F u r t h e r R e m a r k s o n E x t r a R o o t s o f R a y l e i g h E q u a t i o n a n d S o m i g l i a n a W a v e s A . L . L E V S H I N ( * ) - Or. F . P A N ZA ( * * ) R i c e v u t o il 24 Giugno 1970 RIASSUNTO. — L e r a d i c i maggiori d e l l ' u n i t à dell'equazione di Rayleigh a p p l i c a t e a d u n semispazio elastico c o n t r i b u i s c o n o alla soluzione solo p e r v a l o r i a b b a s t a n z a g r a n d i del coefficiente di Poisson (a > 0.309). U n a delle r a d i c i c o r r i s p o n d e ai " l e a k i n g modes " i quali h a n n o velocità di fase m i n o r e della velocità delle o n d e l o n g i t u d i n a l i . U n ' a n a l o g a o n d a con d i v e r s a dispersione p u ò esistere nel caso in cui u n semispazio elastico è c o p e r t o d a u n o s t r a t o c a r a t t e r i z z a t o d a velocità a e fi minori di quelle del semispazio. L o spessore dello s t r a t o n o n d e v e essere t r o p p o piccolo in c o r r i s p o n d e n z a della l u n g h e z z a d ' o n d a . SUMMARY. —• T h e e x t r a r o o t s of t h e R a y l e i g h e q u a t i o n for an elastic h a l f s p a c e c o n t r i b u t e t o t h e solution only for large e n o u g h values of t h e Poisson coefficient (a > 0.309). One of t h e m corresponds t o leaking m o d e s w i t h t h e p h a s e velocity less t h a n t h e velocity of t h e l o n g i t u d i n a l w a v e . A similar w a v e w i t h d i s t i n c t dispersion m a y exists in t h e case w h e r e a n elastic h a l f s p a c e is covered b y a t h i n l a y e r w i t h lower velocities of elastic w a v e s . T h e t h i c k n e s s of a layer should be n o t too small in c o m p a r i s o n w i t h t h e w a v e l e n g t h . I N T R O D U C T I O N . Since the basic paper of Rayleigh (L) on surface waves in an elastic halfspace was published, a significant number of papers have considered the physical meaning of extra roots of the Rayleigh equation. (*) Soviet Geophysical C o m m i t t e e , Moscow ( U S S R ) . (**) I s t i t u t o di Fisica, I s t i t u t o di Geodesia, Bologna ( I t a l i a ) . 4(> L E V S K I N A. L . - P A N Z A G . F . Recently we h a v e t h e papers of Caloi (2), (3), (*), Caloi a n d R o m u a l d i (5) which link a t least one of these roots to t h e Somigliana waves a n d give some physical a r g u m e n t s a n d e x p e r i m e n t a l d a t a for their existence. Iii this review we w a n t to reconsider this problem, basing our a r g u m e n t s on some results of Ewing, J a r d e t z k y and Press (6). P h i n n e y (7), Gilbert a n d L a s t e r (8), Gilbert (9). 1. - Plane wares in a halfspace. Let us consider an elastic homogeneous halfspace a n d let a a n d /? be t h e velocities of P a n d S waves y = | J a n d a be t h e Poisson's ratio. The f u n d a m e n t a l Ravleigh e q u a t i o n for u n k n o w n p h a s e velocity c of surface wave r u n n i n g along a free surface is 2 (0 \2 l/2 (c \2 2 — h — 4 1 — « Like Rayleigh we assume t h a t t h e signs of radicals are positive, t h e n t h e e q u a t i o n h a s oidy one positive real root ci (0 < c i a c (etc = 0.263082) b o t h roots are complex a n d c o n j u g a t e d ; moreover if a > a s (as = 0.309) t h e n t h e real p a r t of 02 a n d C3 is less t h a n a, i.e. with u s u a l n o t a t i o n s Be 02,3 < a; finally if (Jc < a < as t h e n lie 02,3 > a. F o r 0 < (ic b o t h roots are real a n d C3 > C2 > a. These real roots are linked to waves which Caloi called Somigliana waves. Fig. 1 gives t h e roots versus a. I t was shown by F u (10) t h a t t h e p h a s e velocities C2 a n d C3 correspond to such angles of emergence e2 a n d e3 of t h e plane waves a t the free b o u n d a r y for which reflected plane waves of t h e s a m e kind do not exist; this is expressed b y t h e following f o r m u l a e F i g . 2 - D e p e n d e n c e of angles a n d i"s2,n on a. F U R T H E R R E M A R K S ON E X T R A R O O T S O P R A Y L E I G H E Q U A T I O N , E T C . 