DSpace at VNU: Explicit secular equations of Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity

Wave Motion 46 (2009) 427–434 Contents lists available at ScienceDirect Wave Motion journal homepage: www.elsevier.com/locate/wavemoti Explicit secular equations of Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity Pham Chi Vinh a,*, Géza Seriani b a b Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam Istituto Nazionale di Oceanografia e di Geofisica Sperimentale, Borgo Grotta Gigante 42/C, 34100 Sgonico, Trieste, Italy a r t i c l e i n f o Article history: Received 14 November 2008 Received in revised form 26 March 2009 Accepted 21 April 2009 Available online May 2009 Keywords: Rayleigh waves Rayleigh wave velocity Orthotropic Secular equation Non-homogeneous Gravity a b s t r a c t The problem of the Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity is investigated Using an appropriate representation of the solution we derive the secular equation of the wave motion in the explicit form Moreover, following the same approach, we obtain the explicit secular equations for a number of previously investigated Rayleigh wave problems whose dispersion equations were obtained only in the implicit form Ó 2009 Elsevier B.V All rights reserved Introduction Elastic surface waves in isotropic elastic solids, discovered by Lord Rayleigh [1] more than 120 years ago, have been studied extensively and exploited in a wide range of applications in Seismology, Acoustics, Geophysics, Telecommunications Industry and Materials Science, for example It would not be far-fetched to say that Rayleigh’s study of surface waves upon an elastic half-space has had fundamental and far-reaching effects upon modern life and many things that we take for granted today, stretching from mobile phones through to the study of earthquakes, as addressed by Samuel [2] For the Rayleigh waves, their dispersion equations in the explicit form are very significant in practical applications They can be used for solving the direct (forward) problems: studying effects of material parameters on the wave velocity, and especially for the inverse problems: determining material parameters from the measured values of the wave speed Thus, the secular equations in the explicit form are always the main purpose of investigations related to Rayleigh waves The problem on the propagation of Rayleigh waves under the effect of gravity is a significant problem in Seismology and Geophysics, and it has attracted attention of many researchers such as Bromwhich [3], Love [4], Biot [5], Gilbert [6], De and Sengupta [7], Dey and Sengupta [8], Datta [9], Das et al [10], Abd-Alla and Ahmed [12] Bromwhich [3], Gilbert [6] and Love [4] treated the force of gravity as a type of body force, while Biot [5] and the other authors, following him, assumed that the force of gravity to create a type of initial stress of hydrostatic nature Bromwhich [3] assumed that the material is incompressible for the sake of simplicity Love [4] finished Bromwhich’s investigation by considering the compressible case Biot [5] also took the assumption of incompressibility in his study Gilbert [6] used the Bromwich’s secular equation to * Corresponding author Tel.: +84 35532164; fax: +84 38588817 E-mail address: pcvinh@vnu.edu.vn (P.C Vinh) 0165-2125/$ - see front matter Ó 2009 Elsevier B.V All rights reserved doi:10.1016/j.wavemoti.2009.04.003 428 P.C