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Applied Mathematics and Computation 215 (2009) 395–404 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress Pham Chi Vinh Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam a r t i c l e i n f o Keywords: Rayleigh waves Rayleigh wave velocity Gravity Initial stress Orthotropic Secular equation a b s t r a c t The problem of Rayleigh waves in an orthotropic elastic medium under the influence of gravity and initial stress was investigated by Abd-Alla [A M Abd-Alla, Propagation of Rayleigh waves in an elastic half-space of orthotropic material, Appl Math Comput 99 (1999) 61–69], and the secular equation of the wave in the implicit form was derived However, due to the uncorrect representation of the solution, the secular equation is not right The main aim of the present paper is to reconsider this problem We find the secular equation of the wave in explicit form By considering some special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity of some previous studies, in which only implicit secular equations were derived Ó 2009 Elsevier Inc All rights reserved Introduction Elastic surface waves in isotropic elastic solids, discovered by 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 wave, its dispersion equations in the explicit form are very significant in practical applications They can be used for solving the forward (direct) problems, and especially for the inverse problems Thus, the secular equations in the explicit form are always the main purpose of investigations related to the Rayleigh wave 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 [3–11] Bromwhich [3] and Love [4] treated the force of gravity as a type of body force while Biot [5,6] 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 [5,6], Kuipers [10] also took the assumption of incompressibility in their studies The material is assumed to be isotropic in the investigations [3–7,9,10], transversely isotropic in [8], and orthotropic in [11] Following Biot’s approach, Dey and Mahto [12] investigated the influence of gravity on the propagation of the Rayleigh wave in an isotropic elastic medium, taking into account the effect of initial stress The authors have derived the implicit secular equation of the wave Recently, Abd-Alla [13] extended this problem to the orthotropic case He employed two displacement potentials for representing the solution, and has also derived the dispesion equation of Rayleigh waves in the implicit form However, as will be shown, his represention of solution is uncorrect, the secular equation is, thus, not true E-mail address: pcvinh@vnu.edu.vn 0096-3003/$ - see front matter Ó 2009 Elsevier Inc All rights reserved doi:10.1016/j.amc.2009.05.014 396 P.C Vinh / Applied Mathematics and Computation 215 (2009) 395–404 The main purpose of the present paper is to re-investigate the problem on the propagation of Rayleigh waves in an orthotropic elastic medium under the effect of gravity and initial stress Unlike Abd-Alla, we seek the solution directly, not use the displacement potentials Interestingly that, we have found the dispersion equations of the wave in the explicit form From this we obtain the explicit secular equation for Dey and Mahto’s investigation [12] When the initial stress is absent, by considering its special cases, we derive the (exact) explicit secular equations of Rayleigh waves under the effect of gravity of the previous studies [7–9], in which only the implicit secular equations have been