4 9 The dependence of angles of emergence e''2,3 a n d es°,3 on tr < ac is shown in Fig. 2. I t is easy to c o m p u t e t h e components of t h e displacement vector a t free surface as a f u n c t i o n of t h e angle of emergence of plane P or S wave of c o n s t a n t a m p l i t u d e . Such f u n c t i o n s for a = 0,25 are shown in F i g . 3. Tt is seen t h a t no peculiarities exist for angles of emergence e2 a n d e3 in b o t h cases; t h e more complicated behaviour of these f u n c t i o n s for S incident wave is due t o t h e existence of t h e cri- tical angle ec = arcos in this case ( u ) . Therefore plane wave t h e o r y gives no clew for u n d e r s t a n d i n g a p a r t i c u l a r role of these e x t r a roots of t h e Rayleigli equation. 2. - Lamb's problem. Following Gilbert (9) we write t h e solution of n o n s t a t i o n a r y two-dimensional L a m b ' s problem f = Re dick Jo (kr) | [3] 1 F {a>, k) where r is horizontal distance f r o m t h e source to t h e receiver, z is t h e d e p t h of t h e receiver, k is t h e wave n u m b e r , to = ck is t h e circular f r e q u e n c y ; / (to,k,z) depends on t h e source f r e q u e n c y spectrum, on t h e position a n d model (vertical or horizontal) of the receiver. F (to, k) is t h e E a y l e i g h f u n c t i o n , J0(kr) is t h e Bessel function. The e s t i m a t e of t h e inner integral in [3] is usually made by con- t o u r integral m e t h o d . D u e t h e presence of t h e radicals 1 ' CO2 , 1 „ CO2 V = / 1 — — - a n d V = / 1 I a 2 k 2 | ' k2 in / a n d F b o t h these f u n c t i o n are multivalued. To eliminate the m u l t i v a l u e n a t u r e of t h e f u n c t i o n b r a n c h - c u t s are introduced on t h e p l a n e of t h e complex variable co. Usually t h e y are t a k e n along t h e line Rev = 0, Re v' = 0. The B i e m a n surface is formed b y four sheets identified b y t h e signs of Re v a n d Re v'. To close the contour t h e sheet with Rev > 0, Re v' > 0 is used. I t is n o t e d as ( + , + ) sheet. W e are interested only in t h e f o u r t h q u a d r a n t of t h e to plane. F o r real positive k t h e r e is only one F U R T H E R R E M A R K S ON E X T R A ROOTS OP R A Y L E I G H E Q U A T I O N , E T C . 5 1 simple pole of t h e i n t e g r a n d r e l a t e d t o t h e Bayleigli r o o t of tlie e q u a t - ion F(o>,k) = 0 n a m e l y co = lee 1. T h e c o n t r i b u t i o n of t h e b o d y w a v e s is described b y t h e i n t e g r a l a l o n g t h e b r a n c h lines (Fig. 4a). Imu r ,Ko Kp Reu Fig. 4a - B r a n c h p o i n t s , b r a n c h c u t s , initial c o n t o u r of i n t e g r a t i o n 011 c o m p l e x p l a n e . This m e t h o d of describing t h e c o n t r i b u t i o n of t h e b o d y w a v e s is of course n o t u n i q u e . If we c u t t h e b r a n c h e s vertically d o w n f r o m b r a n c h p o i n t s co ka, co — kfi a n d we k e e p t h e u p p e r p a r t of t h e p l a n e 011 t h e s a m e sheet as before, i t is possible t o u n c o v e r p r e v i o u s l y inac- cessible p a r t s of o t h e r sheets w i t h Rev > 0, Rev' < 0 ( + j —) a n d Rev < 0, Rev' < 0 (—, —). T h e n o u r solution m a y b e p r e s e n t e d as a c o n t r i b u t i o n to t h e i n t e g r a l s along t h e n e w b r a n c h c u t s describing m a i n b o d y w a v e s (be- cause of t h e lca a n d kp poles) p l u s t h e c o n t r i b u t i o n of t h e R a y l e i g h pole p l u s t h e c o n t r i b u t i o n of t h e poles f r o m t h e u n c o v e r e d p a r t s of t h e (-{-, —) a n d t h e (—, —) sheets if such poles exist (Fig. 4b). A t our case t h e poles a t t h e u n c o v e r e d p a r t of (—, —) sheet are a b s e n t . F o r a < ac t h e poles co = kc2 a n d co = kcs r e l a t e d t o t h e e x t r a r o o t s of [1] are s i t u a t e d t o t h e r i g h t of co = ka on inacessible p a r t of ( + , —) sheet a n d do n o t c o n t r i b u t e t o our solution. T h e s a m e si- t u a t i o n exists f o r complex poles Rem2,3 ± ilma)2,3 w h e n a < 0,309; only f o r a > 0,309 t h e r e is a complex pole Recoil — iImco2,a a t t h e accessible p a r t of ( + , •—) sheet which c o n t r i b u t e s to t h e solution as so called P pulse or PL wave (e.g. P h i n n e y (7), G i l b e r t a n d L a s t e r (8), 5 2 LEVSILIN A . L . - P A N Z A ( i . F . Manuchov, Agurtsoff (13). This wave is a weak d i s t u r b a n c e behind P wave. I t becomes more a n d more distinct as a increases. As e x t r a roots do n o t c o n t r i b u t e t o tlie solution for a < 0.309. Somigliana waves do n o t exist i n this case. F i g . 