THEORY OF BRILLOUIN SCATTERING BY AN ISOTROPIC ELASTIC PLATE

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THEORY OF BRILLOUIN SCATTERING BY AN ISOTROPIC ELASTIC PLATE

D. Tilley, E. Albuquerque, M. Oliveros

To cite this version:

D. Tilley, E. Albuquerque, M. Oliveros. THEORY OF BRILLOUIN SCATTERING BY AN ISOTROPIC ELASTIC PLATE. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-103-C5-107.

�10.1051/jphyscol:1984514�. �jpa-00224132�

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JOURNAL DE PHYSIQUE

Colloque

C5, supplkment

au

n04,

Tome

45,

avril

1984 page C5-103

THEORY OF B R I L L O U I N SCATTERING

BY AN

I S O T R O P I C E L A S T I C P L A T E

D.R.

T i l l e y , E.L. Albuquerque* and M.C. Oliveros*

Department o f Physics, U n i v e r s i t y of Essex, Colchester COQ

3SQ,

U.K.

* ~ e ~ a r t a m e n t o de Fisica, Universidade Federal do Rio Grande do Norte, 59.000-Natal-RN, Brazil

RLsumL

-

Nous p r d s e n t o n s une t h s o r i e de l a d i f f u s i o n de l a lumiPre p a r l e s ondes s u p e r f i c i e l l e s d'une plaque L l a s t i q u e r 6 s u l t a n t d e s modes d e t o r s i o n

( o n d u l a t i o n ) e t d e s modes l o n g i t u d i n a u x ( p d r i s t a l t i q u e s ) . Les r g s u l t a t s s o n t i l l u s t r L s p a r des c a l c u l s numgriques.

A b s t r a c t

-

The theory of s u r f a c e - r i p p l e l i g h t - s c a t t e r i n g by t h e bending ( u n d u l a t i o n ) and l o n g i t u d i n a l ( p e r i s t a l t i c ) modes of an e l a s t i c p l a t e i s p r e s e n t e d , t o g e t h e r w i t h r e s u l t s of i l l u s t r a t i v e numerical c a l c u l a t i o n s .

The propagating modes i n an e l a s t i c p l a t e of g e n e r a l t h i c k n e s s a r e c a l l e d Lamb waves;

both t h e d e r i v a t i o n of t h e Lamb wave spectrum and i t s d e s c r i p t i o n a r e r e l a t i v e l y complicated./l/ I n t h e l i m i t of t h i c k n e s s L much l e s s than a c o u s t i c wavelength Xa t h e equations of motion can be g r e a t l y s i m p l i f i e d . / 2 / I t i s then found t h a t t h e modes c o n s i s t of a bending, o r u n d u l a t i o n a l , mode i n which t h e two i n t e r f a c e s move

i n phase, and a l o n g i t u d i n a l , o r p e r i s t a l t i c mode i n which t h e i n t e r f a c e s move i n a n t i p h a s e . These a r e i n f a c t t h e lowest o r d e r Lamb waves. / 3 / The bending mode of an i s o t r o p i c p l a t e of d e n s i t y p ' supported on an i d e a l l i q u i d of d e n s i t y p has d i s - p e r s i o n e q u a t i o n

where Qx i s t h e wave v e c t o r i n t h e p l a n e of t h e p l a t e , and

E i s Young's modulus, o P o i s s o n ' s r a t i o and vT t h e b u l k t r a n s v e r s e sound v e l o c i t y . The l o n g i t u d i n a l mode of an unsupported p l a t e has d i s p e r s i o n e q u a t i o n

w = v Q

P x

where v 2 = ~ / ~ ( l - o ~ ) , s o t h a t v < v <

v

where v i s t h e bulk l o n g i t u d i n a l sound

P T P - L' L

v e l o c i t y .

Green f u n c t i o r ~ s and f l u c t u a t i o n power s p e c t r a a r e found by means of l i n e a r response t h e o r y and t h e f l u c t u a t i o n - d i s s i p a t i o n theorem. For t h e bending mode t h e s e q u a n t i - t i e s a r e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984514

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C5-104 JOURNAL DE PHYSIQUE

and

where

2

i s t h e s u r f a c e a r e a of t h e f i l m and w i s t h e frequency g i v e n by eqn. ( 1 ) .