Vinh, G Seriani / Wave Motion 46 (2009) 427–434 investigate the influence of gravity on the Rayleigh wave The material was assumed to be isotropic in the investigations [3– 7,9,10], transversely isotropic in [8] Most of the investigations supposed that the material is homogeneous However, because any realistic model of the earth must take into account continuous changes in the vertical direction of the elastic properties of the material, the problem was extended to the non-homogeneous case by Das et al [10] Das et al assumed that the material is isotropic and they obtained the implicit secular equation Recently, Abd-Alla and Ahmed [12] extended the problem to the orthotropic case Abd-Alla and Ahmed [12] employed two displacement potentials for expressing the solution, and they have derived the secular equation of the wave in the implicit form In the present work we analyze the orthotropic case and using an appropriate representation of the solution we derive an explicit form of the secular equation, which also provides the explicit secular equations for a number of previous investigations related to Rayleigh waves under the gravity, where only the implicit dispersion equations were obtained Note that a secular equation F ¼ is called explicit if F is an explicit function of the wave velocity c, the wave number k, and the parameters characterizing the material and external effects (see for example [13–15]) Otherwise we call it an implicit secular equation Basic equations Consider a non-homogeneous orthotropic elastic body occupying the half-space x3 P subject to the gravity We are interested in a plane motion in ðx1 ; x3 Þ-plane with displacement components u1 ; u2 ; u3 such that: ui ¼ ui ðx1 ; x3 ; tÞ; i ¼ 1; 3; u2  Then the components of the stress tensor tions [12]: 1ị rij ; i; j ẳ 1; are related to the displacement gradients by the following equa- r11 ẳ c11 u1;1 ỵ c13 u3;3 r33 ẳ c13 u1;1 ỵ c33 u3;3 r13 ẳ c55 u1;3 ỵ u3;1 Þ ð2Þ where cij are the material constants Equations of motion are [12]: r11;1 ỵ r13;3 ỵ qgu3;1 ẳ qu1 r13;1 ỵ r33;3 qgu1;1 ẳ qu3 3ị in which q is the mass density of the medium, and g is the acceleration due to gravity, a superposed dot denotes differentiation with respect to t, commas indicate differentiation with respect to spatial variables xi In matrix (operator) form, the Eqs (2) and (3) are written as follows: u0 r ! ẳN u ! 4ị r where: u ¼ ½u1 ; u3 ŠT ; r ¼ ½r13 ; r33 ŠT , the symbol T indicates the transpose of matrices, the prime indicates the derivative with respect to x3 and: N¼ " N1 N2 K N3 ! ; N1 ¼ À@ Àðc13 =c33 Þ@ # ! ; N2 ¼ 1=c55 0 1=c33 ! q@ 2t ỵ ẵc213 c11 c33 ị=c33 @ 21 qg@ ; N ¼ NT1 K¼ qg@ q@ 2t 5ị Here we use the notations: @ ẳ @=@x1 ị; @ 21 ẳ @ =@x21 ị; @ 2t ẳ @ =@t ị In addition to Eq (4), the displacement vector u and the traction vector r are required to satisfy the decay condition: u ¼ 0; r ẳ on x3 ẳ ỵ1 6ị and the free-traction condition at the plane x3 ¼ 0: r ¼ on x3 ¼ ð7Þ Secular equation Assume that the half-space x3 P is made of a material with an exponential depth profile: cij ¼ c0ij e2mx3 ; q ¼ q0 e2mx3 where c0ij ; q0 ; m are constants ð8Þ P.C Vinh, G Seriani / Wave Motion 46 (2009) 427–434 429 Now we consider the propagation of a Rayleigh wave, travelling with velocity c and wave number k in the x1 -direction The components u1 ; u3 of the displacement vector and r13 ; r33 of the traction vector at the planes x3 ¼ const