PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY
Physical and Mathematical Sciences
2012, № 1, p. 33–37
Me cha ni cs
DIFFRACTION OF SHEAR PLANE ELECTRO-ELASTIC WAVE ON THE
SEMI-INFINITE METALLIC LAYER IN THE PIEZOELECTRIC SEMI-SPACE
G. A. SHAKHBAZYAN*, H. A. GHAZARYAN**
Chair of Mechanics YSU, Armenia
The diffraction of plane shear electro-elastic wave on a thin semi-infinite
  ( x,. y ), w ( x, y ) metallic layer piezoelectric-vacuum space is considered in the
presence of a thin infinite (ground shield) metallic layer in vacuum. The problem
is reduced to the solution of a functional equation in the theory of analytical
functions. The problem admits closed solution describing the wave fields in
vacuum and in the piezoelectric medium. The presence of the semi-infinite
metallic layer leads to a diffraction of a waves, a result of which surface electroelastic waves occur.
Keywords: diffraction of waves, metallic layer, piezoelectric, vacuum,
factorization.
In the present paper 6 mm class piezoelectric medium of hexagonal symmetry is considered that occupies y  0 semi-space in the Cartesian coordinate system,
 ( x, y ), w ( x, y ) y  0 semi-space being in vacuum
and y  0 , x  0 semiplane being
metalized with grounded thin layer.
The layer is so thin that its stiffness is
not taken to account, i.e. the layer is
considered to be an electrode on the
mentioned semiplane. OZ axis is
coincident with the piezocrystal.
As y  h plane is screened,


its electric potential  (1)  0 .
A plane shear electro-elastic wave is incident from infinity on the electrode
at  0 angle, where 0  0   / 2 (Fig.). The components of electric and
displacement potential amplitudes of the incident wave are as follows:
e
w ( x, y )  e ikxcos0 ikysin0 ,
  ( x, y )  15 e ikxcos0 ikysin0 ,
(1)
11
Figure.
*
**
E-mail: geg.shahbazyan@gmail.com
E-mail: haykazghazaryan@gmail.com
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2012, № 1, p. 33–37.
34
where k   / c is the wave number, c  c44 (1   ) /  is the velocity of propagating shear electro-elastic wave,  is the frequency of vibrations and  is the
medium density,   e152 / 11c44 is the electromechanical coupling coefficient of the
piezoelectric medium and e15 , 11 , c44 are the piezoelectric, dielectric and elastic
constants respectively.
Here it was taken into account that the dependence of the electro-elastic field
on the time parameter is harmonic: e i t , where t is the time parameter.
Subject to a condition that the piezoelectric medium is in an anti-plane
deformation state, we aimed at determination of wave fields in the piezoelectric
medium and vacuum. Having in view the determination of amplitudes of
displacement and electric potentials, we shall avail of the following equations [1]:
w  k 2 w  0,
  k 2 e15 11 w
(2)
and for vacuum:
(3)
 (1)  0, y  0 ,
where     x 2    y 2 .
Now write out the boundary conditions of the problem under assumption of
the absence of stress at the interface between piezoelectric medium and vacuum:
 yz  c44 w y  e15  y  0, y  0.
(4)
The conditions for electric potential and D2 component of induction vector are:
y  0, x  0,
y  0, x  0,
y  h ,
y  0, x  0,
y  0, x  0,
   (1)  0,
 

(1)
(1)
  0 ( x),
 0,
(5)
(6)
(7)
D2 
D2(1)
D2 
D2(1) ,
 D0 ( x),
D2(1) ( x, y )
(8)
(9)
  0  y , D2 ( x, y )  e15 w y  11  y .
Hence, in the problem under consideration vacuum appears a vacuum layer
of h width. Here  0 ( x), D0 ( x) are the required functions, for which the following
functions are applicable:
where
  ( x)   0 ( x) ( x),   ( x)  D0 ( x ) (  x).
In that case from (5), (6), (8), (9) we obtain:
   (1)    ( x ),
D2  D2(1)   ( x) ,
where  ( x) is the Heaviside function.
Then introducing the following functions
u ( x, y )  w( x, y )  w ( x, y ),
 ( x, y )  ( x, y )    ( x, y ),
we have for stress and induction components:
 yz  c44 u y  e15  y  ikc44 (1   )sin 0 eikxcos0 ikysin0 ,
D2 ( x, y)  e15 u y  11  y .
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2012, № 1, p. 33–37.
35
To determine the functions u ( x, y ), w( x, y ) , we have from equation (2) that must
also satisfy the outgoing wave condition
u  k 2u  0,
  k 2 e15 11 u  0, y  0 .
(10)
Applying the generalized integral Fourier transform [2], we obtain
e
d 2u
d 2
2
2

(


k
)
u

0,
  2  k 2 15 u  0, y  0 ,
(11)
2
2
11
dy
dy
d 2  (1)
  2  (1)  0,
dy 2




y  0,
(12)
where u ( , y )   u ( x, y )ei x dx,  ( , y )    ( x, y )ei x dx, and the boundary
conditions will be
du
d
c44
 e15
 2 ikc44 (1   )sin 0 (  kcos 0 )  0, y  0,
dy
dy
e
y  0,
  2 15  (  kcos 0 )   1    ( ) ,
11
y  0,
e15 du / dy  11d  / dy   0 d    / dy   ( ) ,
1
(13)
(14)
(15)
(16)
y   h,
 1  0.
Here  0 is an electric constant of semi-space y  0 and  ( x ) is the Dirac delta
function. We also considered that
i x
e
dx  2 ( ).