4b - B r a n c h p o i n t s , b r a n c h c u t s , u s e d c o n t o u r of i n t e g r a t i o n on c o m p l e x p l a n e . 3. - Layered half space. L e t us consider t h e effect of t h i n elastic layer overlaying t h e same elastic half space with a < ac. Such a layer as Caloi (') mentioned is necessary t o generate t h e Somigliana waves. The thickness of a layer is H a n d its c o n g r e s s i o n a l a n d shear velocities are a 0 a n d fio respectively (/?» < /3). The f o r m u l a of displacement looks like [3] b u t t h e f u n c t i o n / a n d F shoidd be replaced b y t h e new f u n c t i o n s F* (co, h, z) a n d F* (co, 7c) which t e n d t o / a n d F as E — > 0. L e t us fix 1c > 0. Then for infinitely small IcE we h a v e t h e same situation as before. As TI increases t h e poles of F*(OJ,IC) shift f r o m their positions for H = 0. The Eayleigh pole keeps on t h e accessible p a r t of ( + , + ) sheet moving f r o m u> = Jcci to co = kc0 (here e0 is ve- locity of Eayleigh wave a t t h e halfspace with velocities of body waves a0 a n d fi„ along t h e real axis). T h e behaviour of other poles is much F U R T H E R R E M A R K S ON E X T R A R O O T S OP R A Y L E I G H E Q U A T I O N , E T C . 5 3 more complicated. The couple of poles on the real axis to t h e r i g h t of co = Iri s i t u a t e d on t h e covered p a r t of ( + , —) sheet unites in t h e c o n j u g a t e d couple a n d moves to t h e left as H increases. The pole which moves a t t h e f o u r t h q u a d r a n t eventually crosses t h e P branch-cut lm u KCO K(I0 __TL.KP + , + + , - KQ\KC 2 IJ A+.-) KC 3 ™o— R e u S P Fig. 5 - M o v e m e n t s of pole P + _ for fixed Ic and v a r i a b l e Tl. t o a p p e a r on t h e open p a r t of ( + , —) sheet. Only now it begins to c o n t r i b u t e to t h e solution as dispersive leaking mode PL (P+ -). As H increases more it crosses t h e uncovered stripe of t h e ( + , —) sheet a n d S b r a n c h - c u t , t o t h e covered p a r t of t h e same sheet. H e r e it unites to t h e c o n j u g a t e d pole a t t h e second q u a d r a n t to become real. T h e n one of t h e m still keeps on the same p a r t of ( + , —) sheet, a n d t h e other one overtakes t h e b r a n c h - c u t to a p p e a r 011 accessible p a r t of ( + , + ) sheet to t h e left of co = ft/?. Now it contributes as t h e first shear mode a n d its evolution is limited b y points co = /.•/?<> and co = 1, w a v e we g e t f r o m G i l b e r t ' s figure 23 t h e e s t i m a t e m i n (A:77) «a 1.9. I n f r e q u e n c y d o m a i n it m e a n s t h a t f o r 11 = 35 k m w a v e PL does n o t c a r r y p e r i o d s m o r e t h a n 140 sec a n d PLS w a v e m o r e t h a n 14 sec. A K N O W L E D G M E N T S . A. L . L e v s h i n t h a n k s t h e I t a l i a n N . R . C . f o r t h e g r a n t which m a d e possible his visit t o t h e I n s t i t u t e of Geodesy. B o t h t h e a u t h o r s t h a n k s P r o f . M. C a p u t o a n d P r o f . V . I . Keilis-Borok f o r e n c o u r a g e m e n t a n d h e l p f u l discussion. T h i s r e s e a r c h w a s s u p p o r t e d b y t h e I t a l i a n N.R.U. R E F E R E N C E S ( L ) R A Y L E I G H L . , On Waves Propagated along the Plane Surface of an Elastic Solid. « P r o o . L o n d o n M a t h . Soc. », 17, 4-11, 1885; or «Sci. P a p e r s » , 2, 4 4 1 - 4 4 7 ; « C a m b r i d g e Univ. P r e s s » , L o n d o n ( 1 9 0 0 ) . ( - ) C A L O I P . , L'equazione di Rayleigh e le onde di Somigliana. I I . La teoria di Somigliana rettifiche e conseguenze. « A t t i Acc. Naz. Lincei, Classe Scienze Fis. Mat. e N a t . » , X L I . 5 ( 1 9 6 6 ) . 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