are found in the form (see [16]): fuj ; rj3 gx1 ; x3 ; tị ẳ femx3 U j x3 Þ; iemx3 Rj ðx3 Þgeikðx1 ÀctÞ ; j ¼ 1; ð9Þ Substituting (9) into (4) yields: U0 ! R0 ẳ iM ! U 10ị R where: U ẳ ẵU U T ; R ẳ ẵR1 R3 T , and: M¼ M1 M2 Q M3 ! ; M1 ¼ ! kðX À dÞ ia Q¼ ; Àia kX im=kị D im=kị M3 ẳ ! ; im=kị D ! " M2 ẳ 1=kị 1=c055 0 1=c033 # im=kị 11ị here d ẳ c011 c013 ị2 Þ=c033 , D ¼ c013 =c033 , a ¼ q0 g, X ¼ q0 c2 , the prime indicates the derivative with respect to y ¼ kx3 It is not difficult to verify, by eliminating U from (10), that the traction vector RðyÞ is the solution of the equation: aR00 ibR0 cR ẳ 12ị where the matrices a; b; c are given by: ! Àia=k X ; d ẳ XX dị a=kị2 kd ia=k X dị ! g1 b ẳ M Q ỵ Q M3 ẳ ; g ẳ d ỵ DịX kd g ! h0 img =k ỵ iag =k c ¼ M1 Q À1 M3 À M2 ¼ kd Àimg =k À iag =k h1 a ẳ Q ẳ 13ị 14ị 15ị in which g ẳ d DịX; g2 ẳ D 2 k 2ma d ỵ c055 k d 2maD h1 ẳ X dị þ D X À À c33 k k h0 ẳ m X m2 k m2 ỵ X À dÞ À ð16Þ Now we seek the solution of Eq (12) in the form: Ryị ẳ eipy R0 17ị where R0 is a non-zero constant vector, p is a complex number which must satisfy the condition: Ip > jmj=k ð18Þ in order to ensure the decay condition (6) Substituting (17) into (12) leads to: p2 a pb ỵ cịR0 ẳ 19ị As R0 is a non-zero vector, the determinant of system (19) must vanish This provides the equation for determining p, namely: p4 Sp2 ỵ P ¼ ð20Þ where   d 1 2m2 þ þ XÀ c055 c033 c055 k   ! ðc011 À XÞðc055 À XÞ m2 1 d m4 a2 ỵ 2amc055 c013 ị Pẳ ỵ D X ỵ 0 0 c33 c55 c33 c55 c55 k c033 c055 k k S ẳ 2D 21ị It follows from (20) that: p21 ỵ p22 ẳ S; p21 p22 ẳ P 22ị 430 P.C Vinh, G Seriani / Wave Motion 46 (2009) 427–434 where p21 ; p22 are two roots of the quadratic Eq (20) for p2 It is not difficult to demonstrate that vector R0 ¼ ½A BŠT , the solution of (19), is given by: ia img p ỵ g1 p iag =k k k B ẳ Xp2 ỵ h0 Aẳ 23ị Let p1 , p2 be the two roots of (20) satisfying the condition (18) Then the general solution of Eq (12) is: Ryị ẳ c1 A1 B1 ! A2 eip1 y ỵ c2 B2 ! eip2 y 24ị where Ak ; Bk k ẳ 1; 2ị are given by (23) in which p is replaced by pk , c1 ; c2 (c21 ỵ c22 0) are constants to be determined from the boundary condition (7) that reads: R0ị ẳ 25ị Making use of (24) into (25) yields two equations for c1 ; c2 , namely: " ia=kịp21 ỵ g p1 img =k iag =k ia=kịp22 ỵ g p2 img =k iag =k Xp21 ỵ h0 Xp22 ỵ h0 # ! c1 ẳ0 c2 26ị and vanishing the determinant of the system leads to the secular equation that defines the Rayleigh wave velocity After some algebraic manipulations and removing the factor ðp2 À p1 Þ, the secular equation results in: g Xp1 p2 i=kịẵmg X ỵ ag X ỵ ah0 p1 ỵ p2 ị g h0 ẳ 27ị Suppose that p1 ; p2 are the two roots of (20) satisfying condition (18), we shall show that the following relations hold: P > 0; pffiffiffi P À S > 0; pffiffiffi p1 p2 ¼ À P; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi p1 ỵ p2 ẳ i P S 28ị where S; P are defined by (21) Indeed, from (18) it follows that Imðpi Þ > If the discriminant D of the quadratic Equation (20) for p2 is non-negative, then its two roots must be negative in order that Imðpi Þ > In this case, P ¼ p21 p22 > and the pair p1 ; p2 are of the form: p1 ¼ ir ; p2 ¼ ir where r1 ; r2 are positive If D < 0, Eq (20) for p2 has two conjugate complex roots, again P ¼ p21 p22 > and in order to ensure Imðpi Þ > 0, it must be that p1 ẳ t ỵ ir; p2 ẳ t ỵ ir where r is positive, and t is a real number In both cases, P ¼ p21 p22 > 0, p1 p2 is a negative