Gap Waves in Piezoelectric layered Medium

Ս Ո Թ Ո

60, 11, 2007

539. 3: 537.2

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: ,

– - - .

Key words: Shear electroelastic gap waves, piezoelectric-vacuum-dielectric layered system.

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Z.N. Danoyan, L.A. Atoyan, A.Kh. Berberyan Gap Waves in Piezoelectric layered Medium

In the present paper the conditions of existence of shear electroelastic gap waves in piezoelectric-vacuum-dielectric layered system are found. It is shown that in the discontact layered system the gap electroelastic waves can be propagated. It is considered the limiting case when the thickness of vacuuming layer tends to zero. It is proved that the statement of the problem is true when there is no acoustic contact between piezoelectric and dielectric grounded media.

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3)

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( ):

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1)

x

1

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0

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,

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1 1 1 2 1 1 2

1 2 1

1 1 1 1

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0

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x

x

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u

c

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d e

d

x

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x

x

′

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−

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:

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,

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x

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⎨ϕ →

→ −∞

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3 1

0

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2

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1 3

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1 1 1 1 1 2

1

S

c

c

c

e

c

c

c

d

e

e

e

e

e

e u

x

x

=

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=

+ χ

χ =

ε

=

=

=

=

ε

ε = ε

ε = ε ε = ε

′

ϕ = ϕ −

∆ = ∂ ∂ + ∂ ∂

(8)

1

S

– ,

χ

₁ –

,

c

₁ c₂ – ,

e

₁

d

₁ – , ε₁,ε₂,ε₃ –

,

ρ

₁

ρ

₂ – ,

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( 2 1 )

,

( 2 1 )

1 01 0

i px qx t i px qx t

u

=

U e

+ −ω

ϕ = Φ

e

+ −ω (9)

Ox

₁

x

₂.

q

>

0

p

–

; :

V

=

ω

/

p

(10) (9) (2)-(4)

(7), :

x

₁

>

0

:

[

1 01 1 1 2

]

1 01 1 1 1 01 1 2

exp(

( )

exp (

)

exp(

( ) )

exp(

) exp (

)

u

U

p

V x

i px

t

U e

p

V x

px

i px

t

=

− β

− ω

⎧⎪

⎨ϕ =

− β

+ Φ

−

− ω

⎪⎩

(11)

−

h

<

x

₁

<

0

:

ϕ = Φ

₂

[

+₀₂

exp(

px

₁

)

+ Φ

₀₂−

exp(

−

px

₁

)]exp (

i px

₂

− ω

t

)

(12)

−

∞

<

x

₁

<

−

h

:

.

ϕ

₃

=

Φ

₀₃

exp(

px

₁

)

exp

i

(

px

₂

−

ω

t

)

(13) 03

02 02 01

01

,

Φ

,

Φ

,

Φ

,

Φ

− +

U

– ,

,

β

₁

(

V

)

– ,

β

₁

(

V

)

=

1

−

(

V

2

/

S

₁2

)

(14) (7)

x

₁

→

+∞

,

β

₁

(

V

)

– .

:

0

<

V

_S

<

S

₁ (15)

,

.

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(15),

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(5)-(6),

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(

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(

)

(

)

(

)

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1 1 1 1 01 1 1 01

1 01 1 01 2 02 02

02 02 03

2 02 02 3 03

0

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k k k

e U

c

ie d U

e

id

id U

e

e

e

e

e

e

+ −

+ −

+ − − −

+ − − −

′

+ Φ = Φ + Φ

′

β +

+

+

Φ =

′

ε Φ +

= −ε Φ − Φ

Φ

+ Φ

= Φ

ε Φ

− Φ

= ε Φ

(16)

(16) :

k

=

ph

= π λ

2

h

/

(17)

k

– o ,

λ

– ,

p

–

.

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β

₁

( )

V

=

R k

( )

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2 2

1 2 2 1 1 2

2 2 1 2

( th

1)

(

th )

( )

( th

1)

(

th )

R

k

K

k

R k

k

k

ε ε

+ −

ε ε +

=

ε ε

+ + ε ε +

(19)

ε = ε ε

_{1 1}

/

₃

,

ε = ε ε

_{2 2}

/

₃ (20)

/

,

2

/

_{1 1}

.

1 2 1 1 1 2 1 2

1

e

c

K

d

c

R

=

ε

=

ε

(21) 2

1

R

2

1

K

–

,

R

(

k

)

– ( ).

(16)

k

- .

, :

(11)-(13) , ,

(18)

(k),

V

V

=

_S

(15).

4. .

(18)-(21) ,

(

e

₁

=

d

₁

=

0

)

R

(

k

)

, (18)

[1].

,

(18) :

)

(

k

d

₁

=

0

,

622

,

422

e

₁

=

0

.

:

|

R

(

k

)

|

<<

0

(22) [3].

(18),

)

(

k

R

-k

.

)

(

k

R

[3].

,

R

(

k

)

<

0

,

.

5.

6

mm

,

4

mm

.

,

6

mm

,

4

mm

.

e

₁₄

=

d

₁

=

0

;

e

₁₅

=

e

₁

≠

0

.

)

(

k

R

:

2

1 2 2

2 2 1 2

( th

1)

( )

( th

1)

(

th )

R

k

R k

k

k

ε ε

+

=

ε ε

+ + ε ε +

(23)

(23)

h

→

0

,

k

→

0

.

th

k

→

0

R

(

k

)

:

(

0

)

.

2 1 2 2 2 1

ε

ε

+

ε

ε

=

R

R

(24)

(24) :

3 1 2 1 2 1

ε

ε

1

χ

1

χ

)

0

(

+

+

=

R

, 44 1 2 15 2 1

c

e

ε

=

χ

(25)

[1]. [1]: 2 3 1 4 1 1

)

ε

ε

1

(

æ

1

+

−

=

S

V

_S ,

2 1 2 1 2 1

χ

1

χ

æ

+

=

(26)

,

- - .

,

.

1. . . .: , 1981. 281 .

2. . . .: , 1975. 872 .

3. . ., . . .

: , 1982. 240 .

4. . ., . . . :

. , 2006. 492 .

5. ., . . .: , 1982.

424 .

6. ., . . .: , 1965.

7. . ., . ., . .

. (

. I).// .

. . 2006. .59. № 2. . 43-56.

8. . ., . ., . .

. (

. II).// . .

. 2006. .59. №.3. . 34-44.

9. . ., . .

-. // .: «O ,

» (

), , 2002.

. 97-101.