found 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 parameters characterizing the material and external effects (see for example, [14–16]) Otherwise we call it an implicit secular equation Basic equations Consider a homogeneous orthotropic elastic body occupying the half-space x3 subject to the gravity and an initial compression P along the x1 -direction (see [13]) We are interested in a plane motion in ðx1 ; x3 Þ-plane with displacement components u1 ; u2 ; u3 such that: ui ¼ ui x1 ; x3 ; tị; i ẳ 1; 3; u2 0: ð1Þ Then the components of the stress tensor [13]: rij ; i; j ¼ 1; are related to the displacement gradients by the following equations r11 ¼ c11 ỵ P0 ịu1;1 ỵ c13 ỵ P0 ịu3;3 ; r33 ẳ c13 u1;1 ỵ c33 u3;3 ; r13 ẳ c44 u1;3 ỵ u3;1 ị; 2ị where cij are the material constants Equations of motion are [13]: r11;1 ỵ r13;3 ỵ P0 =2ịu1;3 u3;1 ị;3 qgu3;1 ẳ qu1 ; r13;1 ỵ r33;3 ỵ P0 =2ịu1;3 u3;1 ị;1 ỵ 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 signifies differentiation with respect to the time t, commas indicate differentiation with respect to the spatial variables xi From (2)3 it follows: r31 u3;1 : c44 u1;3 ẳ 4ị Analogously, from (2)2 we have: c r33 À 13 u1;1 : c33 c33 u3;3 ẳ 5ị Employing (3)2 and using (4) yield: r33;3 ẳ qu3 qgu1;1 ỵ P0 u3;11 /r13;1 ; 6ị where / ẳ ỵ P0 =2c44 Þ From (3)1 and taking into account (2)1, (4), (5) we have: r13;3 ẳ q=/ịu1 ẵd ỵ P0 ị=/u1;11 ỵ qg=/ịu3;1 D=/ịr33;1 ; where d ẳ c11 are written as: u0 ! r0 ¼N c213 =c33 , u r 7ị D ẳ c13 =c33 In matrix (operator) form, following the Stroh formalism (see [17,18]), the Eqs (4)(7) ! ; 8ị T where: u ẳ u1 ; u3 ; u3 ẳ u3 =/; r ẳ ẵr13 ; r33 T , the symbol T indicates the transpose of matrices, the prime indicates the derivative with respect to x3 and: N¼ " K¼ N1 N2 K N3 ! ; N1 ẳ /@ D=/ị@ ! q=/ị@ 2t ẵd ỵ P0 ị=/@ 21 qg@ qg@ q@ 2t ỵ P0 @ 21 ; N2 ¼ # ; 1=c44 0 1=ð/c33 Þ ! ; 9ị N ẳ N T1 : Here we use the notations: @ ẳ @=@x1 ị; @ 21 ẳ @ =@x21 ị; @ 2t ẳ @ =@t2 Þ In addition to Eq (8), the displacement vector u and the traction vector r are required to satisfy the decay condition: P.C Vinh / Applied Mathematics and Computation 215 (2009) 395404 r1ị ẳ u1ị ẳ 0; 397 10ị and the free-traction condition at the plane x3 ¼ 0: r0ị ẳ 0: 11ị Secular equation Now we consider the propagation of a Rayleigh wave, travelling with velocity c and wave number k in the x1 -direction The components u1 ; uÃ3 of the displacement vector and r13 ; r33 of the traction vector at the planes x3 ¼ const are found in the form: È É u1 ; u3 ; rj3 x1 ; x3 ; tị ẳ fU ðx3 Þ; U ðx3 Þ; iRj ðx3 Þgeikðx1 ctị ; j ẳ 1; 3: 12ị Substituting (12) into (8) yields: U0 ! R0 ¼ iM where: U ¼ ẵU Mẳ M1 U R ! ; 13ị U T ; R ẳ ẵR M2 ! ; R3 T , and: M1 ¼ À/ ! ; M ẳ 1=kị Q M3 D=/ ! Àia ðX À d À P0 Þ=/ ; M ẳ M T1 ; Q ẳk ia /X ỵ P Þ 1=c44 0 1=ð/c33 Þ ! ; ð14Þ here a ¼ qg=k, X ¼ qc2 , the prime indicates the derivative with respect to y ¼ kx3 Following the approach employed in [16,19,20], by eliminating U from (13), we obtain the equation for the traction vector RðyÞ, namely: a^ R00 ib^R0 c^R ẳ 0; 15ị ^ c ^ ; b; ^ are given by: where the matrices a ! ia /X ỵ P0 ị ; kd Àia ðX À d À P Þ=/ ! ^ ẳ M Q ỵ Q M3 ¼ g ; b kd g a^ ¼ Q À1 ¼ d ¼ ðX þ P ÞðX À d À P Þ a2 ; 16ị 17ị where g ẳ X d P0 ị DX ỵ P0 ị; c^ ¼ M1 Q À1 h0 ÀiDa M3 À M2 ẳ kd iDa h1 ! 