ОТРАЖЕНИЕ И ПРЕЛОМЛЕНИЕ ЭЛЕКТРОУПРУГИХ ВОЛН НА ГРАНИЦЕ РАЗДЕЛА ПЬЕЗОЭЛЕКТРИЧЕСКИХ И ПЬЕЗОМАГНИТНЫХ СРЕД

(1)

Ա Ա Ա Ի Ի Ո Թ Ո ԻԱ Ա Ի Ա Ա ԻԱ Ի Ղ Ա Ի

59, 14, 2006

539.3

. ., . .

. . , . .

և

Ա 6mm 4mm

և

և : ,

`

և

և և `

և

-և :

V.G. Garakov, Z.N. Danoyan

Reflection and refraction of electroelastic waves at the interface of piezoelectric and piezomagnetic solids

In the paper reflection and refraction of plane electroelastic wave is disscussed at the interface of piezoelectric and piezomagnetic solids with symmetry 6mm or 4mm. It is shown that the incidend wave interacting with the piezomagnetic medium generates in the piezoelectric a reflected homogeneous electroelastic wave and any attendant nonhomogeneous electro-magnetostatic surface wave and in the piezomagnetic a refracted homogeneous magnetoelastic wave and an attendant nonhomogeneous electromagnetostatic surface wave.

6mm 4mm. ,

, ,

- .

1.

[1-4], – [5-7], [8]

-- .

- .

Ox

₁

x

₂

x

₃

0

2

>

x

,

x

2

=

0

,

x

₂

<

0

.

(2)

59

- . ,

6mm ( 4mm),

3

Ox

.

Ox

₁

( .1).

[4]. ,

1

x

₂

t

:

(

,

)

,

(

,

)

,

(

,

)

,

0

₃ ₁ ₂ ₁ ₂ ₁ ₂

2

1

u

x

t

x

t

x

t

u

=

≡

=

ϕ

=

ϕ

ψ

=

ψ

(1.1)

ϕ

ψ

– :

grad ,

grad

E

r

= −

ϕ

H

r

= −

ψ

(1.2)

E

r

– ,

H

r

–

.

[4,8], (1.2),

:

1.

x

₂

>

0

( ):

2 1

1 1 1 2 1 1

1 1 1 1

1

0

4

0

0 u

c u

e

t

e u

∂

∆ − ρ

+ ∆ϕ =

∂

π ∆ − ε ∆ϕ =

∆ψ =

(1.3)

,

:

µ

₁

,

e

₁

,

c

₁

,

ρ

₁

,

ε

₁

฀

L

6

(

L

4

)

x

3

x

1

x

2 Cьеƒ%м=г…е2,*

(6mm ,л, 4mm)

Cьеƒ%._ле*2!,*

(6mm ,л, 4mm)

t,г.1

(3)

( ) ( ) ( ) 1 1 11 1 1 15 1 1 44

1

,

=

,

=

,

ε

=

ε

ρ

c

e

(1.4)

2.

x

₂

<

0

( ):

2 2

2 2 2 2 2 2

2 2 2 2

2

0

4

0 0,

u

c

u

b

t

b u

∂

∆ − ρ

+ ∆ψ =

∂

π ∆ − µ ∆ψ =

∆ϕ =

(1.5)

,

:

( ) ( ) ( )

2 2 22 2 2 15 2 2 44

2

,

=

,

=

,

µ

=

µ

ρ

c

b

(1.6)

3.

x

₂

=

0

:

1 2 1 2 1 2

1 1 2

2 2 2

2 2 1

2 2 2

1 1 2 2

2 2 2 2

,

4

4 u

u

e

x

u

b

x

u

c

e

c

b

x

=

ϕ = ϕ ψ = ψ

∂ϕ

∂

∂ϕ

−ε

+ π

= −ε

∂

∂ψ

∂

∂ψ

−µ

+ π

= −µ

∂

₊

∂ϕ

₌

∂

₊

∂ψ

∂

(1.7)

:

( ) ( ) 1 1 11 2 2

11

=

ε

,

µ

=

µ

ε

(1.8)

«1» , «2» –

.

