Wave propagation in piezoelectric medium of hexagonal symmetry

Prec. Indian Acad. Sci., Vet. 87 A (Mathematical Sciences-2), No. 5, May 1978, pp. 125-136, @ printed in India.

Wave propagation in piezoelectric medium of hexagonal symmetry

B SRINIVASA RAO Department of Mathematics, Government College, Rajahmundry 533 104

M S received 1 April 1977; revised 31 January 1978

Abslraet. The wave propagation at large distances from a source of disturbance (isolated harmonic electric charge or body force of fixed frequency) in an infinite piezoelectric medium belonging to classes (6), (6 m m) or ceramic (a m) and (6 2 2) is discussed by means of asymptotic evaluation (at large distances) of Fourier integrals. Numerical results are given for the (6 m m) crystal class using the constants of cadmium selenide crystal.

Keywords. Wave surface; wave propagation ; Fourier integrals; piezoelectric medium.

1. Introduction

The geometry of wave propagation in elastic medium has been investigated by Musgrave [4], Synge [6] and others. The asymptotic behaviour (at large distances) of solutions of linear partial differential equations with constant coefficients has been demonstrated by Lighthill [3]. He has also studied the geometry of wave propagation using the asymptotic evaluation of Fourier integrals in many dimen- sions for large arguments. Buchwald [2] has discussed the wave propagation at large distances in elastic anisotropic medium due to radiation from a source of body force. The ' Forcing Funct ion ' which is necessary to represent the radiation from a localized disturbance could be a body force or any other source which vanishes outside a finite region. Paldas [5] has studied the propagation of waves through piezoelectric a-quartz crystals when the source of disturbance is an isolated harmonic body force. He has studied wave surfaces due to propagation of longitudinal waves and obtained slowness surface following the analysis of Buchwald [2]. In the present paper the propagation of waves at large distances from a source of disturbance in an infinite piezoelectric medium of hexagonal symmetry belonging to classes (6), (6 m m) or ceramic ( ~ m) and (6 2 2) is investigated. Two cases are analysed in which (a) the source of disturbance is an isolated harmonic electric charge of fixed frequency and (b) the source of disturbance is an isolated harmonic body force of fixed frequency. We have followed the analysis of Lighthill [3] to obtain the wave surface from the slowness surface and studied the wave amplitudes as well. Numerical results are presented for cadmium selenide crystal which belongs to (6 m m) class. It is interesting to note that the source of disturbance due to electric charge generates longitudinal waves in (6 m m) class and torsional waves in (6 2 2) class whereas the source of disturbance due to body force generates flexural waves in both (6 m m) class and (6 2 2) class.

125

126 ~ Srtniv~a Rao

2. Som'ee of electric charge

We choose a rectangular coordinate system with the axis of symmetry of (6) crystal class as z-axis. The stresses T~j and electric displacements Dl for crystal class (6) can be expressed in terms of displaceemnt components (u, v, w) and electric potential

as

Txx : Cl I /4 ,x + Cl~O,y --~ c13w, z "If- esl~,.

Ty, = cx2u,,, + ealv,. + c13w,. + esl~, ,

T.. : c13 (u,x + v,y) + c3aw,. + eaa~,.

T,. : c . (v,. q- w,.) q- el,~, x q- e15~, ,

T,,~, = c~ (u,~, -t- v,x) (1)

D~ -~ e14 (w,. -t- v,.) + exs (w,~ -F u,.) - - En~,.

D, : els (w,, -I- v,.) - - e14 (w, x -t- u,~) - - "n~,,

D. : e31 (u., + v,~) + e3a w,~ - - ~ , .

where c~j, eo and ~o are elastic, piezoelectric and dielectric constants respectively. Comma followed by subscript denotes partial differentiation with respect to the subscript variable. Also cee = (clx - - Clz)/2.

The equations of motion (in the absence of a body force) are

cxx u ,~ + ce0 u , . + c~ u , . + (c1~ + c66) v,~

+ (c18 + ca,) w,x. + (e15 + e81) $,x. - - el4$,n = su, . (2)

(c12 + c~6) u,x. + cee v,xx + cl1 v , . + e~ v,..