To satisfy the condition of the outgoing wave, we assume that
 ( )   2  k 2 |  | when |  |  and  2  k 2  i k 2   2 , i.e. in the
complex plane     i the real axis bypasses the branching points –k from top
and k from bottom [3].
Whereas u,  0 when y  , the solutions of equations (11) and (12),
will have the form
u  A( ) e y ,
e
  B ( ) e | | y  15 A( ) e  y ,
y  0,
(17)
11
 (1)  C ( ) ch |  | y  D( )sh |  | y, y  0.
From conditions (14)–(16) we obtain:
ch |  | h
C ( )    ( ),
D ( )    ( )
,
sh |  | h

ch |  | h
1
B ( )   0   ( )

  ( ),
11
sh |  | h 11 |  |

ch |  | h   
1
A( )   11   0

   2 (  kcos 0 ).

sh
|

|
h
e
e
|

|

 15
15
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2012, № 1, p. 33–37.
36
From condition (13) we derive the following functional equation for
unknown functions   ( ) and   ( ) :
1  ( )  (1/ |  |) K ( )  ( )  2 (  kcos 0 )  0,
(1   )e15 itg 0
where 1  (1   )11   0 , λ  2
,
K 2 (kcos 0 )
(18)
K ( )  K1 ( ) / K 2 ( ), K1 ( )  (1   )  2  k 2 / |  |   ,
1
 2  k2
 (1  æ )
1 
| |
and K1,2 ( )  1 when |  | .
K 2 ( ) 

ch |  | h 
ch |  | h 

 11   0
  0 
sh |  | h 
sh |  | h 


It is shown that K1 ( ) function has a zero when    1 , where  1 is the
only root of equation K1 ( )  0 when   k , and K 2 ( ) function has a zero when
   2 , where  2 is the only root of equation K 2 ( )  0 when k   2   1 , so
K i ( k )  0, Ki ()  0, Ki( )  0 when   k (i  1, 2), K 2 ( )  0 .
The functional equation can be considered as the Riemann boundary
problem in the theory of analytical functions.
The real axis bypasses also points    1 ,    2 from the top and
   1 ,    2 from the bottom, ensuring the observance of the conditions of
outgoing wave. So, as the dispersion equations Ki ( )  0 (i  1, 2) have roots
 i , the localized (surface) waves propagate in the piezoelectric medium.
Let us find the solution of functional equation (18) by means of factorization
method, representing K () and || in the following form [2]:
K ( )  K  ( ) K  ( ), |  | (  i0)1/ 2 (  i0)1/ 2 ,
(19)
where K  ( ) are regular functions that have no zeros in the upper and lower semispaces Im   0 and Im   0 of the complex plane, where     i . Note, that
K  ( )  1 are in their regularity domains when |  |  . Then equation (18) is
presented in the following form:
(kcos0  i0)1/ 2
1 (  i0)1/ 2 
1



(

)

K
(

)

(

)

2

 (  kcos0 )  0.
K  ( )
(  i0)1/ 2
K  (kcos0 )
Taking into consideration also that
1
1
2 i (  kcos 0 ) 

,
  kcos0  i 0   kcos 0  i0
equation (18) will be written in the following form when      :
 (  i 0)1/ 2 
 ( kcos 0  i 0)1/ 2
1
J  ( )  1 
 ( ) 



K ( )
iK (kcos 0 )   kcos0  i 0

 (kcos 0  i0)1/ 2
1
1


K
(

)

(

)


 J  ( ).
(  i 0)1/ 2
iK  ( kcos 0 )   kcos 0  i0
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2012, № 1, p. 33–37.
37
As this equation is solved by means of the same method as that in [2, 4],
therefore, the equality J  ( )  J  ( ) holds, only if J  ( )  J  ( )  0.
Thus, we obtain for the unknown functions:
 ( kcos 0  i 0)1/ 2
1
K  ( )
  ( ) 


,

i1K (kcos 0 )   kcos0  i0 (  i0)1/ 2
 (kcos 0  i0)1/ 2
1
(  i0)1/ 2


.

  kcos 0  i0 K  ( )
iK ( kcos 0 )
Hence, the problem of binding the expressions for amplitudes of displacement and electric field will have solutions in the following form:
1 
 i x
w( x, y )  w ( x, y ) 
 u ( , y ) e d ,
2 
  ( ) 
 ( x, y )    ( x, y ) 
1
2

  ( , y ) e
 i x
d ,
(20)

1  (1)
 i x
  ( , y ) e d .
2 
The wave field in the piezoelectric medium is presented by the diffracted,
incident, reflected and localized (surface) waves.
 (1) ( x, y ) 
Received 11.01.2012
R EFE R EN C ES
1. Balakirev M.K., Gilisnkiy I.A. Waves in Piezocrystals. Novosibirsk: Nauka, 1982, 240 p.
(in Russian).
2. Grigoryan E.Kh., Jilavyan S.H. // Izv. NAN RA. Mekhanica, 2005, v. 58, № 1,
p. 38–50 (in Russian).
3. Noble B. Wiener–Hopf Method. M.: Mir, 1962, 279 p. (in Russian).
4. Grigoryan E.Kh. // Uchenie Zapiski EGU, 1979, № 3, p. 29–34 (in Russian).
38
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2012, № 1, p. 33–37.
Գ. Ա. Շահբազյան, Հ. Ա. Ղազարյան. Սահքի հարթ էլեկտրաառաձգական ալիքի
դիֆրակցիան
կիսաանվերջ
շերտի
վրա
պիեզոէլեկտրական
կիսատարածությունում
էջ. 33–37
Դիտարկված է պիեզոէլեկտրիկ–վակուումային միջավայրում սահքի հարթ
ալիքի դիֆրակցիան` բարակ կիսաանվերջ մետաղական շերտի վրա,
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