

α
2
g
β
1
g
β
2
M
6
= g
β
1
(1 + g
2
β
1
)(−1 + 2r
d
c
v
− g
2
β
2
)
M
7
= 2g
α

2
g
β
1
(−2 + r
d
c
v
(1 + g
2
β
1
))g
β
2
M
8
= 2g
β
1
(1 −r
d
c
v
(1 + g
2
β
1
) + g
2

β
2
)
n
4n
(n − 1)
m ∆
m
m ω
m
∆
m
=
∂u
∂x
+
∂w
∂z
= exp[i(pt −kx)].[∆

m
exp(−ikg
α
m
z) + ∆

m
exp(ikg
α
m

z)]
ω
m
=
1
2
(
∂u
∂z
−
∂w
∂x
) = exp[i(pt −kx)].[ω

m
exp(−ikg
β
m
z) + ω

m
exp(ikg
β
m
z)]
g
β
m
=


(c/β
m
)
2
− 1 g
α
m
=

(c/α
m
)
2
− 1 ∆

m
∆

m
ω

m
ω

m
u = −(α
m
/p)
2
(∂∆

m
/∂x) −2(β
m
/p)
2
(∂ω
m
/∂z)
w = −(α
m
/p)
2
(∂∆
m
/∂x) + 2(β
m
/p)
2
(∂ω
m
/∂z)
σ = ρ
m
[α
2
m
∆
m
+ 2β
2

m
{(α
m
/p)
2
(∂
2
∆
m
/∂x
2
) + 2(β
m
/p)(∂
2
ω
2
m
/∂x∂z)}]
τ = 2ρ
m
β
2
m
[−(α
m
/p)
2
(∂∆
2

m
/∂x∂z) + (β
m
/p)
2
{(∂
2
ω
m
/∂x
2
) −(∂
2
ω
m
/∂z
2
)}]
∆
m
ω
m
˙u/c = −(α
m
/c)
2
[(∆

m
+ ∆


m
) cos(kg
α
m
z) −i(∆

m
− ∆

m
) sin(kg
α
m
z)]
− γ
m
g
β
m
[(ω

m
− ω

m
) cos(kg
β
m
z) −i(ω


m
− ω

m
) sin(kg
β
m
z)]
˙w/c = −(α
m
/c)
2
g
α
m
[(∆

m
+ ∆

m
) sin(kg
α
m
z) + i(∆

m
− ∆


m
) cos(kg
α
m
z)]
+ γ
m
[−i(ω

m
− ω

m
) sin(kg
β
m
z) + (ω

m
− ω

m
) cos(kg
β
m
z)]
σ = −ρ
m
α
2

m
(G
m
− 1)[(∆

m
+ ∆

m
) cos(kg
α
m
z) −i(∆

m
− ∆

m
) sin(kg
α
m
z)]
− ρ
m
c
2
G
2
m
g

β
m
[(ω

m
− ω

m
) cos(kg
β
m
z) −i(ω

m
− ω

m
) sin(kg
β
m
z)]
τ =ρ
m
(α
m
/c)
2
G
m
g

α
m
[(∆

m
+ ∆

m
) sin(kg
α
m
z) + i(∆

m
− ∆

m
) cos(kg
α
m
z)]
− ρ
m
c
2
m
[−i(ω

m
− ω


m
) sin(kg
β
m
z) + (ω

m
− ω

m
) cos(kg
β
m
z)]
m (m − 1)
(m) (m − 1)
˙u/c, ˙v/c, σ, τ (∆

m
+ ∆

m
) (∆

m
−∆

m
) (ω


m
+ ω

m
) (ω

m
−ω

m
)
( ˙u
m−1
/c, ˙w
m−1
/c, σ
m−1
, τ
m−1
) = E
m
((∆

m
+∆

m
), (∆


m
−∆

m
), (ω

m
+ω

m
), (ω

m
−ω

m
))
E
m
E
m
=




−(α
m
/c)
2

0 −G
m
g
β
m
0
0 −(α
m
/c)
2
g
α
m
0 G
m
−ρ
m
α
2
m
(G
m
− 1) 0 −ρ
m
c
2
G
m
g
β

m
0
0 ρ
m
α
2
m
G
m
g
α
m
0 −ρ
m
c
2
G
m
(G
m
− 1)