2. (1.3), (1.5), (1.7) .

:

1 1 1 1 1 1 1

2 2 2 2 2 2 2

,

4 ,

4 e u

e

b u

b

′

ϕ = ϕ −

= π ε

′

ψ = ψ −

= π

µ

(2.1)

1

u

ϕ′

₁

u

₂

ψ′

₂

:

2 1

1 2 1 1

,

1

0,

1

0 u

c u

t

∂

_′

ρ

= ∆

∆ϕ =

∆ψ =

∂

(2.2)

:

(

2

)

2

1 1

1

,

1 1 1 1

c

=

c

+ χ

χ =

e e c

(2.3)

2 2

2 2 2 2

,

2

0,

2

0 u

c

u

t

∂

_′

ρ

= ∆

∆ψ =

∆ϕ =

∂

(2.4)

:

(

2

)

2

2 2

1

2

,

2 2 2 2

c

=

c

+ χ

χ =

b b c

(2.5)

1

(4)

61

1 1 2 2

1 2 1 1 2 2

2 2 2 2

1 1 1 2 1 2 2 2

1 2 2 1

2 2 2 2

,

u

c

e

c

b

x

e u

b u

x

′

∂

∂ϕ

∂

∂ψ

=

+

=

+

∂

′

ϕ +

= ϕ ψ = ψ +

′

∂ϕ

∂ψ

ε

= ε

µ

= µ

∂

(2.6)

(2.2)

( 1 1 2 ) ( 1 2 2 ) ( 1 3 2 )

1 1

,

1 1

,

1 1

i px q x t i px q x t i px q x t

u

=

u e

%

+ −ω

ϕ = ϕ

′

%

′

e

+ −ω

ψ = ψ

%

e

+ −ω (2.7) :

(

)

(

)

(

)

2 2 2

1 1 1

2 2 2 2

1 1

0 0,

0 c

p

q

u

p

q

p

q

⎡

+

− ρ ω

⎤

=

⎣

⎦

′

+

ϕ =

+

ψ =

%

(2.8)

p

q

– ,

ω

–

. (2.8) :

2

2 2 2 2 2 2

1 2 2 3

1

,

q

p

q

p

q

p

s

ω

=

−

= −

(2.9)

:

(

)

1 2 1 1

1 1 2

1 2 1

1 ρ

χ

+

=

ρ

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ω

=

c

k

s

(2.10)

ρ

ω

, ,

x

₂

=

0 t

. ,

s

₁

. ( ) :

2 2 1 2

10

1

p

s

q

=

m

=

m

ω

−

(2.11)

1

θ

, . .

( )

p

q

k

r

₁

,

Ox

₁ ( .2).

1

cos

1

,

1 1

sin

1

tg

1 10

p

=

k

θ

q

= −

k

θ = −

p

θ = −

q

(2.12)

2 1 2

1

p

q

k

=

+

– ,

q

₁₀ (2.11). ,

10

1

q

=

−

– ,

q

₁

=

q

₁₀ –

(5)

(2.9)

2

,

3

q

= ±

i p

q

= ±

i p

(2.13)

,

1

Ox

.

(2.13)

(

). ,

,

.

( ) [4]. ,

- . ,

:

(

)

1 10 11

,

1 1 1 10 11

,

1 1

u

=

u

+

u

ϕ = ϕ +

′

e u

+

u

ψ = ψ

(2.14)

( ) ( )

1 10 2 1 10 2

2 2

1 1

10 10 11 11

1 1 1 1

,

i px q x t i px q x t

p x p x

i px t i px t

u

u e

u

u e

e

+ −ω − −ω

− −

−ω −ω

=

′

ϕ = ϕ

ψ = ψ

%

(2.15)

(2.14) ,

,

.

(

0

2

<

x

).

-:

t

_,

г

.