+ (Cla + c~t) w,,. + (exs + esl) $,,. + exir x, ~-- so,. (3)

(c~3 + c~) (u,,,. + v,~.) + c~ (w,xx + w , . ) + c ~ w , .

+ ex5 ($,x,~ + ~ , . ) + es~$,.z = sw,,, (4)

where s is the density of the medium. When electric charge is present, Gauss's equation

Div D = 4rr q gives

(ex6 + ezl ) u,~. - - e14 u,~ + e14 v, x, + (e~ + esO v,~, + e~ (w ,~ + w , . )

§ eaa w,.. - - ~11 (~,xx Jr ~ ,w) - - ~83 ~,z , : 4rrq (5)

W a v e propagat ion in p iezoe lec t r ic med ium

From (2) and (3) we get

a 4 ~71 s A + as A, , , + as V1 ~ 2: = A, , ,

at Vz ~ A + as A ,~ + a3 Vz 2 r + as Vz 1 27 = A,, , .

From (4) and (5) we obtain

as A,~, + as \71 s r + as r,~, + as ~71 s 27 + a 7 27,z, = r m �9

as A,, , + as A, , , + as V1 ~ r + a~ r , , ,

- ato Vx z Z' -- atx 27,,, = 4 ~" q,z-

In eqs (6) to (9) we have taken

A = (v,~ - - u , , ) ; A = (u,~ + v,,);

r=w,, ; 27 = $ , ,

s a t : c u ; saa = c s s

s a4 = ces; s as = c 4 4 ;

s aT = e~; s as ---- (els+e31);

S a t o : e n ; s a t 1 = ~ss

The displacements u, v, w

and

s as = c,t+cxs;

s as ---- el5 ;

s as = e14 ;

127

(6)

(7)

(8)

(9)

(10)

and potential ~ are completely the variables A, A, 1 ~ and 27.

We first consider the case (case 1) when the point source is an isolated harmonic electric charge of fixed frequency oJ acting at the origin; that is,

q ---- qo 3(x) 80,) 8(z) exp (-- i w t) (11)

where qo is constant and 8(x) is Dirac's delta function. We now express the variables A, A, F, 27 and q in terms of Fourier integrals in

the form

p ___ foo foo foo P exp [i ( a x + fly + y z - - cat)] d~ dfl dT. (12) j - - o o j - - o o j - - o o

Substituting integrals of the above form for the variables in eqs (6) to (9) and considering the Fourier transform of the source term, we obtain

[a, ( ~ ' + ~ ) + as 7 ~ - - ,~q X + as ( ~ + ~ ) ,~ = 0

[a t ( a 2 + / ~ ) + a s ~ - o ~ s ] A + a s ( a s + ~ ) F W a s ( a a + / ~ ) ~ = 0

determined by

128 B Srinivasa Rao

a 3 # -A + [a 5 (t~" + ~ ) + a, z : - - o; ~]

+ [as(~' + ~) + a, : ] ~ = o

a, : X + as ~" A + [a, (a" + / ~ ) + av : ]

- [axo (~' + ~ ) + axl ~,'] s = - iq, ~,/(2 : ) (13)

3. Slowness surface

Solutions of eq. (13) are

A = X d ~ ;

where

H

Mxl

M ~

M~4

M3~

M~

M0z

-~ = x d ~ ; r = X~l~ and ~=KflH (14)

---- H (a, 13, V, o,) = det. I Mu[ (i, j = 1 to 4)

= a 4 ( ~ ~ + / ~ ) + as y' - o~', Mxs = Mxs = 0

= ao (~' + ~ ) , gsx = 0

---- al ( a~ + fro + a5 y~ - - ~ , M2a = ~ (a ~ + / ~ ) ;

= a s (a ' + fl~), Max = 0

= aa y2; Maa = a5 (a ~ + ff~) + aa yz _ to~

= a s (a' + ~ ) + a 7 ~,'; M a = a 9 ~,'; M,2 = as ~,'

= Mu; M ~ = - - [a~o ( ~ 2 + / ~ ) + au ~ ] (15)

and K~ ----- El (a, 13, ~,, oJ) is the determinant obtained from H, by replacing ith column with the column whose elements are respectively 0, 0, 0, - - i qo y/(2 ~r2). From (14) and 02), we find

