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Applied Mathematics and Computation 215 (2010) 3515–3525 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Explicit secular equations of Stoneley 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 Keywords: Stoneley waves Stoneley wave velocity Orthotropic Secular equation Non-homogeneous Gravity a b s t r a c t The problem of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity was studied recently by Abd-Alla and Ahmed [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], who derived the secular equation of the wave in the implicit form In this paper, by using an appropriate representation of the solution, we obtain the secular equation of the wave in the explicit form Moreover, considering its special cases, we derive explicit secular equations for a number of investigations of Stoneley waves under the influence of gravity, for which only the implicit dispersion equations were previously obtained Ó 2009 Elsevier Inc All rights reserved Introduction The propagation of Stoneley waves under the effect of gravity is a significant problem in Seismology and Geophysics, which has attracted the attention of researchers such as De and Sengupta [1], Dey and Sengupta [2], Das et al [3] These authors, following Biot [4], all assumed the force of gravity to create a type of initial stress of hydrostatic nature, and derived the secular equation of the wave in the implicit form De and Sengupta [1] assumed the material is isotropic elastic, while Dey and Sengupta [2] considered the case of transversely isotropic elastic materials All supposed that the material is homogeneous However, because any realistic model of the earth must take into account continuous changes of elastic properties in the vertical direction, the problem was extended to the (exponentially) non-homogeneous case by Das et al [3], who assumed that the material is isotropic Recently, Abd-Alla and Ahmed [5] extended the problem to the orthotropic case; these authors employed two displacement potentials for expressing the solution, and derived the secular equation of the wave in the implicit form For Rayleigh and Stoneley waves, dispersion equations in the explicit form are very significant in practical applications They can be used for solving direct (forward) problems, i.e studying the effects of material parameters on the wave velocity; and especially the inverse problems, i.e determining material parameters from the measured values of the wave speed The main purpose of this paper is to obtain the explicit secular equation of Stoneley waves under the effect of gravity for inhomogeneous orthotropic elastic materials This equation is identified in the explicit form by using an appropriate representation of solution From this we derive the explicit secular equations for the particular cases investigated by De and Sengupta [1], Dey and Sengupta [2], Das et al [3], and Pal and Acharya [6], in which only 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 parameters characterizing materials and external effects (see for example, [7–9]) Otherwise we call it an implicit secular equation * Corresponding author E-mail address: pcvinh@vnu.edu.vn (P.C Vinh) 0096-3003/$ - see front matter Ó 2009 Elsevier