18ị 19ị in which h0 ẳ /X d P ị d=c44 ; h1 ẳ D2 X ỵ P ị=/ À d=ð/c33 Þ: ð20Þ Now we seek the solution of the Eq (15) in the form: Ryị ẳ eipy R0 ; ð21Þ where R0 is a non-zero constant vector, p is a complex number which must satisfy the condition: Ip < ð22Þ in order to ensure the decay condition (10) Substituting (21) into (15) leads to: ^ỵc ^ pb ^ịR0 ẳ 0: p2 a 23ị As R0 is a non-zero vector, the determinant of the system (23) must vanish This provides the equation for determining p, namely: p4 Sp2 ỵ P ẳ 0; where 24ị 398 P.C Vinh / Applied Mathematics and Computation 215 (2009) 395–404 1 S ẳ 2D ỵ X d P ị ỵ X ỵ P ị; /c44 c33 c11 X P X ỵ P0 a2 P¼ À : 1À À c33 c33 /c44 /c33 c44 25ị It follows from (24) that: p21 ỵ p22 ẳ S; p21 p22 ẳ P; 26ị p21 ; p22 where are two roots of the quadratic equation (24) for p It is not difficult to demonstrate that vector R0 ẳ ẵA solution of (23), is given by: T B , the A ¼ g p ỵ iaD p2 ị; B ẳ /X ỵ P ịp2 ỵ h0 : 27ị Let p1 , p2 be the two roots of (24) satisfying the condition (22) Then the general solution of the equation (15) is: RðyÞ ẳ c1 A1 B1 ! eip1 y ỵ c2 A2 B2 ! eip2 y ; ð28Þ where Ak ; Bk k ẳ 1; 2ị are given by (27) in which p is replaced by pk , c1 ; c2 are non-zero constants to be determined from the boundary condition (11) that reads: R0ị ẳ 0: 29ị Making use of (28) into (29) yields two equations for c1 , c2 : " g p1 ỵ iaD p21 ị g p2 ỵ iaD p22 ị X ỵ P0 ịp21 ỵ h0 X ỵ P ịp22 ỵ h0 # ! c1 ẳ0 c2 30ị 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 is: g /X ỵ P0 ịp1 p2 ỵ iaẵh0 ỵ D/X ỵ P0 ịp1 ỵ p2 ị g h0 ẳ 0: 31ị Suppose p1 ; p2 are the two roots of (24) satisfying the condition (22) We shall show that: P > 0; pffiffiffi P À S > 0; pffiffiffi p1 p2 ¼ À P ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p1 ỵ p2 ẳ i P S; ð32Þ where S; P are defined by (25) Indeed, if the discriminant D of the quadratic Eq (24) for p2 is non-negative, then its two roots must be negative in order that (22) is to be satisfied In this case, P ¼ p21 p22 > and the pair p1 ; p2 are of the form: p1 ¼ Àir ; p2 ¼ Àir where r ; r are positive If D < 0, the quadratic Eq (24) for p2 has two conjugate complex roots, again P ¼ p21 p22 > 0, and in order to ensure the condition (22): p1 ¼ t À ir; p2 ¼ Àt À 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 negative imaginary part, hence p1 ỵ p2 ị2 is a negative number Therefore, with the help of (26), it follows that the relations (32) are true Taking into account (32) Eq (31) becomes: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffi pffiffiffi g /ðX ỵ P0 ị P ỵ h0 aẵh0 ỵ D/X ỵ P0 ị P S ẳ 0: 33ị Eq (33) is the (exact) secular equation of Rayleigh waves in orthotropic elastic media under the gravity and the initial compression Since P; S, g , h0 , /, d, D are explicitly expressed in terms of c; k; , it is clear that the secular Eq (33) is fully explicit As a depends on the wave number k, so does the Rayleigh wave velocity defined by Eq (33) Thus, the Rayleigh wave in orthotropic elastic media under the gravity and the initial compression is dispersive When the pre-stress is absent, i.e P ¼ 0, the Eq (33) coincides with equation (2.15) in [21] with a ¼ However, it should be observed, as above, that the expressions in the square roots of equation (2.15) in [21] have positive values Special cases 4.1 Rayleigh waves in isotropic elastic half-spaces under gravity and initial stresses The problem was considered by Dey and Mahto [12], and the authors have been derived the secular equation in the implicit form In their notations, the explicit secular equation for this problem is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffiffiffi pffiffiffiffiffi ffi g /X ỵ Pị P ỵ h0 aẵh0 ỵ D/X ỵ Pị P S ẳ 0; 34ị P.C Vinh / Applied Mathematics and