O n t h e a n a l y s i s o f R a y l e i g h w a v e i n e l a s t i c m e d i a I t . G . B H A T I A ( * ) - J . C . D R A K O P O U L O S ( * * ) R e c e i v e d o n O c t o b e r 20, 1971 SUMMARY. — I n t h i s p a p e r t h e R a y l e i g h w a v e e q u a t i o n h a s b e e n s i m p l i f i e d t o t w o s i m p l e e q u a t i o n s , t h e b i f u r c a t i o n p o i n t b e i n g t h e r a t i o of s h e a r t o l o n g i t u d i n a l w a v e v e l o c i t y . T h e p u r p o s e of t h e s t u d y is t o ob- t a i n t h e p h a s e v e l o c i t y w i t h o u t b e i n g i n v o l v e d i n t o t h e s o l u t i o n of s i x d e g r e e e q u a t i o n w h e n t h e c o n s t a n t s of t h e m e d i a a r e k n o w n . RIASSUNTO. •— I n q u e s t a n o t a l ' e q u a z i o n e delle o n d e di R a y l e i g h è s t a t a r i d o t t a a d u e s e m p l i c i e q u a z i o n i , il cui p u n t o di i n t e r s e z i o n e r a p p r e - s e n t a il r a p p o r t o f r a la v e l o c i t à delle o n d e s u p e r f i c i a l i ( t r a s v e r s a l i ) e q u e l l a delle o n d e l o n g i t u d i n a l i . S c o p o dello s t u d i o è di o t t e n e r e l a v e l o c i t à d i f a s e , q u a n d o lo c o s t a n t i dei m e z z i s o n o c o n o s c i u t e , e v i t a n d o cosi di risol- v e r e u n ' e q u a z i o n e di s e s t o g r a d o . 1 . I N T R O D U C T I O N I t is shown in treatises on theory of elasticity t h a t two types of wave motion are propagated in an isotropic homogeneous and un- bounded elastic medium. These are the longitudinal or t h e compres- sional (P) waves and t h e transverse or the shear (S) waves. Their velocities are respectively given by are Lame's constants of the media and o is t h e density. I n any media the P waves travel faster t h a n t h e 8 waves. All the e a r t h q u a k e re- cords show the presence of surface waves in addition to these. 1/ and where A, ¡i (*) School of R e s e a r c h a n d T r a i n i n g i n E a r t h q u a k e E n g i n e e r i n g , U n i - v e r s i t y R o o r k e e , I n d i a . (**) S e i s i n o l o g i c a l L a b o r a t o r y , A t h e n s U n i v e r s i t y , G r e e c e . 2 2 K . G . B H A T I A - J . C . D R A K O P O U L O S The surface waves comprising t h e Rayleigli and Love waves are distinguished f r o m t h e bodily P a n d S waves in being more or less confined to t h e surface. Thus, their a m p l i t u d e s decrease v e r y r a p i d l y with d e p t h below t h e surface. I t has always been t h e necessity to o b t a i n t h e p h a s e velocities of surface waves in order to analyse t h e behaviour of these waves in a n y media. I n this p a p e r t h e a u t h o r s have paid p a r t i c u l a r a t t e n t i o n on t h e Rayleigli t y p e of wave only. 2 . F O R M U L A T I O N O F T H E P R O B L E M A N D C O N C L U S I O N The well k n o w n Rayleigli wave equation i s f 1 - 2 ) : f 2 [ | « — 8 f 4 + | 2 (24 — 16?;2) + 16 { r ¡ ~ 1)] = 0 [1] c b where f = a n d »? = , c is t h e p h a s e velocity of Rayleigli wave in b a elastic media and r¡ is t h e ratio of shear wave velocity to compres- sional w a v e velocity, a a n d b are t h e longitudinal a n d shear wave -• 12 u -}- "k -j / velocities in t h e m e d i a given bv a = 1/- - a n d b = / . F r o m I Q I S equation [1] we see t h a t one root of f is zero b u t this gives p h a s e velo- city equal t o zero a n d hence is of not use. Therefore f r o m [1] we o b t a i n |o _ + p (24 — 1 6 ^ ) + 16 (7f- — 1) = 0 [2] We m a y write it as r¡2 (16 — 16£2) + — 814 + 2 4 f 2 — 1 6 = 0 or 16 — 24I2 + 8f 4 — 11 = 16 ( I - ! 