Inc All rights reserved doi:10.1016/j.amc.2009.10.047 3516 P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525 Also note that, due to Abd-Alla and Ahmed’s incorrect representation of the solution (see [12]), the secular equation in the implicit form derived by them in [5] for Stoneley waves is not valid Basic equations Let us consider the two non-homogeneous orthotropic elastic bodies, X and Xà , occupying the half-space x3 P 0; x3 0, respectively, subject to gravity They are in welded contact with each other at the plane x3 ¼ These two media extend to an infinitely great distance from the origin and Xà is to be taken above X Same quantities related to X and Xà have the same symbol but are systematically distinguished by an asterisk if pertaining to Xà We are interested in planar motion in the 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 rij ; i; j ¼ 1; are related to the displacement gradients by the following equations [5]: 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 of the form [5]: r11;1 ỵ r13;3 þ qgu3;1 ¼ qu€1 ; r13;1 þ r33;3 À qgu1;1 ¼ qu€3 ð3Þ 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, following the Stroh formalism (see [10,11]), the Eqs (2) and (3) are written as follows: u0 r ! ẳN u ! r 4ị ; where u ẳ ẵu1 ; u3 T ; r ẳ ẵr13 ; r33 T , the symbol T indicates the transpose of a matrix, the prime indicates the derivative with respect to x3 and: N¼ " K¼ N1 K ! N2 ; N3 N1 ¼ ! À@ 1=c55 ; N2 ¼ 0 # Àqg@ ; N ¼ NT1 : q@ 2t Àðc13 =c33 Þ@ q@ 2t ỵ ẵc213 c11 c33 ị=c33 @ 21 qg@ ! ; 1=c33 ð5Þ Here we use the notations: @ ẳ @=@x1 ị; @ 21 ẳ @ =@x21 ị; @ 2t ẳ @ =@t Þ 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 ¼ at x3 ẳ ỵ1: 6ị For X we have equations similar to (1)–(5) in which the quantities are asterisked, and the decay condition (6) is replaced by: uà ¼ 0; rà ¼ on x3 ¼ À1: ð7Þ Since the half-spaces are in welded contact with each other at the plane x3 ¼ 0, the displacement vector and the traction vector must satisfy the continuous condition: u ¼ uà ; r ¼ rà on x3 ¼ 0: ð8Þ Explicit secular equation Assume that the half-spaces X and Xà are made of materials with an exponential depth profile: cij ¼ c0ij e2mx3 ; c0ij ; à à 2m x3 q ¼ q0 e2mx3 ; cÃij ¼ cÃ0 ; qà ¼ qÃ0 e2m x3 ; ij e ð9Þ where q; m q ; m are constants Now we consider the propagation of a wave, travelling with velocity c and wave number k in the x1 -direction, being mostly confined to the neighbourhood of the interface x3 ¼ Then 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 [13]): cÃ0 ij ; Ã0 à P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525 fuk ; rk3 gðx1 ; x3 ; tị ẳ femx3 U k x3 ị; iemx3 Rk x3 ịgeikx1 ctị ; k ẳ 1; 3: 3517 10ị Substituting (10) into (4) yields: U0 ! ẳ iM R0 Qẳ ! 11ị ; R U T ; R ẳ ẵ R where U ẳ ẵ U M¼ U M1 M2 Q M3 ! M1 ¼ ; kðX À dÞ ia Àia kX ! ; R3 T , and: Àiðm=kÞ À1 ÀD Àiðm=kÞ " ! ; ! im=kị D ; M3 ẳ im=kị 1=c055 M ẳ 1=kị # ; 1=c033 12ị c013 Þ2 Þ=c033 ; c011 here d ¼ À D ¼ c013 =c033 ; a ¼ q0 g; X ¼ q0 c2 , the prime indicates the derivative with respect to y ¼ kx3 Following the approach employed in [9,14,15], by eliminating U from (11), the traction vector RðyÞ is the solution of the equation: aR00 À ibR0 À cR ¼ 0; ð13Þ where the matrices a; b; c are given by: ! X Àia=k ; d ¼ XðX À dÞ À ða=kÞ2 kd ia=k ðX À dÞ ! g1 ; g ¼ d À ð1 ỵ DịX b ẳ M Q ỵ Q À1 M3 ¼ kd g ! h0 img =k ỵ iag =k c ẳ M1 Q À1 M3 À M2 ¼ kd Àimg =k À iag =k h1 a ¼ Q À1 ¼ 14ị 15ị 16ị in which g ẳ d DịX; m2 X g2 ẳ D m2 k 2ma ; d ỵ ; c055 k d 2maD h1 ẳ X dị ỵ D2 X À À : c33 k k h0 ẳ k m2 ỵ X dị 17ị The displacement vector U is determined in terms of R by: U ¼ ÀiQ À1 R0 À Q À1 M3 R: ð18Þ Now we seek the solution of the Eq (13) in the form: Ryị ẳ eipy R0 ; 19ị where R0 is a non-zero constant vector, p is a complex number which must satisfy the condition: Ip > jmj=k ð20Þ in order to ensure the decay condition (6) Substituting (19) into (13) leads to: p2 a pb ỵ cịR0 ẳ 0: 21ị As R0 is a non-zero vector, the determinant of the system (21) must vanish This provides an equation for determining p, namely: p4 Sp2 ỵ P ẳ 0; 22ị where d 1 2m2 X ; ỵ ỵ 0 c55 c33 c55 k ! ðc011 À XÞðc055 À Xị m2 1 d Pẳ ỵ X À 2D À 0 0 c33 c55 c33 c55 c55 k S ẳ 2D ỵ m4 k a2 ỵ 2amc055 c013 ị k c033 c055 : ð23Þ 3518 P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525 It follows from (22) that: p21 ỵ p22 ẳ S; p21 ; p21 p22 ẳ P; 24ị p22 T where are two roots of the quadratic Eq (22) for p It is not difficult to demonstrate that the vector R0 ¼ ½A B , the solution of (21), is given by: ia img p ỵ g1 p iag =k; k k B ẳ Xp2 ỵ h0 : Aẳ 25ị Let p1 ; p2 be the two roots of (22) satisfying the condition (20) Then the general solution of the Eq (13) is: ! ! A1 ip1 y A2 ip2 y e ỵ c2 e ; B1 B2 Ryị ẳ c1 26ị where Ak ; Bk k ẳ 1; 2ị are given by (25), in which p is replaced by pk , and pk ; c1 , and c2 c21 ỵ c22 0ị are constants to be determined We have the following result: Proposition Suppose p1 ; p2 are the two roots of (22) satisfying the condition (20) Then we have: P > 0; pffiffiffi P S > 0; q p p1 ỵ p2 ¼ i P À S; pffiffiffi p1 p2 ¼ À P ; ð27Þ where S; P are defined by (23) Indeed, from (20) it follows that Imðpi Þ > If the discriminant D of the quadratic Eq (22) 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, the Eq (22) for p2 has two conjugate complex roots, again P ¼ p21 p22 > 0, and in order to ensure Imðpi ị > 0, it must be p1 ẳ t ỵ ir and p2 ẳ t ỵ ir, where r is positive 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 a positive imaginary part, thus are p1 ỵ p2 ị2 is a negative number Therefore, with the help of (24), it follows that the relations (27) p ffiffiffi true It is noted that the result (27)3, (27)4 were obtained in [13], but without showing that P > 0; P À S > From (18) and (26) we have: E1 Uyị ẳ c1 F1 ! E2 eip1 y ỵ c2 ! F2 eip2 y ; 28ị where Ek ẳ e2 p2k ie1 pk ỵ e0 ; F k ẳ f3 p3k þ if2 p2k þ f1 pk þ if0 ; ð29Þ k ¼ 1; 2; where ej ¼ bj =kd; j ¼ 0; 1; 2; fj ¼ wj =kd; j ¼ 0; 1; 2; 3, and: 2 b0 ¼ h0 DX am=k ị 1=k ịa ỵ Xmịmg ỵ ag ị; b1 ẳ 1=kịẵg a ỵ Xmị ỵ Xmg ỵ ag ị ỵ ah0 ; 2 b2 ẳ 1=k ịaa ỵ Xmị ỵ g X ỵ XDX am=k ị; w0 ẳ 1=kịh0 ẵaD mX dị 1=kịmg þ ag ÞðX À d þ ma=k Þ; 30ị w1 ẳ g X d ỵ ma=k ị ỵ 1=k ịamg ỵ ag ị ỵ X dịh0 ; w2 ẳ 1=kịaX d ỵ ma=k ị ỵ 1=kịag ỵ X=kịẵaD mX dị; w3 ẳ d: Similarly, vectors u and rà are sought in the form: Ãx fuÃk ; rÃk3 gx1 ; x3 ; tị ẳ fem x U Ãk ðx3 Þ; iem RÃk ðx3 Þgeikðx1 ÀctÞ ; k ẳ 1; 31ị in which U ẳ c1 ! ! E eip1 y ỵ c2 2à eip2 y ; F2 ð32Þ ! ! Ẫ1 ipà y A2 ip y ỵ c ; à e à e B1 B2 ð33Þ Ễ1 F Ã1 and Rà ¼ cÃ1 3519 