Similar integrals hold for A, F and 27, the integrand containing (KdH) (i = 2, 3, 4 respectively) instead of Kx/tt. The only singularities of the integrand are the roots of H ---- 0. It is known (Buchwald [2]) that for large arguments, important contributions in evaluating integrals (16) arise only from points on the real surface H = 0 in a, 13, y space (a-space). Further outward travelling waves from source will have large contribution to the integrals. H = 0 regarded as an equation of a,/3, y gives the slowness surface for A, /k , F and 27 for real values of a, fl, y. We find that it is a surface of revolution. The plane y = 0 is a plane of symmetry and ~,-axis is the

Wave propagation in piezoelectric medium 129

axis of revolution. This surface can be represented as follows: Consider the section of H = 0 with plane/3 = 0 given by

H(a, 0, ~,) = 0. (17)

Since H i s homogeneous in a, fl, ~ and ~o we can put ~, = ka where k is a constant. The resulting quartic equation in a~ can be solved. One of the four roots is found to be zero and the corresponding sheet o f the surface collapses to a point viz. the origin. This corresponds to a wave with infinite velocity generated at the origin. Corresponding to each real positive root of the quartie, one real sheet of the surface is obtained. By giving arbitrary values of k at suitable intervals, we obtain a set of values (a/oJ, y/oJ) corresponding to each non-zero positive root, which when plotted on the plane/3 = 0 gives a curve. Rotating this curve about r-axis, a sheet of the slowness surface is obtained. Hence we find that in a piezoelectric medium of class (6), the slowness surface has in general three sheets.

In the case of piezoelectric medium of class (6 m m) only a 9 = 0. Corresponding to eqs (14) we have

A = 0

: [i q0/(21r~)] y (a ~ q- t33) {(asa s -- azao) (a ~ + ~ )

q- (aza s - - aaaT) ya - - a s co z}/H1

B

F = [i q0l(2~rz)] y {[a x (a s q- ~ ) -q- a 5 Va--eo ~] (18)

x [a~ (~,~ + ~'.) + a, r ~] - a~a3 (~, + f~) r~}l~r,

2 = [iq,l(2~)] r H , I ~

where H 1 is det [ Mu l, (i, j = 2, 3, 4)

and /r = ~ ' (a '~ + / ~ ) y~' - [ a i (o~" + / ~ ) + a5 }" - - ,.,.;']

x [a~ (,~' + / ~ ) + a, ~,~ - ,,.,~]. (19)

We conclude that rotational wave A does not propagate in the medium and the waves of A, F, and 27 are coupled, the slowness surface being H 1 = 0. We find that this surface has two real sheets corresponding to the two real positive roots of the quadratic equation in a s given by H 1 (a, 0, y) = 0.

I f we consider piezoelectric medium of class (6 2 2) we have a s = a 7 = a s = 0 and a 8 ~ 0. In this case, eq. (18) becomes

A = [i q0/(2=')] a , ~ (a ' + ~ ) / / &

= F -~ 0 and (20)

2 = [i q0/(2=~)] ~, I-Z411-&

130

where

B Srinivasa Rao

~r8 = [a4 (~' + ~ ) + as ~' - o:] [a~0 (~' + ~ ) + a~x ~1

+ a~ ~- (~, + ~) ~.

and / /4 = a 4 (r + ~ ) + as ~ - - ~

We find that waves of dilation A and F do not propagate. the slowness surface H s -: 0 which has only one sheet.

(21)

Also A and 27 have

4. Wave surface

The wave surface is the envelope (at time t = 1) of plane waves originating from a source at the origin at time t : 0. A point (a,/3, y) of a-space corresponds to a plane wave passing through a point (x, y, z) of x-space provided the vector (x, y, z) is parallel to the normal at (~, /3, 7) to the slowness surface and the parametric equation of the wave surface is

x : - - H , a / H , ~ o ; y : - - H , [ J / H , co; z=: - - -H,~ , /H , co. (22)

We obtain the wave surface corresponding to the slowness surface/at 1 = 0. This is the surface of revolution obtained by rotating about the z-axis in plane y : 0, the curve

where

and

R x : aoJ [ b l a 4 - - b 2 a S y s - - b e 7 4 - - b4aSoJs - - bs?So~ 2 + alooJ 4]