real number, and p1 ỵ p2 is a purely imaginary number with positive imaginary part, hence p1 ỵ p2 ị2 is a negative number Therefore, with the help of (22), it follows that pffiffiffithe relations (28) are true It is noted that the result (28)3, (28)4 were obtained in [16], but without showing that P > 0, P À S > Taking into account (28), (27) becomes: p g X P ỵ h0 Þ À ðm=kÞg X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi P S a=kịh0 ỵ g Xị P S ẳ 29ị Since g i , h0 , P; S are explicitly expressed in terms of X ¼ q c , Eq (29) is fully explicit in terms of the Rayleigh wave speed Eq (29) is the (exact) secular equation, in the explicit form, of Rayleigh waves in non-homogeneous orthotropic elastic media under the influence of gravity, where g , g , g , h0 , S, P are defined by (16)1, (14)2, (16)2, (16)3, (21)1, (21)2, respectively Remark (i) One can obtain the quadratic Equation (20) for p2 by another way that has been used by Kulkarni and Achenbach [17] First, by substituting (2) into (3) and taking into account the assumption (8), an equation for u is derived and we call it  j eipy eikðx1 ÀctÞ ðj ¼ 1; 3), into this equathe equation for the displacement vector Then, substituting u, defined as uj ¼ A tion yields a homogeneous system of two linear equations for constants Aj The vanishing of the determinant of the  that is quite complicated as remarked by Kulkarni and Achenbach system leads to a fully quartic equation for p [17] By a (linear) transformation that cancels the cubic term of the equation, the authors obtain the quadratic Equation (20) for p2 This has also been pointed out in [16] It should be noted that, the quantities cj (j ¼ 1; 2) of the paper [17] are not always real number, they can be complex numbers Thus, they are required to have positive real parts, rather than to be positive numbers, in order that the decay condition is satisfied Also note that the results (28)1,2 (with a ¼ 0) ensure that the expressions in the square roots of formula (29) in [17] have positive values e j eipy ð A e j being constantÞ, into the equation for the displacement vector will (ii) Substituting u defined by (9), where U j ¼ A also yields immediately the bi-quadratic Equation (20) However, the use of the equation for the traction vector (12) is better than that of the equation for the displacement vector, since the boundary condition is expressed in terms of the traction vector The secular equation is derived more quickly if we use the equation for the traction vector This can be seen by comparing the secular Eq (27) in which a ¼ with the corresponding Equation (39) in [17] (iii) The representation of solution (9) indicates clearly the decay behaviours of the displacement vector u and the traction vector r Unlike the homogeneous case, they are quite different from each other P.C Vinh, G Seriani / Wave Motion 46 (2009) 427–434 431 (iv) One can arrive at the explicit secular equation of the wave by following the procedure carried out by Kulkarni and Achenbach [17] Explicit secular equations of Rayleigh waves under the effect of gravity in special cases 4.1 Case of a non-homogeneous isotropic elastic half-space under gravity This problem was consider Das et al [10], but the authors only have derived the implicit form of the secular equation From (29) and (14)2, (16), (21), and taking into account that the material is isotropic, i.e c011 ẳ c033 ẳ k0 ỵ 2l0 ; c013 ẳ k0 ; c055 ¼ l0 here k0 ; l0 are constants, the explicit secular equation for this case is: h pffiffiffi i 2ð1 À cÞð2 À xÞ x P þ ðx þ 4c À 4Þð1 À xÞ þ m2 x=k ỵ 2m=k ỵ 2 q p ẩ ẫ 2mx2 2c cxị=k ỵ  x2 ỵ 6x 1ị ỵ 4c 1ị ỵ 2 PSẳ0 2 02 2 30ị where x ¼ X=l0 ¼ c2 =c02 ; c ¼ c02 =c01 ;  ẳ g=kc2 ; c01 ẳ k0 ỵ 2l0 Þ=q0 ; c02 ¼ l0 =q0 , and S ¼ ỵ cịx 2m2 =k 2 P ẳ xị1 cxị m2 ẵ1 ỵ cịx ỵ 24c 3ị=k ỵ m4 =k þ 2mð1 À 3cÞ=k À c2 ð31Þ 4.2 Case of a non-homogeneous orthotropic elastic half-space without gravity When the gravity is absent, i.e a ¼ 0, Eq (29) becomes: pffiffiffi g X P ỵ h0 ị m=kịg X q p PSẳ0 32ị in which g ¼ d À ð1 À DÞX; g ¼ d ỵ DịX X=c055 ị h0 ẳ X dị1 ỵ m2 X=k   d 1 2m2 S ẳ 2D ỵ ỵ X À c55 c33 c55 k   ! ðc011 À XÞðc055 À XÞ m2 1 d m4 Pẳ ỵ X 2D ỵ 0 c33 c55 c33 c55 c55 k k ð33Þ Eq (32) coincides with the secular equation derived recently by Destrade [16] Note that in [16] it is not shown P > and pffiffiffi P À S > 4.3 Case of a non-homogeneous transversely isotropic elastic half-space without gravity This problem was considered by Pal and Acharya [11], but only the implicit form of the secular equation has been derived in their work In their notations, the explicit secular equation for the problem is Eq (32), where m is replaced by m=2, the functions g ; g ; h0 ; S; P (in terms of X) are given by (33) in which c011 ; c013 ; c033 ; c055 ; q0 are replaced by A1 ; F ; C ; L1 ; q1 , respectively Here A1 ; F ; C ; L1 are the material constants (see [18]) 4.4 Case of a homogeneous orthotropic elastic half-space under gravity In this case m ¼ 0, Eq (29) thus is simplified to: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffiffi i pffiffiffi g X P ỵ h0 a=kịẵh0 ỵ DX P S ẳ 34ị in which: S ẳ 2D ỵ X dị=c55 ỵ X=c33 ; g ẳ d ỵ DịX; Pẳ    c11 X X a2 1À À À c55 c33 c33 k c33 c55 h0 ¼ ðX À dị1 X=c55 ị ỵ a2 =k c55 ị 35ị ð36Þ 432 P.C Vinh, G Seriani / Wave Motion 46 (2009) 427–434 Eq (34) is the explicit secular equation of Rayleigh waves in orthotropic elastic media under the effect of gravity In this case one can show that the Rayleigh wave velocity is limited by: < X ¼ qc2 < minðc55 ; c11 Þ ð37Þ 4.5 Case of a homogeneous transversely isotropic elastic half-space under gravity This problem was considered by Dey and Sengupta [8], and only the implicit form of the secular equation was derived In their notations, the explicit secular equation for this problem is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h p i p g X P ỵ h0 a=kịẵh0 ỵ DX P S ẳ 38ị in which: Sẳ   2F 2F 2A X; ỵ ỵ ỵ C L L C CL g ẳ ỵ F=CịX ỵ A À F =C;  A X À C C P¼   2X 2a2 1À À L CL h0 ẳ X A ỵ F =Cị1 2X=Lị þ 2a2 =L ð39Þ ð40Þ here A; C; F; L are the material constants It is noted that Eq (38) is Eq (34) in which c11 ; c33 ; c13 ; c55 are replaced by A; C; F; L=2, respectively The Rayleigh wave velocity is also subjected to the limitation (37) in which c11 ; c55 are, respectively, replaced by A; L=2 4.6 Case of a homogeneous isotropic elastic half-space under gravity By putting m ¼ in Eq (30) and replacing c0ij ; q0 by cij ; q we obtain the explicit secular equation of Rayleigh waves in homogeneous isotropic elastic half-space under gravity, namely: ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 À cÞð2 À xÞ x ð1 À xÞð1 À cxÞ À c2 2 ỵ x ỵ 4c 4ị1 xị þ 2  þ ðx þ 4c À 4Þð1 xị ỵ 2cịx ỵ  r ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 À xÞð1 À cxÞ c  ỵ ỵ cịx ẳ 41ị where x ẳ c2 =c22 ; c ¼ c22 =c21 ;  ¼ g=kc2 ; c21 ¼ k ỵ 