Q

The r e s u l t s a r e g i v e n i n terms of u t h e displacement of t h e median p l a n e i n t h e z'

normal d i r e c t i o n , s i n c e a l l s t r a i n and displacement components can b e found from u

.I2/

The main q u a l i t a t i v e f e a t u r e s of eqn. (5) a r e t h e p r o p o r t i o n a l i t i e s t o Q - ~

X

and L - ~ .

For t h e l o n g i t u d i n a l mode of an unsupported f i l m we f i n d

and

The t h e o r y of B r i l l o u i n s c a t t e r i n g by Lamb modes has been t r e a t e d i n a g e n e r a l way by B o r t o l a n i e t a1./4/ They i n c l u d e coupling v i a t h e "bulk" acousto-optic mechanism a s w e l l a s by t h e s u r f a c e r i p p l e mechanism. I n p r i n c i p l e t h e r e s u l t s f o r t h e c r o s s s e c t i o n f o r s c a t t e r i n g by t h e bending and l o n g i t u d i n a l modes should be o b t a i n a b l e a s l i m i t i n g cases of t h e i r e x p r e s s i o n s . I n p r a c t i c e , however, i t i s much simpler t o d e r i v e t h e r e s u l t s d i r e c t l y w i t h t h e u s e of t h e power s p e c t r a of eqns. (5) and ( 7 ) . We r e t a i n only t h e s u r f a c e r i p p l e mechanism, which a s w i l l b e seen, i s dominant f o r s c a t t e r i n g by t h e bending mode of a supported p l a t e , and f o r s c a t t e r i n g by t h e l o n g i t u d i n a l mode whether t h e p l a t e i s supported o r n o t .

S c a t t e r i n g by t h e r i p p l e s on a s i n g l e i n t e r f a c e was t r e a t e d by Marvin e t a l . , / 5 / a n d t h e method i s r e a d i l y g e n e r a l i s e d f o r two i n t e r f a c e s . P a r t i c u l a r importance a t t a c h e s t o t h e asymptotic forms of t h e s c a t t e r e d f i e l d s f o r small L. We f i n d t h a t f o r an unsupported p l a t e b o t h t h e back s c a t t e r e d and t h e forward s c a t t e r e d f i e l d s a r e of o r d e r L f o r s c a t t e r i n g by t h e bending mode b u t independent of L f o r s c a t t e r i n g by t h e l o n g i t u d i n a l mode. These r e s u l t s hold f o r both s and p ~ o l a r i z a t i o n ; they a r i s e , l o o s e l y speaking, because t h e i n t e r f e r e n c e between l i g h t s c a t t e r e d o f f t h e two i n t e r - f a c e s i s d e s t r u c t i v e f o r t h e bending mode but c o n s t r u c t i v e f o r t h e l o n g i t u d i n a l mode.

For a supported p l a t e , i n which t h e r e f r a c t i v e i n d i c e s on e i t h e r s i d e of t h e p l a t e a r e d i f f e r e n t , t h e s c a t t e r e d f i e l d s a r e independent of L i n a l l c a s e s . This f a c t s u b s t a n t i a t e s t h e a s s e r t i o n t h a t t h e r i p p l e mechanism i s dominant f o r s c a t t e r i n g by a supported p l a t e , s i n c e t h e acousto-optic c o n t r i b u t i o n i s of o r d e r L.

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Given t h e s c a t t e r e d f i e l d s , t h e c a l c u l a t i o n of t h e d i f f e r e n t i a l c r o s s s e c t i o n i s s t a n d a r d . / 6 / We c o n s i d e r i n p a r t i c u l a r s c a t t e r i n g by an unsupported f i l m i n s p o l a r i z a t i o n when t h e p l a n e of s c a t t e r i n g c o i n c i d e s w i t h t h e p l a n e of incidence.