R = o: ( a , ~ ~ + (a:~ + , ~ ) ~

+ (a~o ~ + a ~ : ) [(a~ + a~) ~ + (a~ + as) : - 2o:1}

b l = 3al (asalo + ao 2)

b~. : 2 [ 2 a e ( a a a a - - aTa 0 - - asas 2

- - a s ( a s a l o + a t a l l q- as z) - - e l0 (ala,z - - as e) ]

b a = a~ [2 ( a a a 8 - - % a s ) - - a la~]

b 4 = 2

b 5 = 2

b e = 3

by : 2

a,a (asal 0 Jr- ala n -~- a82) -- all (as s - - a32)

[alo (al + as) + aeq

aea~ + asau + a~alo + asalo + alan + a s 2

a5 (a~all + a:)

[a n (a s -a t- as) + a7 2]

(23)

(24)


The method of Lighthill [3] gives us the asymptotic contributions (in 6 m In) medium for A, F and 27 for large r as

A , ~ (2/r) ~ N= 1 An qo i An y.(an 2 +/3.2) X

[(asa 5 - - azae) (a. 2 +/3. 2) + (a~aa - - aaaT)7. 2 - - asia2]

V '~ (2/r) ~nN= 1 A. q0 iA. y. { [ax (=2 +/3.2) + as 72 _ ojg]

• [a e (r +/3.2) + aTy=) __ azas (a2 +/3z) y2} (25)

27 ~-~ (2/r) ~ N = 1 A. qo i ~ny. ( ~2 ( a , +/3.2) y s __

_ lax (c~2 +/3.2) + an 7 2 _ oJ21 [a. (~.~ + / 3 . 2) + a 7 y. 2] }

where summation is taken for points (a.,/3., 7.) on the slowness surface at which the normal is parallel to OP, the point P(x, y, z) lying on the wave surface. A. is phase constant. For an axisymmetric ease, it is enough if we evaluate amplitude coefficient )~. corresponding to fl=0. Then ~2 is obtained as

)~2 n =(/-/'2= + H2y)/[FI## (H,22 H,~, r --2/-/',= //,y /-/,=y +/-/'2y Ha=)]

(26) For (6 2 2) medium the asymptotic contributions for A, 27 are

A , ~ ( 2 / r ) ~ N = 1 A"q~ A" a ' Y" (="2 +/3"2)

and , ~ (2/r ) ~ N = 1 An qo i An Yn [a4 (an2+/3 .2)+as 7.~--~ �9

5. Source of body force

The source of disturbance could be a body force instead of an electric charge. We now consider (case 2) an isolated harmonic body force of fixed frequency acting at the origin along the x-axis, the electric charge q being absent. The body force is given by

X = X 0 3(x) 8(y) 3(z) exp (-- ioJt) (27)

Y = Z = 0 .

Gauss's equation in this case is Div D = 0 and eqs (13) are modified as

[a 4 (a2+~) +as72--to 2] -A+a~ (a~'+/32)ff:--iflXo/(8 ='~)

[ a x ( a ' + ~ ) -I- asy 2-oJ21 A --[- aa( ~2 + fl.z) ~"

+ a8( 2 +/32) f = i x0/(8 : ) (lY)

132 B Srinivasa Rao

az?' A + Is+ ( ~ + fl~) + aa ~' - - oJ'] P-"+ [a n (a n +/32) + aT y,) ~, = 0

a9 '~ A + a s ?~ A + [as (~" + / ~ ) + aT y~) P

- [,+1o (,~" + ~ ) + a~ ~'-') z v = 0.

Solving the above equations we obtain

= L d ~ ; -A = L,/Zr F = q / ~ r and U = z ~ / ~ (28)

where L~ -~ Ll (a, /3, ?, +,o) is the determinant obtained from H by replacing ith column with the column whose elements are respectively - - i / 3 X o / (8~), l a x o / (8rr3), 0,0.

I f we consider the medium of class (6), the slowness surface is the same as in case 1. In (6 m m ) piezoelectric medium we obtain

A = - i#a-o (8 ,p zr

and

A = - - iaXo {[a5 ((~2 +/32) + a.z ~,a - - to z] [alo (a ' -4- ~ ) -4- axx ~t)]

+ [a+ (~z + t32) + aT ?212}/(8~rsH1 )

= i~Xo?~[(aaalo + asas)(aa + fin) + (a3axl + a~as) ~'z]/(8rrn/-/'x)

(29)

,8 : iaXoy2 [(aaaa - - ass 8 ) ( ~ +/3a) + (aaa 7 _ a2aa)y2 + as~2] / (8 ~r all1).