2lị=q; c22 ẳ l=q and S ẳ ỵ cịx 2; P ẳ xị1 cxÞ À c2 ð42Þ here k; l are Lame’s constants The Eq (41) provides the exact secular equation in the explicit form for the investigations by De and Sengupta [7] and Datta [9] 4.7 Case of a homogeneous orthotropic elastic half-space without gravity When the material is homogeneous and the gravity is absent we have: m ¼ a ¼ Then Eq (29) is simplified to (see also [19,20]): ðc55 Xịẵc213 c33 c11 Xị ỵ p p c33 c55 X c11 Xịc55 Xị ẳ ð43Þ In this case we can obtain the explicit formula for the Rayleigh wave velocity (see [20]), namely: qc2 =c55 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffi pffiffiffiffi b1 b2 b3 b1 =3ịb2 b3 ỵ 2ị ỵ R ỵ D ỵ R D 44ị where b1 ¼ c033 =c011 ; b2 ¼ d=c011 ; b3 ¼ c011 =c055 ; R and D are given by: hðb1 ; b2 ; b3 Þ 54 i h pffiffiffiffiffi b1 ð1 À b2 Þhðb1 ; b2 ; b3 ị ỵ 27b1 b2 ị2 ỵ b1 b2 b3 ị2 ỵ Dẳ 108 Rẳ 45ị in which hb1 ; b2 ; b3 ị ẳ p b1 ẵ2b1 b2 b3 ị3 ỵ 93b2 À b2 b3 À 2ފ ð46Þ and the roots in (44) taking their principal values It is clear that the speed of Raleigh waves in homogeneous orthotropic elastic solids is a continuous function of three dimensionless parameters b1 ; b2 ; b3 433 P.C Vinh, G Seriani / Wave Motion 46 (2009) 427–434 0.9 0.8 0.7 x=ρ c /c 55 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −m/k Fig Dependence of squared dimensionless Rayleigh wave velocity x ¼ q0 c2 =c055 on the parameter Àm=k with different values of  ¼ 0:3 (dashed line),  ¼ 0:5 (dash-dot line),  ¼ 0:8 (dotted line)  :  ¼ (solid line), 0.9 x=ρ c /c55 0.8 0.7 0.6 0.5 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ε 0.8 Fig Dependence of squared dimensionless Rayleigh wave velocity x ¼ q0 c2 =c055 on the parameter  with different values of Àm=k : Àm=k ¼ (solid line), Àm=k ¼ 0:1 (dotted line), Àm=k ¼ 0:2 (dashed line), Àm=k ¼ 0:4 (dash-dot line) A numerical example As an example, we consider a non-homogeneous orthotropic elastic half-space whose elastic constants and mass density are defined by (8), in which m and (see [16]): c011 =q0 ¼ km=sị2 ; c013 =q0 ẳ 3:6 km=sị2 c033 =q0 ẳ 9:89 km=sị2 ; c055 =q0 ẳ 2:182 km=sị2 47ị Taking into account (47), it is easy to numerically solve the secular Equation (29), and the dependence of squared dimension0 less Rayleigh wave velocity x ¼ q0 c2 =c055 on Àm=k and  ¼ q0 g=kc55 are shown in Figs and It appears that the influence of the inhomogeneity on the Rayleigh wave velocity is stronger than that of the gravity 434 P.C Vinh, G Seriani / Wave Motion 46 (2009) 427–434 Conclusions The problem of the Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity is considered and the secular equation of the wave motion in the explicit form is derived Furthermore, by considering various special cases, the explicit secular equations is obtained for the Rayleigh wave motions under the effect of inhomogeneity and/or gravity, corresponding to a number of previous studies in which only the implicit dispersion equations were given The explicit secular equations derived in this work may be useful in practical applications Acknowledgements The authors wish to thank Prof J.D Achenbach for helpful discussions They also would like to give thanks to an anonymous reviewer for recommending the paper by F Gilbert The first author undertook this work during his visit to the OGS (Istituto Nazionale di Oceanografia e Geofisica Sperimentale) with the support of the ICTP Programme for Training and