2 ) [ 3 ] Therefore 16 — 2 4 | 2 + 8£4 — £« 16 (1 — I2) w Since c is always less t h a n b, f is always less t h a n u n i t y . U N T H E A N A L Y S I S O F R A Y L E I G H W A V E I N E L A S T I C M E D I A 2 3 F r o m [4] we can obtain a plot of •>] versus We can also solve t h e e q u a t i o n [2] for f for different values of rj. A simple computer program was written for this and t h e results were obtained. M a x i m u m value of i] has been t a k e n as 0.70 because for ?/ higher t h a n this, Poisson's ratio becomes negative a n d no m a t e r i a l y e t has been obtained with negative Poisson's ratio (3). T a b l e I = 0 . 9 5 5 2 9 1 £ t e s 0.00 0 . 9 5 5 2 9 0.00 0.05 0 . 9 5 5 2 0 — 0 . 0 0 0 0 9 0.10 0 . 9 5 4 6 5 — 0 . 0 0 0 6 4 0.15 0 . 9 5 3 9 3 — 0 . 0 0 1 3 6 0.20 0 . 9 5 2 7 4 — 0 . 0 0 2 5 5 0.25 0.9511 1 — 0 . 0 0 4 1 8 0.30 0 . 9 4 9 0 4 — 0 . 0 0 6 2 5 0.35 0 . 9 4 6 3 4 — 0 . 0 0 8 9 5 0.40 0 . 9 4 2 8 6 — 0 . 0 1 2 4 3 0.45 0 . 9 3 8 3 5 — 0 . 0 1 6 9 4 0.50 0 . 9 3 2 4 9 — 0 . 0 2 2 8 0 0.55 0 . 9 2 4 7 3 — 0 . 0 3 0 5 6 0.00 0 . 9 1 4 1 6 — 0 . 0 4 1 1 3 0.65 0 . 8 9 9 3 6 — 0 . 0 5 5 9 3 0.70 0.87789 — 0 . 0 7 7 4 0 In order to m a k e t h e first value equal to zero, £„ = 0.95529 has been s u b t r a c t e d f r o m each value. A log-log plot is obtained for (£ — £„) vs 7) (see fig. 1). F o r 0 < ?/ ^ 0.40 we get one s t r a i g h t line a n d for values 0.40 < r) < 0.70 we get another s t r a i g h t line. E a c h s t r a i g h t line will give a n equation of t h e t y p e y = h xsec i.e f —1„ = k r]sec. E x a c t results are obtained by a d j u s t i n g t h e values of c o n s t a n t 7c. F r o m graph for 0 < 7] < 0.40 f = 0.95529 — (0.10 + 0.05?;) [5] a n d for 0.40 < < 0.70 £ = 0.95529 — (0.12 + 0.10?;) r f ™ [6] 2 4 K . G . B H A T I A - J . C . D R A K 0 P 0 U L 0 S Therefore for 0 < rj < 0.4 c = 0.95529 b — (0.10 + 0.5??) btf-™ [7] a n d for 0.4 < rj < 0.7 c = 0.95529 b — (0.12 + 0.10?;) bif -83 [8] F i g . 1 - R e d u c e d R a y l e i g h w a v e e q u a t i o n . l i q u a t i o n s [5] a n d [6] or more practically [7] a n d [8] are reduced R a y - leigh wave e q u a t i o n a n d these give results correct to t h i r d decimal place. T h e values of h = I / ^ a n d n = I/,-,-:— are calculated f r o m t h e I Q I A + 2^ given values a n d hence t h e p h a s e velocity can i n s t a n t l y , be obtained. I t can be n o t e d t h a t the p h a s e velocity v a r i a t i o n in both of t h e zones on ?/ i.e 0 < rj < 0.4 a n d 0.4 < •>] < 0.7 is different. I t is expected t h a t such equations will help i n t h e analysis of Rayleigh wave under certain strict b o u n d a r y conditions. Also, in order to analyse t h e Rayleigh wave i n anelastic media, where t h e dissipation accompanies v i b r a t i o n , equations of these two t y p e will help in obtaining t h e numerical solution of t h e problem. ON T H E A N A L Y S I S O F R A Y L E I G H W A V E I N E L A S T I C M E D I A 25 R E F E R E N C E S ( 1 ) BHATIA K . G . , a n d DKAKOPOULOS J . C . , 1 9 7 1 . - A study on the Displace- ment Components oj Bayleigh wave. " A n n a l i di G e o f i s i c a " , X X I V , 1, p p . 89-101. (2) BULLEN K . E . , 1963. - An Introduction to theory oj Seismology. C a n i b . U n i v . P r e s s . (3) HUANG Y. T . , 1968. - Effect oj an Elastically Restrained Boundary on SV-Wave radiation patterns. " B u l l . Seisin. S o c . A m . " , p p . 4 9 7 - 5 2 0 .