P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525 where cÃ1 ; cÃ2 are constants to be determined, AÃk ; BÃk ; Ek ; F k k ẳ 1; 2ị are dened by: à à ia à im g Ã0 à pk ị ỵ g pk ia g Ã2 =k; k k à BÃk ¼ X à ðpÃk ị2 ỵ h0 ; Ak ẳ 34ị Ek ẳ e2 pk ị2 ie1 pk ỵ e0 ; F k ẳ f3 pk ị3 ỵ if pk ị2 ỵ f1 pk ỵ if ; k ¼ 1; à à à à à in which aà ¼ qÃ0 g; g Ãj ; h0 are defined by formulas similar to those for g j ; h0 , and eÃj ¼ bj =kd ; fjà ¼ wÃj =kd and bj ; wÃj are exà pressed by those similar to (30); furthermore, d ðDà ; dà ; X Ã Þ are given by the expressions similar to those for d ðD; d; XÞ, à à and p1 ; p2 are two roots of the equation: p4 À Sà p2 ỵ P ẳ 35ị satisfying: Ipj < jm j ; k j ẳ 1; 2; 36ị here Sà ; P à are determined by those similar to (23) Analogously as above, one can show that: Pà > 0; pffiffiffiffiffi P à À Sà > 0; pffiffiffiffiffi p1 p2 ẳ P ; q p p1 ỵ p2 ẳ i P S : 37ị From the continuity conditions (8) we have: U k 0ị ẳ U k 0ị; Rk 0ị ẳ Rk 0ị; k ẳ 1; 3: ð38Þ Eq (38) yield a homogeneous linear system for c1 ; c2 ; cÃ1 ; cÃ2 The secular equation, determining the Stoneley wave velocity c, is obtained by vanishing the determinant of the system: E1 F A1 B E2 EÃ1 F2 F Ã1 AÃ1 BÃ1 A2 B2 EÃ2 à F2 ¼ 0: AÃ2 BÃ2 ð39Þ After some algebraic manipulations, taking into account (25), (29) and (34) and removing the factor ðp1 À p2 ÞðpÃ1 À pÃ2 Þ , the dispersion Eq (39) is equivalent to: q11 q 21 q31 q 41 q12 q22 qÃ11 qÃ21 q32 qÃ31 q42 qÃ41 qÃ12 q22 ẳ 0; q32 q 40ị 42 where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi P À S À b1 ; q p ẳ b2 S ỵ b1 P S ỵ 2b0 ; q p p ẳ w3 S Pị w2 P S ỵ w1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi ¼ Àw3 ðS þ PÞ P À S À w2 S À w1 P À S À 2w0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! p ẳ d a=kị P S ỵ g ; q11 ¼ b2 q12 q21 q22 q31 q32 ẳ d 2=kịmg ỵ ag ị a=kịS À g q41 ¼ d X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffi PS ; q! p PS ; q42 ẳ dẵXS þ 2h0 ; ð41Þ 3520 P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi à Pà À Sà À b1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi à à à ¼ b2 Sà À b1 Pà À S ỵ 2b0 ; q p p ẳ w3 S P ị ỵ w2 P S ỵ wÃ1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi ¼ wÃ3 ðSà þ PÃ Þ Pà À Sà À wÃ2 Sà þ wÃ1 P à À Sà À 2wÃ0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! p ẳ d a =kị P S ỵ g q! p ẳ d 2=kịm g ỵ a g ị a =kịS ỵ g P S ; q! ! pffiffiffiffiffi à à ¼ d ÀX à Pà S ; q42 ẳ X S ỵ 2h0 : à qÃ11 ¼ Àb2 qÃ12 qÃ21 qÃ22 qÃ31 qÃ32 qÃ41 ð42Þ It is clear that qij ; qÃij are explicit functions of c; k; , thus, the secular Eq (40) is fully explicit Special cases 4.1 Explicit secular equation for inhomogeneous isotropic media subject to gravity The propagation of Stoneley waves in inhomogeneous isotropic solids under the effect of gravity was investigated by Das et al [3], but the authors did not derived the dispersion equation in the explicit form due to the characteristic equations for p and pà being fully quartic When the half-spaces are isotropic we have: c011 ẳ c033 ẳ k0 ỵ 2l0 ; 0à 0à 0à c0à 11 ¼ c 33 ¼ k ỵ 2l ; c055 ẳ l0 ; c0 55 ẳ l ; c13 ẳ k0 ; 43ị c0à 13 ¼ k : On view of (43), Eq (40) becomes: q11 q 21 q31 q 41 q12 qÃ11 q22 qÃ21 q32 q42 hqÃ31 hqÃ41 qÃ12 qÃ22 ¼ 0; hqÃ32 à hq42 ð44Þ where h ¼ lÃ0 =l0 The elements qij are