Research in Italian Laboratories, Trieste, Italy References [1] L Rayleigh, On waves propagating along the plane surface of an elastic solid, Proc Roy Soc Lond A 17 (1885) 4–11 [2] D Samuel et al, Rayleigh waves guided by topography, Proc Roy Soc A 463 (2007) 531–550 [3] T.J I’A Bromwhich, On the influence of gravity on elastic waves, and, in particular, on the vibrations of an elastic globe, Proc Lond Math Soc 30 (1898) 98–120 [4] A.E Love, Some Problems of Geodynamics, Dover, New York, 1957 [5] M.A Biot, Mechanics of Incremental Deformation, Wiley, New York, 1965 [6] F Gilbert, Gravitationally perturbed elastic waves, Bull Seism Soc Am 57 (1967) 783–794 [7] S.N De, P.R Sengupta, Surface waves under the influence of gravity, Gerlands Beitr Geophys 85 (1976) 311–318 [8] S.K Dey, P.R Sengupta, Effects of anisotropy on surface waves under the influence of gravity, Acta Geophys Polonica XXVI (1978) 291–298 [9] B.K Datta, Some observation on interaction of Rayleigh waves in an elastic solid medium with the gravity field, Rev Roumaine Sci Tech Ser Mec Appl 31 (1986) 369–374 [10] S.C Das, D.P Acharya, D.R Sengupta, Surface waves in an inhomogeneous elastic medium under the influence of gravity, Rev Roumaine Sci Tech Ser Mec Appl 37 (1992) 539–551 [11] P.K Pal, D Acharya, Effects of inhomogeneity on surface waves in anisotropic media, Sadhana 23 (1998) 247–258 [12] A.M Abd-Alla, S.M Ahmed, Stoneley waves and Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity, Appl Math Compt 135 (2003) 187–200 [13] T.C.T Ting, An explicit secular equation for surface waves in an elastic material of general anisotropy, Quart J Mech Appl Math 55 (2) (2002) 297– 311 [14] T.C.T Ting, Explicit secular equations for surface waves in an anisotropic elastic half-space from Rayleigh to today Surface waves in anisotropic and laminated bodies and defects detection, NATO Sci Ser II Math Phys Chem., vol 163, Kluwer Academic Publisher, Dordrecht, 2004, pp 95–116 [15] M Destrade, The explicit secular equation for surface acoustic waves in monoclinic elastic crystals, J Acoust Soc Am 109 (2001) 1338–1402 [16] M Destrade, Seismic Rayleigh waves on an exponentially graded, orthotropic elastic half-space, Proc Roy Soc A 463 (2007) 495–502 [17] S.S Kulkarni, J.D Achenbach, Application of the reciprocity theorem to determine line-load-generated surface waves on an inhomogeneous transversely isotropic half-space, Wave Motion 45 (2008) 350–360 [18] W.M Ewing, W.S Jardetzky, F Press, Elastic Waves in Layered Media, McGraw-Hill Book Comp., New York-Toronto-London, 1957 [19] P Chadwick, The existence of pure surface modes in elastic materials with orthorhombic symmetry, J Sound Vib 47 (1) (1976) 39–52 [20] C.V Pham, R.W Ogden, Formulas for the Rayleigh wave speed in orthotropic elastic solids, Ach Mech 56 (3) (2004) 247–265  ... can arrive at the explicit secular equation of the wave by following the procedure carried out by Kulkarni and Achenbach [17] Explicit secular equations of Rayleigh waves under the effect of gravity. .. system of two linear equations for constants Aj The vanishing of the determinant of the  that is quite complicated as remarked by Kulkarni and Achenbach system leads to a fully quartic equation... Conclusions The problem of the Rayleigh waves in a non-homogeneous orthotropic elastic medium under the in uence of gravity is considered and the secular equation of the wave motion in the explicit