For a n g l e of i n c i d e n c e OI and angle of s c a t t e r i n g ^€is t h e frequency s h i f t Aw f o r Stokes s c a t t e r i n g o f f t h e bending mode i s given by

and t h e d i f f e r e n t i a l c r o s s s e c t i o n i s

d20 cos2e

- a - S

lr,I2

dCldwS cose S(w-0

I L(sin0 _I

-

^{s i n e}_S^{) 4}

^Q

where wI i s t h e i n c i d e n t frequency and

TS

i s a slowly varying f u n c t i o n of €IS and 8 I' The corresponding r e s u l t s f o r t h e l o n g i t u d i n a l mode a r e

I I

cAw/vTwI = 2 ' / ( ~ - 0 ) ' l s i n e ~

-

s i n e S

I

⁽¹⁰⁾

and

The c r o s s s e c t i o n s a r e i l l u s t r a t e d i n Figs. 1 and 2 which show t h e i n t e g r a t e d i n t e n - s i t i e s i n each case. As i s s e e n from eqn. (9) and Fig. 1, t h e c r o s s s e c t i o n f o r s c a t t e r i n g o f f t h e bending mode i s dominated by t h e s t r o n g s i n g u l a r i t y due t o t h e term ( s i n 0

-

i n t h e denominator, and i t i n c r e a s e s w i t h d e c r e a s i n g L.

I

Both t h e s e f e a t u r e s stem from t h e Q ~ ~ L - ~ dependence of t h e power spectrum noted under eqn. ( 5 ) . The c r o s s s e c t i o n f o r s c a t t e r i n g o f f t h e l o n g i t u d i n a l mode, con- v e r s e l y , d e c r e a s e s w i t h d e c r e a s i n g L. The f a c t o r L i n eqn. (11) i s t h e product of t h e L-' shown i n eqn. (9) w i t h L2 due t o t h e conversion of t h e power spectrum from

Acknowledgements

We thank N. A. Clark, R. Loudon and J. F. S c o t t f o r h e l p f u l d i s c u s s i o n s , and t h e B r i t i s h Council, CNPq and UFRN f o r f i n a n c i a l support.

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JOURNAL

DE PHYSIQUE

F i g . 1

-

I n t e g r a t e d i n t e n s i t y v e r s u s s c a t t e r i n g a n g l e €IS f o r B r i l l o u i n s c a t t e r i n g o f f t h e bending mode of a n unsupported f i l m w i t h o p t i c a l d i e l e c t r i c c o n s t a n t ^E= 1.5 a t a n g l e o f i n c i d e n c e '81 = a / 4 . The c u r v e s c o r r e s p o n d t o d i f f e r e n t v a l u e s of LwIfC:

- - -

^{0 . 1 ;}

- . - . - .

^0.2;

- .. - .. -

^{0 . 3 ;}

- ... - ,..

^{0 . 4 ;}

- .... -

^{0 . 5 ,} t h e l a s t 3 b e i n g i n d i s t i n g u i s h a b l e o v e r p a r t of t h e range. The v e r t i c a l s c a l e i s l i n e a r .

Fig. 2

-

As F i g . 1 f o r t h e l o n g i t u d i n a l mode.

References

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Academic P r e s s , New York 1964) 112.

/2/ LANDAU L.D. and LIFSHITZ E.M., Theory o f E l a s t i c i t y (Pergamon, Oxford 1970).

/3/ AULD B.A., A c o u s t i c F i e l d s and Waves i n S o l i d s , Vol. I1 (John Wiley, London 1973).

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/4/ BORTOLANI V., MARVIN A.M., NIZZOLI F. a n d SANTORO G . , J. Phys. C: S o l i d S t a t e Phys. l&( 1 9 8 3 ) 1 7 5 7 .

/ 5 / MARVIN A . , TOIGO F. a n d CELL1 V . , Phys. Rev. ( 1 9 7 5 ) , 2 7 7 7 .

/ 6 / ALBUQUERQUE E.L., J. Phys. C: S o l i d S t . Phys.

2,

( 1 9 8 0 ) , 2 6 2 3 , f o r e x a m p l e .