Computation 215 (2009) 395–404 399 where: 1 S ẳ 2D ỵ X d Pị ỵ X ỵ Pị; /Q B22 B11 X XỵP a2 P ẳ À À ; /Q B22 B22 /Q B22 P B12 P B12 Pị2 ; Dẳ ; d ¼ B11 À P À ; 2Q B22 B22 g ẳ d ỵ P Xị DX ỵ Pị; XỵP a2 ; h0 ẳ X d Pị / ỵ Q2 Q2 /ẳ1ỵ 35ị B11 , B12 , B22 , Q are given by the formula (8a) in [12] (or by (13) in [22]) It is noted that Eq (34) is derived from Eq (33) in which c11 , c33 , c13 , c44 , P , P are replaced by B11 À P, B22 , B12 À P, Q , P, P Ã respectively 4.2 Rayleigh waves in orthotropic elastic half-spaces under the gravity When the initial stress is absent, i.e P0 ¼ 0, the Eq (33) becomes: q h p i p g X P ỵ h0 aẵh0 ỵ DX P S ẳ 36ị in which: S ẳ 2D ỵ X dị=c44 ỵ X=c33 ; Pẳ c11 X X a2 À ; 1À À c44 c33 c33 c33 c44 g ẳ ỵ DịX ỵ d; h0 ẳ X dị1 X=c44 ị ỵ a2 =c44 : ð37Þ ð38Þ Eq (36) is the (exact) explicit secular equation of Rayleigh waves in orthotropic elastic media under the effect of gravity In this case we can show that the Rayleigh wave velocity is limited by: < X ẳ qc2 < minc44 ; c11 ị: 39ị Indeed, first we rewrite (37)1 as follows: h i S ¼ c33 X c11 ị ỵ c44 X c44 ị ỵ c13 ỵ c44 ị2 =c33 c44 ị: 40ị It follows from (32)1 and (37)2 that ðc11 À XÞ and ðc44 À XÞ must have the same sign This yields: < X < minðc11 c44 Þ or X > maxðc11 ; c44 Þ: ð41Þ On use of (40) we see that the discriminant D ¼ S À 4P of Eq (24) is given by: o n 4a2 D ẳ c13 ỵ c44 ị4 ỵ 2c13 ỵ c44 ị2 ẵc33 X c11 ị ỵ c44 X c44 ị ỵ ẵc33 X c11 ị c44 X c44 ị2 =c33 c44 ị2 ỵ : c33 c44 ð42Þ Now, if the (41)2 exists, then it follows from (42) that D P 0, so Eq (24) for this case has two real roots p21 ; p22 with the same sign, according to (32)1 On the other hand, it is clear from (40) and (41)2 that S ¼ p21 ỵ p22 > Thus, both p21 and p22 are positive This leads to the contradiction to the requirement that p1 ; p2 must have negative imaginary part The inequalities (39) are proved 4.3 Rayleigh waves in transversely isotropic elastic media under the effect of gravity This problem was considered by Dey and Sengupta [8], but only the implicit form of the secular equation has been derived in their work In their notations, the explicit secular equation for this problem is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffiffi i pffiffiffi g X P ỵ h0 aẵh0 ỵ DX P S ẳ 43ị in which: Sẳ 2F 2F 2A X; ỵ ỵ ỵ À C L L C CL g ¼ Àð1 ỵ F=CịX ỵ A F =C; Pẳ A X À C C 1À 2X 2a2 ; L CL h0 ẳ X A ỵ F =Cị1 2X=Lị ỵ 2a2 =L; 44ị 45ị here A; C; F; L are the material constants (see also [23]) It is noted that Eq (43) is Eq (36) in which c11 , c33 , c13 , c44 are replaced by A, C, F, L=2, respectively The Rayleigh wave velocity is also subjected to the limitation (39) in which c11 , c44 are replaced by A, L=2, respectively 400 P.C Vinh / Applied Mathematics and Computation 215 (2009) 395–404 4.4 Rayleigh waves in isotropic elastic half-spaces under the gravity When the material is isotropic we have: c11 ¼ c33 ẳ k ỵ 2l; where k, c44 ẳ l; c13 ẳ k; 46ị l are Lames constants On view of (46) the limitation (39) becomes: < x < 1; 47ị p where x ẳ c2 =c22 (dimensionless Rayleigh wave speed), c2 ¼ l=q (the shear wave velocity), and the expression (37)1 of S is simplified to: S ¼ ỵ cịx 2; < c ẳ c22 =c21 < 1; 48ị p where c1 ẳ k ỵ 2lÞ=q is the longitudinal wave velocity Now, making use of (46) along with (37)2, (38), (48) into (36) we have: ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 À cÞð2 À xÞ x ð1 À xị1 cxị c2 2 ỵ x ỵ 4c 4ị1 xị ỵ 2 r q x ỵ 4c 4ị1 xị þ ð1 À 2cÞx þ 2 ð1 À xÞð1 cxị c2 2 ỵ ỵ cịx ẳ 0; 49ị where ẳ g=kc2 ị Eq (49) is the (exact) secular