given by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi P À S À b1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ẳ b2 S ỵ b1 P S þ 2b0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi ¼ w3 ðS À Pị w2 P S ỵ w1 ; q q p p p ẳ w3 S ỵ Pị P À S À w2 S À w1 P À S À 2w0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi ¼ d P S ỵ g ; q! p ỵ g ị S À g P À S ; ¼ d 2ðmg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi ¼ d x P À S ; q42 ẳ dẵxS ỵ 2h0 ; q11 ẳ b2 q12 q21 q22 q31 q32 q41 where x ¼ X=l0 x ẳ X =l0 ị and: ị ỵ xmị mg ỵ g ị; b0 ẳ h0 Dx m ỵ xmg ỵ g ị ỵ h0 ; b1 ẳ ẵg ỵ xmị ỵ g x ỵ xDx mị; b2 ẳ þ xmÞ À dÞ À ðmg þ g ịx d ỵ m ị; w0 ¼ h0 ½D À mðx w1 ¼ g x d ỵ mị ỵ mg ỵ g ị ỵ x dịh0 ; ị ỵ g ỵ xẵD mx dị; w2 ẳ x d ỵ m g ẳ 2ẵ21 c0 ị c0 x; w3 ẳ d; ð45Þ 3521 P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525 2; g ¼ 2ð1 c0 ị2 xị; g ẳ 2c0 À m À Á x ỵ ẵx 41 c ị1 xị ỵ ỵ 2m ; h0 ẳ m 2; d ẳ xẵx 41 c0 ị 2 ; S ẳ ỵ c0 ịx 2m ẵ24c0 3ị ỵ þ c0 Þx þ m À c0 P ¼ ð1 À xÞð1 À c0 xÞ À m c0 ẳ l k ỵ 2l 0 ; ¼ qg ; kl0 ¼ m 2 c0 À 1Þ; À 2mð3 m ; k ð46Þ where D ¼ À 2c0 ; d ¼ 4l0 ð1 À c0 Þ: ð47Þ ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffi The elements qÃij are expressed by formulas similar to (45) in which P À S is replaced by À P à À Sà The quantities à à à à ; Dà ; dà are given by formulas similar to (46) and (47) Eq (44), along with (45)–(47), estabbi ; wÃi ; g Ãi ; h0 ; d ; Sà ; P à ; cÃ0 ; à ; m lishes the explicit secular equation of Stoneley waves for the case of inhomogeneous isotropic elastic half-spaces subject to gravity Note that qij and qÃij are dimensionless quantities 4.2 Explicit secular equation for homogeneous transversely isotropic half-spaces subject to gravity When two half-spaces are homogeneous, i.e m ¼ mà ¼ 0, one can see that the explicit secular equation of the wave is of the form (40), in which the elements qij are simplified to: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi q11 ẳ P S ỵ a=kc55 ị; q pffiffiffi a q12 ¼ ÀS À P À S ỵ 2D1 X=c055 ị; kc p 55 q21 ¼ S À P À D À ðX À dÞ=c055 ; q h pi p q22 ẳ D ỵ ðX À dÞ=c055 À S À P P À S ỵ 2aDị=kc55 ị; q p q31 ẳ a=kị P S ỵ d ỵ DịX; q p q32 ẳ a=kị2D Sị ẵd þ DÞX P À S; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi q41 ẳ X P S; q42 ẳ XS ỵ 2X dị1 X=c055 ị ỵ 2a2 =k c055 Þ; ð48Þ where S ¼ 2D À P¼ d 1 ỵ ỵ X; c55 c33 c55 ð49Þ ðc011 À XÞðc055 À XÞ a2 À : c033 c055 k c033 c055 ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffi Elements qÃij ; i ¼ 1; 2; 3; 4; j ¼ 1; are defined by formulas similar to (48) in which P À S is replaced by À P à À Sà Note that, in this case, it can be shown that the Rayleigh wave velocity is limited by: Ã0 Ã0 Ã0 < c2 < minðc055 =q0 ; c011 =q0 ; cÃ0 55 =q ; c 11 =q Þ: ð50Þ Indeed, in view of (49)1 we have: h i S ¼ c033 ðX c011 ị ỵ c055 X c055 ị ỵ c013 ỵ c055 ị2 =c033 c055 ị: 51ị It follows from (27)1 and (49)2 that ðc011 À XÞ and ðc055 À XÞ must have the same sign This yields: < X < minðc011 ; c055 Þ or X > maxðc011 ; c055 Þ: ð52Þ Using (51) we see that the discriminant D ¼ S À 4P of Eq (22) is given by: n  à  Ã2 o D ẳ c013 ỵ c055 ị4 ỵ 2c013 ỵ c055 ị2 c033 X c011 ị ỵ c055 X c055 ị ỵ c033 X c011 ị c055 X c055 ị =c033 c055 ị2 ỵ 4a2 k c033 c055 : ð53Þ Now, if (52)2 exists, then it follows from (53) that