(m) m
( ˙u
m
/c, ˙w
m

/c, σ
m
, τ
m
) = D
m
((∆

m
+ ∆

m
), (∆

m
−∆

m
), (ω

m
+ ω

m
), (ω

m
−ω

m

))
D
m
D
m
=


−(α
m
/c)
2
cos(p
m
) i(α
m
/c)
2
sin(p
m
) −G
m
g
β
m
cos(q
m
) iG
m
g

β
m
sin(q
m
)
i(α
m
/c)
2
g
α
m
sin(p
m
) −(α
m
/c)
2
g
α
m
cos(p
m
) −iG
m
sin(q
m
) G
m
cos(q

m
)
−ρ
m
α
2
m
(G
m
−1) cos(p
m
) ρ
m
α
2
m
(G
m
−1) sin(p
m
) −ρ
m
c
2
G
2
m
g
β
m

cos(q
m
) iρ
m
c
2
G
2
m
g
β
m
sin(q
m
)
−iρ
m
α
2
m
G
m
g
α
m
sin(p
m
) ρ
m
α

2
m
G
m
g
α
m
cos(p
m
) iρ
m
c
2
G
m
(G
m
−1) sin(p
m
) −ρ
m
c
2
G
m
(G
m
−1) cos(q
m
)



(m −1) (m) (∆

m
+ ∆

m
) (∆

m
−∆

m
)
(ω

m
+ ω

m
) (ω

m
− ω

m
)
(m) (m − 1)
m

( ˙u
m
/c, ˙w
m
/c, σ
m
, τ
m
) = D
m
E
−1
m
( ˙u
m−1
/c, ˙w
m−1
/c, σ
m−1
, τ
m−1
)
a
m
= D
m
E
−1
m
a

m
(a
m
)
11
= G
m
cos(p
m
) −(G
m
− 1) cos(q
m
)
(a
m
)
12
= i[(G
m
− 1)g
−1
αm
sin(p
m
) + G
m
g
βm
sin(q

m
)]
(a
m
)
13
= −(ρ
m
c
2
)
−
1(cos(p
m
) −cos(q
m
))
(a
m
)
14
= i(ρ
m
c
2
)
−
1(g
−1
αm

sin(p
m
) + g
βm
sin(q
m
))
(a
m
)
21
= −i[G
m
g
αm
sin(p
m
) + (G
m
− 1)g
−1
βm
sin(q
m
)]
(a
m
)
22
= −(G

m
− 1) cos(p
m
) + G
m
cos(q
m
)
(a
m
)
23
= i(ρ
m
c
2
)
−1
(g
αm
sin(p
m
) + g
−1
βm
sin(q
m
))
(a
m

)
24
= (a
m
)
13
(a
m
)
31
= ρ
m
c
2
G
m
(G
m
− 1)(cos(p
m
) −cos(q
m
))
(a
m
)
32
= iρ
m
c

2
[(G
m
− 1)
2
g
−1
αm
sin(p
m
) + G
2
m
g
βm
sin(q
m
)]
(a
m
)
33
= a
22
, a
34
= a
12
(a
m

)
41
= iρ
m
c
2
[G
2
m
g
αm
sin(p
m
) + (G
m
− 1)
2
g
−1
βm
sin(q
m
)]
(a
m
)
42
= (a
m
)

31
(a
m
)
43
= (a
m
)
21
(a
m
)
44
= (a
m
)
11
G
m
= 2(β
m
/c)
2
= 2/(1 + g
2
β
m
)
( ˙u
m

/c, ˙w
m
/c, σ
m
, τ
m
) = a
m
( ˙u
m−1
/c, ˙w
m−1
/c, σ
m−1
, τ
m−1
).
(m −1)
(m) m
( ˙u
n−1
/c, ˙w
n−1
/c, σ
n−1
, τ
n−1
) = a
n−1
a

n−2
a
1
( ˙u
0
/c, ˙w
0
/c, σ
0
, τ
0
)
χ =
u(0)
w(0)
x z
u w
u = −(α
m
/p)
2
(∂∆
m
/∂x) −2(β
m
/p)
2
(∂ω
m
/∂z)

w = −(α
m
/p)
2
(∂∆
m
/∂x) + 2(β
m
/p)
2
(∂ω
m
/∂z)
u w exp[pt −kx]
χ =
u(0)
w(0)
=
˙u(0)
˙w(0)
(n−1) (0)
( ˙u
n−1
/c, ˙w
n−1
/c, σ
n−1
, τ
n−1
) = a

n−1
.a
n−2
a
1
( ˙u
0
/c, ˙w
0
/c, σ
0
, τ
0
).
d
1
ν
1
ρ
1
α
1
β
1
(x0z)
( ˙u/c, ˙w/c, σ, τ) = a
1
( ˙u
0
/c, ˙w

0
/c, σ
0
, τ
0
).