equation, in the explicit form, of Rayleigh waves in isotropic elastic halfspaces under the influence of gravity It is noted that this problem was considered by De and Sengupta [7] and Datta [9], but only the implicit dispersion equation of the wave have been derived Now suppose that ẳ g=kc2 ị is much small comparison with the unit Then by omitting the powers of order bigger than one in terms of , from the exact secular Eq (49) we have immediately: pffiffiffiffiffiffiffiffiffiffiffih pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffii 2ðc À 1Þð2 À xÞ À x ð2 À xÞ2 À À x À cx ỵ xẵx ỵ 4c 4ị1 xị ỵ 2cịx ẳ 0: 50ị Eq (50) is an approximate dispersion equation of Rayleigh waves in isotropic elastic media under the effect of gravity, in the case < ẳ g=kc2 ị ( Remark (i) As noted in Remark 2.3i (Section 5), when the material is isotropic and the stress is absent P0 ẳ 0ị we can express the displacement components u1 , u3 in terms of the potentilas u, w by (71), in which u, w satisfy (81) It is not difficult to see that corresponding to a surface wave travelling with velocity c and wave number k in the x1 -direction and decaying in the x3 -direction, the potentials u, w are given by: u ẳ A1 eikp1 x3 ỵ A2 eikp2 x3 eikx1 ctị ; w ẳ ik A1 m1 eikpx3 ỵ A2 m2 eikp2 x3 eikx1 ctị ; 51ị A1 , c21 p2j A2 where mj ẳ 1ị; j ẳ 1; 2, p1 , p2 are roots of Eq (24) with negative imaginary parts, are non-zero constants On use of (51), (71) into (2)2,3 in which c13 ẳ k, c33 ẳ k ỵ 2l, c44 ẳ l, and taking into account (12), we have: " Ryị ẳ c1 A1 B1 # e " ip1 y 2 ỵc A2 B2 # eip2 y ; 52ị where: Aj ẳ lẵ2pj ỵ ikmj p2j ị; Bj ẳ k1 þ p2j Þ þ 2lðp2j þ ikmj pj Þ; j ¼ 1; 2; ð53Þ c1 , c2 , are non-zero constants Substituting (52) into (29) leads to a homogeneous linear system for c1 , c2 Vanishing the determinant of this system provides: A1 B1 A2 ẳ 0: B2 54ị With the help of (32), it is not difficult to verify that the Eq (54) is equivalent to the equation: ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 À xÞð1 À cxị c2 ỵ ỵ cịx x À 2Þ2 À ð1 À xÞð1 À cxÞ À c2 2 q! ỵ 4 c xị1 cxị c2 ẳ 55ị p in the interval x ð0; 1Þ Note that, since the limiting velocity (see, for instance, [24,25]) in this case is c2 ¼ l=q, the imaginary parts p1 , p2 are negative for the values of x ð0; 1Þ Consequently, the relations (32) hold for values of x belonging to this interval P.C Vinh / Applied Mathematics and Computation 215 (2009) 395–404 401 (2i) Eqs (49) and (55) are different in form, but they are equivalent to each other in the interval (0, 1) This is proved as follows First, we recall that Eq (54) is equivalent to Eq (55) in the interval (0, 1) It is clear from (27)–(29) that Eq (49) is equivalent to the equation: e A ¼0 e2 B e A1 e1 B ð56Þ in the interval (0, 1), where: e j ẳ g p ỵ ia D À p2 ; A j j e j ẳ Xp2 ỵ h0 ; B j j ẳ 1; 2; ð57Þ g , h0 , D correspond to the isotropic elastic solids without pre-stress P0 ẳ 0ị Now, suppose that x is a root of (55) and < x < 1, then x is a root of (54) and RðyÞ given by (52) is a solution of Eq (16), i.e.: a^ R00 À ib^R0 À c^R 0; ð58Þ ^ c ^ , b, ^ are correspond to the isotropic elastic solids without pre-stress ðP ¼ 0Þ Since functions eip1 y , eip2 y are linearly where a independent of each other (noting that p1 – p2 ), from (58) it follows: ^ p2j a " # A j ^ ^ ¼ 0; À bpj ỵ c Bj j ẳ 1; 2: 59ị From (59) it deduces that: Bj ¼ Aj ej B ej A or Bj Aj ¼ ¼ Lj – 0; ej A ej B j ẳ 1; 2: 60ị On view of (60) it is clear that Eqs (54) and (56) are equivalent to each other, therefore, x is a