D P 0, so Eq (22) for this case has two real roots p21 ; p22 with the same sign, according to (27)1 On the other hand, it is clear from (51) and (52)2 that S ẳ p21 ỵ p22 > Thus, both p21 and p22 are positive This contradicts the fact that p1 ; p2 must have a positive imaginary part 3522 P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525 Thus, it must be: < X ¼ q0 c2 < minðc055 ; c011 Þ: ð54Þ Similarly, we have: Ã0 < X à ¼ qÃ0 c2 < minðcÃ0 55 ; c 11 Þ: ð55Þ Then, from (54) and (55) we deduce (50) Eq (40) in which qij ; qÃij given by (48) is the explicit secular equation of Stoneley waves for orthotropic homogeneous elastic half-spaces subject to gravity It provides the explicit secular equation of Stoneley waves for the transversely isotropic case which was investigated by Dey and Sengupta [2] When the media are isotropic we have the relations (43) From (40), (43), (48) and (49) one can see that the explicit secular equation for this case is: q11 q 21 q31 q 41 q12 qÃ11 q22 qÃ21 q32 qÃ31 q42 qÃ41 qÃ12 qÃ22 ¼ 0; qÃ32 qÃ42 ð56Þ where qij are given by: q p q11 ẳ 21 ỵ Pị ỵ c0 ịx ỵ ; q p q12 ẳ c0 ị4 3xị 21 ỵ Pị ỵ c0 ịx; p q21 ẳ 2c0 ỵ c0 x P; p q p q22 ¼ ð2c0 À À c0 x À P Þ 21 ỵ Pị ỵ c0 ịx ỵ 21 2c0 ị; q p q31 ẳ 21 ỵ Pị ỵ c0 ịx ỵ 21 c0 ị2 xị; q p q32 ẳ ẵ41 c0 ị ỵ c0 ịx 21 c0 ị2 xị 21 ỵ Pị ỵ c0 ịx; q p q41 ẳ x 21 ỵ Pị ỵ c0 ịx; 57ị q42 ẳ xẵ1 ỵ c0 ịx ỵ 2ẵx 41 c0 ị1 xị ỵ 22 in which: P ẳ xÞð1 À c0 xÞ À c0 Elements qÃij 2 : ð58Þ are determined by similar formulas to (57), in which q q p p 21 ỵ Pị ỵ c0 ịx is replaced by 21 þ PÃ Þ À ð1 þ cÃ0 Þxà : Eqs (56)–(58) establish the explicit secular equation for the investigation [1] 4.3 Explicit secular equation for inhomogeneous transversely isotropic half-spaces When gravity is absent, i.e a ¼ aà ¼ 0, the dispersion equation of Stoneley waves is Eq (40), in which qij are reduced to (qÃij determined by the similar formulas): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi q11 ¼ À P À S þ 2m; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi P À S þ 2D1 X=c055 ị ỵ 2m 2; q12 ẳ S 2m q p p PSDỵm X dị=c055 ; q21 ẳ S P ỵ m q h pi p ỵ X dị=c055 S P P S ỵ mS X dị=c055 ; ỵ 2mị Dỵm q22 ẳ D m q31 ẳ d ỵ DịX; q p DịX ẵd ỵ DịX P S; q32 ẳ 2mẵd q pffiffiffi X; q41 ¼ X P À S; q42 ẳ XS ỵ 2X dị1 X=c055 ị ỵ 2m where S is calculated by (23)1 and: ð59Þ 3523 P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 35153525 Pẳ c011 Xịc055 Xị 2 Àm c033 c055 ! 1 d þ D À X À : c033 c055 c055 ð60Þ Eqs (40), (59) and (60) provide the explicit secular equation for the investigation [6] 4.4 Explicit secular equation for homogeneous isotropic half-spaces Now we consider the case when two half-spaces are homogeneous isotropic elastic and they are not subject to gravity For this case we have: c11 ¼ c33 ẳ k ỵ 2l; c55 ẳ l; c11 ẳ c33 ẳ k ỵ 2l ; c13 ẳ k; c13 ẳ k ; c55 ẳ l ; 61ị m ẳ m ¼ a ¼ aà ¼ 0; where k; l; kà ; là are Lame constants of the half-spaces It is easy to verify that the roots pj ; pÃj ðj ¼ 1; 2Þ of the characteristic equations (22) and (35) are: sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 p1 ¼ i À ; c1 sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 p2 ¼ i À ; c2 pÃ1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ¼ Ài À Ã2 ; c1 pÃ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ¼ Ài À Ã2 c2 ð62Þ Ã Ã Ã Ã2 à à in which c21 ẳ k ỵ 2lị=q; c22 ẳ l=q; c2 ẳ k ỵ 2l ị=q ; c ẳ l =q It is clear that the secular equation (39) is equivalent to: 1 À1 À1 à à à à F =ðiE Þ F =ðiE Þ ÀF =ðiE Þ ÀF =ðiE Þ 1 2 2 à à à à ¼ 0: A1 =ðiklE1 Þ A2 =ðiklE2 Þ ÀA1 =ðiklE1 Þ ÀA2 =ðiklE2 Þ B =ðÀklE Þ B =ðÀklE Þ ÀBà =ðÀklEÃ Þ ÀBà =ðÀklEÃ Þ 1 2 1 2 ð63Þ By employing (25), (29), (34), (61), (62) one can show that: F =E1 ẳ p1 ; A1 =kE1 ị ẳ 2lp1 ; ¼ ¼ ¼ Àlð2 À c2 =c22 Þ=p2 ; B2 =kE2 ị ẳ 2l; l p1 ; B1 =kE1 ị ¼ Àlà ð2 À c2 =cÃ2 Þ; à à à Ã2 à ¼ Àl ð2 À c =c2 Þ=p2 ; B2 =ðkE2 Þ ¼ À2là : F =E2 ¼ À1=p2 ; F Ã1 =Ễ1 F Ã2 =Ễ2 B1 =kE1 ị ẳ l2 c2 =c22 ị; A2 =kE2 ị p1 ; A1 =kE1 ị ẳ à À1=pÃ2 ; Ẫ2 =ðkE2 Þ ð64Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi à à On use of (64) into (63) and setting pj ¼ Àbj =ðikÞ; pÃj ¼ bj =ðikÞ; j ¼ 1; ðbj ¼ k À c2 =c2j ; bj ¼ k À c2 =cÃ2 j Þ we have: b =k 2b1 =k À c2 =c2 à à k=b2 b1 =k k=b2 à à ¼ 2 à à Ã2 ð2 À c =c2 Þðk=b2 Þ 2ðl =lÞðb1 =kÞ ðl =lÞð2 À c =c2 Þðk=b2 Þ Àðlà =lÞð2 À c2 =cÃ2 À2ðlà =lÞ Þ À1 À1 ð65Þ 0.8 x=ρ0c2/c0 55 0.75 0.7 0.65 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 * m à Here D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:7; Dà ¼ 0:4; dÃ1 ¼ 2:5; dÃ2 ¼ 0:6; Fig Dependence of the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 on m ¼ À0:1; / ¼ 0; d3 ¼ 0:25; d4 ¼ 0:3 m 3524 P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525 This is the secular equation of Stoneley waves for the case of two homogeneous isotropic elastic half-spaces without the effect of gravity, that coincides with the one published in [16] Numerical results and discussion It is readily to see from (40) that the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 depends on 11 dimensionless à à Ã0 Ã0 Ã0 Ã0 ẳ m=k; / ẳ g=kc22 ị; D ẳ cÃ0 parameters, namely, D ¼ c013 =c033 ; d1 ¼ c011 =c055 ; d2 ¼ c055 =c033 ; m 13 =c33 ; d1 ¼ c11 =c55 ; d2 ¼ c55 =c 33 ; 0 à ¼ mà =k; d3 ¼ qÃ0 =q0 ; d4 ¼ cÃ0 =c , here c ¼ c = q Given 11 these dimensionless parameters, it is not difficult to m 55 55 55 numerically calculate the dimensionless Stoneley wave velocity x using the dispersion Eq (40) It is well known that, for the homogeneous isotropic half-spaces not being subject to the gravity, the Stoneley wave exists if shear velocities c2 ¼ l=q and cÃ2 ¼ là =qà differ only slightly (see, for example, [16,17]), we therefore take d3 ¼ 0:25; d4 ¼ 0:3 à in three cases: Xà is isotropic with à Fig shows the variation of x with m Fig shows the dependence of x on m à à à à Dà ¼ 0:5; d1 ¼ 4; d2 ¼ 0:25 ðkà ¼ 2là Þ; Xà is orthotropic with Dà ¼ 0:5; d1 ¼ 6; d2 ¼ 0:25; Xà is orthotropic with à à à for different values of the parameter /, while Fig D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:25 Fig presents the dependence of x on m shows the variation with / of x for distinct values of the parameter m It is clear from Figs 1, and that the dimensionless Stoneley wave velocity x depends strongly on the inhomogeneity and the gravity Especially, the Fig shows that the orthotropy also strongly affects on the dimensionless Stoneley wave 0.9 0.85 x=ρ0c2/c0 55 0.8 0.75 0.7 0.65 0.6 0.55 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 * m à for three cases: Xà is isotropic (dashed line) with Fig