The slowness surface for A exists and is given by H4=0. In this ease the slowness surface is identical with that of transversely isotropie elastic medium. Here the slowness surface is a spheroid and the corresponding wave surface is also a spheroid. The slowness surface for A, F and 27 is the same as in case 1.

The asymptotic contributions for large r can be obtained as in case 1. Considering medium of class (6 2 2), we obtain

A = i/3 X 0 [ax0 ( a~ + t32) + axl y~] / ( 8 ~ Ha)

= i a X 0 [a 6 (a z + / 3 2 ) + a2 ?2 _ t o ~ ] / (8~ra Ha )

F- : i a X o a z ~9. ] (8rt3 H2), and

~, -= i/3 X o a 9 ~ ] (8~ a Ha)

(30)

A and 27 have the same slowness surface as in case 1. The surface for /k and P exists and is given by hrs. = 0. The asymptotic contributions for large r can be obtained as in case 1.


6. N u m e r i c a l resu l t s

Since all the constants for a class (6) crystal are not available we shall consider (6 ram) piezoelectric medium in this section. We choose the surface H x = 0 for obtaining numerical results and use the constants of cadmium selenide crystal (Berlincourt etal[l])

c n = 7"41 x 101~ N / m s

cia ~ - 4"51

cia - - 3 . 93

can : 8 -36

c44 : 1"32

exs = - - 0 . 1 3 8 c o u l / m ~

eax = - - 0 . 1 5 9

ea3 = - - 0"347

s = 5 6 8 4 k g / m a

~zt = 84 .4 • 10 - i2 F / m

eaa = 90"3

(31)

We consider the section of the surface H t (a, fl, y ) = 0 with the plane fl=0. In Hi(a,0, ~,)=0 we put 9,=ka and evaluate the non-zero -oots of the resulting quadratic equation in ea for different values o f k ~. The corresponding values of y~ are obtained.

Table 1. Piezoelectric case

Coordinates of Slowness Coordinates of wave Amplitude coefficient surface with 8 = 0 surface with y = 0 x 10 TM

• 10 -4 see]met • 10 a met/see kg2/(Sec Coul)~

~/~ y/a~ x z a

First sheet

0.00 2.59 O'OO 3"86 6"6 1"09 2.44 1"04 3"63 5.2 1"23 2.40 1-21 3"54 4-9 1.36 2"35 1"38 3"45 4"7

1"46 2"31 1.52 3"37 4"5 1.67 2.20 1.83 3-16 4.2 1-98 1-98 2"34 2.70 3"9 2.04 1"93 2.44 2-61 3"8

2.10 1.87 2.54 2"50 3"8 2.16 1"81 2"65 2.37 3"8 2"66 0.84 3.46 0"93 4"3 2-77 0.00 3-61 0-00 4-6

Second sheet

0.00 6.56 0.00 1.52 0-2 2.37 5"30 1.34 1.29 0.8 2-60 5"06 1"40 1"26 1' 1 2.79 4"84 1-42 1"25 1.8

2.95 4.66 1.43 1.24 oo 3-77 3.77 1.32 1-34 1-7 3.87 3.67 1.30 1.36 1.8 3.98 3.56 1.28 1.38 1.9

4-04 3-50 1.27 1-39 2.0 4.55 3.05 1.24 1.43 5.80 1.83 1.38 1-09 0.8 6.51 0-00 1.54 0.00 0.5

P. (A)----4

134 B Srinivasa Rao

Table 2. Elastic case

Coordinates of slowness Coordinates of wave Amplitude surface with 8=0 surface with y = 0 coefficient all x 10 -4 see/met all x l0 s met]see all x 10-' a/a~ ~,#o x z sec~/met*