root of Eq (56) Since (56) is equivalent to (49), x is a root of (49) also Thus, it has been observed that if x is a root of Eq (55) then it is a root of Eq (49) Now let x be a root of (49) and < x < 1, then it is a solution of (56), and the corresponding potentials u, w given by (51) satisfy (81) Therefore RðyÞ defined by (52) is a solution of (16) This again leads to the relations (60), and as its consequence, x is a root of (54), thus x is a root of (55), because (54) is equivalent to (55) The proof is finished (3i) In the case that < ( 1, by neglecting the powers of order bigger than one in terms of , and taking into account the equalities: ffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 xị1 cxị ỵ ỵ cịx ẳ x ỵ cx; p p À x À À cx pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ẳ : xc 1ị x ỵ À cx ð61Þ ð62Þ Eq (55) becomes: h pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffii ðx À 2Þ2 À À x À cx þ Replacing x by c2 =c22 , c by c22 =c21 , À À c2 =c22 À Á2 pffiffiffiffiffiffiffiffiffiffiffi pi 4 h ỵ c cxị x ỵ c xị cx ¼ 0: xðc À 1Þ ð63Þ by g=ðkc22 Þ, from (63) we have: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! À À c2 =c21 À c2 =c22 ! À Áqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 À q2 4g 2 =c c ỵ c À c c =c 2 =c c c ỵ c À c c 2 2 ẳ 0: c2 c21 c22 k 64ị Eq (64) is an approximate secular equation of Rayleigh waves in isotropic elastic media under the effect of gravity, in the case < ¼ g= kc2 ( 1, and it was first derived by Love [4] in a different way Interestingly that, the (original) exact Eqs (49) and (55) give the same roots, while the corresponding approximate Eqs (50) and (63) give different solutions (see Fig 1) 4.5 Rayleigh waves in orthotropic elastic media without the effect of gravity and initial stress When both initial stress and gravity are absent, i.e P ¼ a ¼ 0, the Eq (33) simplifies to (see also [26,27]): Â Ã pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc44 À XÞ c213 À c33 c11 Xị ỵ c33 c44 X c11 Xịc44 Xị ẳ 0: 65ị 402 P.C Vinh / Applied Mathematics and Computation 215 (2009) 395–404 0.8 0.7 γ=0.5 0.5 x=c /c2 0.6 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 ε Fig Dependence on ¼ g=kc2 of x ¼ c2 =c22 defined by the exact secular Eqs (49) and (55) (solid line), approximate Eq (50) (dashed line) and approximate one (63) (dash-dot line), with c ¼ 0:5 In this case we can obtain the explicit formula for the Rayleigh wave velocity (see [27]), namely: pffiffiffi pffiffiffi qc2 =c44 ẳ b1 b2 b3 = b1 =3ịb2 b3 ỵ 2ị ỵ q! q p p 3 Rỵ Dỵ R D ; 66ị where b1 ẳ c33 =c11 , b2 ¼ d=c11 , b3 ¼ c11 =c44 , R and D are given by: hðb1 ; b2 ; b3 Þ; 54 i h pffiffiffiffiffi b1 ð1 b2 ịhb1 ; b2 ; b3 ị ỵ 27b1 b2 ị2 ỵ b1 b2 b3 ị2 ỵ Dẳ 108 Rẳ 67ị in which hb1 ; b2 ; b3 ị ẳ i p h b1 2b1 b2 b3 ị3 ỵ 93b2 b2 b3 À 2Þ ð68Þ and the roots in (65) taking their principal values It is clear that the speed of Rayleigh waves in orthotropic elastic solids is a continuous function of three dimensionless parameters On Abd-Alla’s representation of solution We recall briefly Abd-Alla’s representation of solution, and then show that it is uncorrect First, substituting (2) into (3) and taking into account the assumption: c44 ¼ ðc11 À c13 ị=2, he obtained: ; c11 ỵ P0 ị2u1;11 ỵ u1;33 ỵ u3;13 ị ỵ c13 u3;13 u1;33 ị 2qgu3;1 ẳ 2qu 69ị : c11 u1;13 ỵ u3;11 ị ỵ c13 ỵ P0 ịu1;13 u3;11 ị ỵ 2c33 u3;33 ỵ 2qgu1;1 ẳ 2qu 70ị According to his argument, by expressing the displacement components u1 , u3 in terms of the displacement potentials uðx1 ; x3 ; tị and wx1 ; x3 ; tị as: u1 ẳ u;1 w;3 ; u3 ẳ u;3 ỵ w;1 : ð71Þ Eqs (69) and (70) reduce, respectively, to: €; ðc11 ỵ P0 ịO2 u qgw;1 ẳ qu 72ị c13 c11 P0 ịO w ỵ 2qg u;1 ẳ 2qw 73ị ; c11 u;11 ỵ c33 u;33 qgw;1 ẳ qu 74ị c11 w;11 w;33 ị c13 ỵ P0 ịO w ỵ 2c33 w;33 ỵ 2qg u;1 ẳ 2qw; 75ị and where O2 f ẳ f;11 ỵ f;33 403 P.C Vinh / Applied Mathematics and Computation 215 (2009) 395–404 Also by his argument, Eqs (72) and (74) represent the compressive wave along the x1 and x3 directions, respectively, and Eqs (73) and (75) represent the shear wave along those directions According to him, since he considered the propagation of Rayleigh waves in the direction of x1 only, thus he restricted his attention only to Eqs (72) and (75) Now we show that u1 , u3 expressed by (71) in which u and w is a solution of the system (72) and (75) does not satisfy the system (69), (70), in general Indeed, substituting (71) into (69) and (70) leads, respectively, to: h i ẳ0 ;1 ỵ c13 c11 P0 ịO2 w 2qg u;1 ỵ 2qw c11 ỵ P0 ịO2 u qgw;1 qu 76ị ;3 and h h i i ỵ c11 u ỵ c33 u qgw qu ẳ 0: c11 w;11 w;33 ị c13 ỵ P0 ịO2 w ỵ 2c33 w;33 ỵ 2qg u;1 2qw ;1 ;11 ;33 ;1 ;3 ð77Þ Since u, w is a solution of the system (72) and (75), the first terms of the left-hand sides of (76) and (77) vanish, thus they become: h i € ¼0 ðc13 À c11 P0 ịO2 w 2qg u;1 ỵ 2qw 78ị ;3 and h c11 u;11 ỵ c33 u;33 qgw;1 qu i ;3 ẳ 0; 79ị respectively It is clear that a pair u and w which is a solution of the system (72) and (75) does not necessarily satisfy the Eqs (78) and (79) The observation is demonstrated Remark (i) It is not difficult to verify that if c33 c11 ỵ P (being valid in general) and u, w given by: w 0; u ¼ Aeipx3 eixt ; p2 ¼ qx2 =ðc11 ỵ P ị; x ẳ const 0; A ¼ const – 0; ð80Þ then u, w is a solution of the system (72) and (75), however, they not satisfy the Eqs (78) and (79) (ii) We can say, from (76) and (77), that Eq (69) [Eq (70)] is satisfied if u1 , u3 is defined by (71) in which u and w is a solution of the system (72) and (78) [system (75) and (79)].From these, it is clear that the system (69), (70) is satisfied if u1 , u3 is given by (71) in which u and w is a solution of a system of four equations, namely: (72), (75), (78) and (79) (3i) When the initial stress is absent (P ¼ 0) and the material is isotrpic, Eqs (78) and (79) are satisfied if u and w is a solution of the system (72) and (75), which now is: O2 u À g € ¼ 0; w À u c21 ;1 c21 O2 w ỵ g u w ẳ 0: c22 ;1 c22 ð81Þ That means, the representation of solution (71), (72) and (75) is valid for this case (4i) If unknown functions u and w are sought in the form (as in [13]): u ẳ Aeikpx3 eikx1 ctị ; w ¼ BeÀikpx3 eikðx1 ÀctÞ ; ð82Þ 2 where A; B are constants satisfying A ỵ B 0, then p has to satisfy a system of quadratic equations that has no solution in general (5i) The assumption: c44 ẳ c11 c13 ị=2 is not taken in the present paper Conclusions In this paper the propagation of Rayleigh waves in homogeneous orthotropic elastic media under the influence of gravity and initial stress is investigated We have found the exact secular equation in the explicit form, and it is a new result By considering its special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity, corresponding to some previous studies in which only implicitdispersion equations have been found In the case that the material is isotropic, the initial stress is absent, and < ¼ g= kc2 ( 1, we have derived directly, from the exact secular equations, approximate dispersion equations, and one of them coincides with the one obtained by