2 ฀

θ

1

ϕ′

1

,

ψ

1

x

1

x

2

Cьеƒ%м=г…е2_,* (2)

Cьеƒ%._ле_*2_!,* (1)

θ

2

θ

1

u

21

u

11

ϕ

2

,

ψ

2

′

(6)

63

2 21

,

2 2 2 21

,

2 2

u

=

u

ψ = ψ +

′

b u

ϕ = ϕ

(2.16)

( )

( ) 1 20 2

2 1 2 1 21 21 2 2 2 2

i px q x t

p x i px t

u

u e

e

+ −ω −ω −ω

=

′

ψ = ψ

ϕ = ϕ

%

(2.17) 2 2 2 2 2 2 2 2 2 2

20

⎟

=

ρ

⎠

⎞

⎜

⎝

⎛ ω

=

−

ω

=

c

k

s

p

s

q

(2.18)

2 2 2 2 2 2 20

2 2 2

2 2

cos

,

sin

tg

p

k

q

k

p

q

k

p

q

=

θ

= −

θ = −

=

+

2

s

–

,

k

₂– o ,

θ

₂–

. (2.16), . 2 2 21 1 1 11

~

,

~

,

~

,

~

,

~

,

~

_ϕ′

_ψ

_u

_ψ′

_ϕ

u

( 21 2 21 11 1 11 10 1 10

~

,

~

,

~

₌

_e

_u

_ϕ

₌

_e

_u

_ψ

₌

_b

_u

ϕ

)

(2.6):

(

)

(

)

10 11 21 1 2 2 21

1 10 11 10 1 1 2 20 21 2 2

1 1 10 11 2

1 1 2 2 1 1 2 2

,

u

b u

ic q

u

e p

ic q u

b p

e u

u

′

+

=

ψ = ψ +

′

−

+

ϕ =

−

ψ

′

ϕ +

+

= ϕ

′

ε ϕ = −ε ϕ µ ψ = −µ ψ

%

(2.19)

(2.19) (2.12) (2.18), :

1 2 1

11

10 1 2 1

21 1

10 1 2 1

2 1 2 2

1 10 1 10

1 2 1 2

1 1 1 2

2 10 2 10

1 2 1 2

sin

cos

sin

cos

2 sin

1 sin

sin

cos

,

r

i

u

R

u

r

i

u

T

R

u

r

i

e

b

Tu

e

b

Tu

θ −

θ + χ

θ

=

θ +

θ − χ

θ

=

= + =

θ +

θ − χ

θ

ε

µ

′

ϕ = −

ψ =

ε + ε

µ + µ

ε

_′

µ

ϕ =

ψ = −

ε + ε

µ + µ

(7)

(

)

(

)

2 2 2 2

1 1 1 1

2 1 1 1 2 2

1 2 1 1 2 1

c k

c

r

c k

c

e e

b b

c

ρ

=

ρ

ε

µ

χ =

+

ε + ε

µ + µ

(2.21)

21 11 10

,

ϕ

,

ψ

ϕ

:

10 1 10 11 1 11 1 10

21 2 21 2 10

,

e u

e Ru

b u

b Tu

ϕ =

=

ψ =

=

%

(2.22)

,

: –

, –

, ,

– .

. :

1

2 2 1 1

cos

θ =

s s

−

cos

θ

(2.23)

1 2

s

.

θ

₂

=

0

, :

2 1 1

cos

s

=

θ

∗

2 1 1

arccos

s

=

θ

∗

(2.24)

∗

θ

₁

cos

,

s

₁

<

s

₂.

, ,

, . .

R

e

. (2.24),

∗

θ

₁ ,

.

,

– ,

∗

θ

₁. ,

. ,

,

R

–

.

,

x

₂

>

0

,

x

₂

=

0

. , ,

(8)

65

1. . .

. // . . . . 1985. .38. N1. .12-19.

2. . ., . . ,

. // . .

. 1994. .47. N3-4. .78-82.

3. . ., . ., . .

. //

. . . : 1991. .49-54.

4. . ., . . . :

. 1982. 240 .

5. . ., . ., . .

. // III .

« ». : . .

1984. .22-25.

6. . ., . ., . .

.

// II . - - « ,

». : . . 1984. .1. .92-96.

7. . ., . ., . .

. // , . . . . . 5. . 1986. .102-109.

8. . ., . .

- . // IV

« ». : . . 1989. .67-72.