0.00 1.21 1.40 1.46

1.59 1.76 1.98 2.16

2-23 2.44 2.66 2.77

0.00 2.59 2.89 2.99

3.18 3-43 3.80 4.01

4.28 4.80 5.81 6-56

First sheet

2.61 0-00 3.85 6.4 2.42 1-23 3.53 5.3 2.34 1.47 3.39 5.0 2.31 1-55 3.34 4.9

2.25 1.73 3-22 4.7 2-15 1.98 3.03 4.5 1.98 2.34 2.70 4.2 1.81 2.64 2.38 4.0

1.73 2-76 2.23 4.0 1.40 3.12 1.69 4.0 0.84 3.46 0.93 4.2 0.00 3.61 0.00 4.4

Second sheet

6.56 0.00 1.52 1.5 5.17 1.34 1.26 5.0 4.84 1.40 1.23 8.7 4"73 1.41 1.22 11"5

4.50 1.42 1.22 oo 4.21 1.41 1.23 15.2 3.80 1-35 1.28 26.9 3.59 1-32 1.31 24.1

3.32 1.28 1.36 15"4 2.84 1.26 1.39 oo 1-84 1.37 1.12 4.0 0.00 1.52 0.00 2"3

~'4-

FigureTl. Slowness surfaces obtained by rotating the above curves about the ~-axis. (1) First sheet: (2) Second sheet.

x and z are found with the help of eqs (23). The ampl i tude coefficient An is obta ined f rom eq. (26). The results for the two sheets in the case of piezoelectric med ium are given in table 1. The corresponding results, t reat ing the medium as elastic, are given in table 2. The numerical results are also represented graphically in figures 1 and 2.

Wave propagation in piezoeleetriv medium 135

�9 . : ' : 4 0

F i g u r e 2. Wave surfaces obtained by rotating the above curves about the z-axis. (1) First sheet; (2) Second sheet.

The portions of curves where results for elastic medium are different from piezoelectric medium are shown by dotted line.

7. Conclusions

Case I : Source of electric charge: Rotational wave A is not propagated ifi (6 m m) piezoelectric medium whereas A, F and 27 waves possess a slowness surface of two sheets. The waves A and F are not propagated in (6 2 2) piezoelectric medium whereas A and 27 waves have a slowness surface of only one sheet. We put y = 0 in (18)

and (20) and find that A, A, F and 27 are zero in both media. Hence there is no wave propagation in the plane z ---- 0.

Case 2: Source of body force: A wave has a slowness surface in (6 m m) piezoelectric medium and A, F and 27 waves have the same slowness surface as in case 1. The waves a and F have a slowness surface in (6 2 2) piezoelectric medium. We note that in both the cases where the slowness surface does not exist for the source of electric charge, the surface obtained for the source of body force is the same as in elastic (6 m m) and (6 2 2) media. A and 27 in (6 2 2) medium have the same slowness

surface as in case 1. We put a=fl=O in equations (29) and (31) and find that A, A,

and f l a r e all zero. Therefore there is no wave propagation along the z-axis for both the media. The numerical results obtained for cadmium selenide (6 m m) medium enable us to observe the following. The slowness and wave surfaces are very nearly the same when the medium is considered piezoelectric or purely elastic. However, the amplitude coefficients vary considerably in the two media. Hence the electromechanical interaction does not affect much the slowness and wave surfaces but has considerable influence on the amplitude coefficient. It may be pointed out that tlie change will be appreciable if the medium of piezoelectric ceramic is considered instead of that of piezoelectric crystal since the electromechanical coupling factor for ceramic (oo m) is large compared to a crystal. It may be mentioned that the

136 B Srinivasa Rao

piezoelectric constants noted in the table given in [5] are wrong. results given in [5] cannot be relied upon.

Hence numerical

Acknowledgements

The au thor expresses his grateful acknowledgements to Dr H S Paul for his useful suggestions. The author also wishes to thank a referee for his valuable suggestions.

References

[1] Berlincourt D, Jaffe H and Shiozawa L R 1963 Phys. Rev. 129 1009 [2] Buchwald V T 1959 Proc. R. Soc. A253 563 [3] Lighthill M J 1959-60 Philos. Trans. A252 397 [4] Musgrave M J P 1954 Proc. R. Soc. A226 p. 339 [5] Paldas M M 1971 Czech. J. Phys. B21 673 [6] Synge J L 1957 J. Math. Phys. 35 323

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Wave propagation in piezoelectric medium of hexagonal symmetry