Love Acknowledgement The author wish to thank an anonymous reviewer for recommending him some references useful with the research 404 P.C Vinh / Applied Mathematics and Computation 215 (2009) 395–404 References [1] L Rayleigh, On waves propagating along the plane surface of an elastic solid, Proc Roy Soc Lond A17 (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, The influence of initial stress on elastic waves, J Appl Phys 11 (1940) 522–530 [6] M.A Biot, Mechanics of Incremental Deformation, Wiley, New York, 1965 [7] S.N De, P.R Sengupta, Surface waves under the influence of gravity, Gerlands Beitr, Geophysik 85 (1976) 311–318 [8] S.K Dey, P.R Sengupta, Effects of anisotropy on surface waves under the influence of gravity, Acta Geophys Pol 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 Mech Appl 31 (1986) 369–374 [10] M Kuipers, A.A.F van de Ven, Rayleigh-gravity waves in a heavy elastic medium, Acta Mech 81 (1990) 181–190 [11] 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 Comput 135 (2003) 187–200 [12] S Dey, P Mahto, Surface waves in a highly pre-stressed medium, Acta Geophys Pol 36 (1988) 89–99 [13] A.M Abd-Alla, Propagation of Rayleigh waves in an elastic half-space of orthotropic material, Appl Math Comput 99 (1999) 61–69 [14] 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 [15] 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 Acad Publ., Dordrecht, 2004 pp 95–116 [16] M Destrade, The explicit secular equation for surface acoustic waves in monoclinic elastic crystals, J Acoust Soc Am 109 (2001) 1398–1402 [17] A.N Stroh, Dislocations and cracks in anisotropic elasticity, Philos Mag (1958) 625–646 [18] A.N Stroh, Steady state problems in anisotropic elasticity, J Math Phys 41 (1962) 77–103 [19] M Destrade, Surface waves in orthotropic incompressible materials, J Acoust Soc Am 110 (2001) 837–840 [20] M Destrade, Rayleigh waves in symmetry planes of crystals: explicit secular equations and some explicit wave speeds, Mech Mater 35 (2003) 931– 939 [21] M Destrade, Seismic Rayleigh waves on an exponentially graded, orthotropic elastic half-space, Proc Roy Soc A 463 (2007) 495–502 [22] I Tolstoy, On elastic waves in prestressed solids, J Geophys Res 87 (1982) 6823–6827 [23] W.M Ewing, W.S Jardetzky, F Press, Elastic Waves in Layered Media, McGraw-Hill Book Comp., New York–Toronto–London, 1957 [24] D.M Barnett, J Lothe, Consideration of the existence of surface wave (Rayleigh wave) solutions in anisotropic elastic crystals, J Phys F: Metal Phys (1974) 671–686 [25] P Chadwick, Interfacial and surface waves in pre-stressed isotropic elastic media, Z Angew Math Phys 46 (1995) S51–S71 [26] P Chadwick, The existence of pure surface modes in elastic materials with orthorhombic symmetry, J Sound Vib 47 (1) (1976) 39–52 [27] Pham Chi Vinh, R.W Ogden, Formulas for the Rayleigh wave speed in orthotropic elastic solids, Ach Mech 56 (3) (2004) 247–265 ... Conclusions In this paper the propagation of Rayleigh waves in homogeneous orthotropic elastic media under the in uence of gravity and initial stress is investigated We have found the exact secular equation... derive the (exact) explicit secular equations of Rayleigh waves under the effect of gravity of the previous studies [7–9], in which only the implicit secular equations have been found Note that a secular. .. is the (exact) secular equation of Rayleigh waves in orthotropic elastic media under the gravity and the initial compression Since P; S, g , h0 , /, d, D are explicitly expressed in terms of