Dependence of the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 on m à à à à à à Dà ¼ 0:5; d1 ¼ 4; d2 ¼ 0:25 k ẳ 2l ị; X is orthotropic with D ẳ 0:5; d1 ¼ 6; d2 ¼ 0:25 (solid line); Xà is orthotropic with Dà ¼ 0:5; d1 ¼ 2; d2 ¼ 0:25 ¼ À0:1; / ¼ 0; d3 ¼ 0:25; d4 ¼ 0:3 (dash-dot line) For all cases X is orthotropic with D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:7, and m x=ρ c /c55 0.8 0.75 0.7 0.65 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 m for different values of the parameter / : / ¼ (dash-dot line), / ¼ 0:2 Fig Dependence of the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 on m à à à ¼ 0:1; d3 ¼ 0:25; d4 ¼ 0:3 (solid line), / ¼ 0:5 (dashed line) For three cases: D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:7; Dà ¼ 0:5; d1 ¼ 4; d2 ¼ 0:25; m P.C Vinh, G Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525 3525 0.9 0.8 x=ρ0c2/c0 55 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 φ : m ¼ (solid line), m ¼ À0:1 Fig Dependence of the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 on / for different values of the parameter m à ¼ 0:2; d3 ¼ 0:25; d4 ¼ 0:3 ¼ À0:4 (dash-dot line) Here D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:7; Dà ¼ 0:4; dÃ1 ¼ 2:5; dÃ2 ¼ 0:6; m (dashed line), m velocity x This is unlike the conclusion of Alla-Abd and Ahmed [5] which stated that the effect of orthotropy on the Stoneley wave is small and can be neglected Conclusions In this paper, we have derived the explicit secular equation of Stoneley waves in a non-homogeneous orthotropic elastic medium, under the influence of gravity, using an appropriate representation of the solution From this secular equation, we also derive explicit secular equations for a number of cases previously investigated for Stoneley waves under the influence of gravity, for both homogeneous and non-homogeneous elastic media These explicit secular equations will be useful in practical applications The numerical results show that the Stoneley wave velocity depends strongly on the inhomogeneity, the anisotropy and the gravity Acknowledgements The first author undertook this work during his visit to 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] S.N De, P.R Sengupta, Surface waves under the influence of gravity, Gerlands Beitr, Geophysik 85 (1976) 311–318 [2] S.K Dey, P.R Sengupta, Effects of anisotropy on surface waves under the influence of gravity, Acta Geophys Polonica XXVI (1978) 291–298 [3] S.C Das, D.P Acharya, D.R Sengupta, Surface waves in an inhomogeneous elastic medium under the influence of gravity, Rev Roumaine Sci Technol Ser Mech Appl 37 (1992) 359–368 [4] M.A Biot, Mechanics of Incremental Deformation, Wiley, New York, 1965 [5] 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 [6] P.K Pal, D Acharya, Effects of inhomogeneity on surface waves in anisotropic media, Sadhana 23 (1998) 247–258 [7] 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 [8] T.C.T Ting, Explicit secular equations for surface waves in an anisotropic elastic half-space from Rayleigh to today, in: 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 [9] M Destrade, The explicit secular equation for surface acoustic waves in monoclinic elastic crystals, J Acoust Soc Am 109 (2001) 1338–1402 [10] A.N Stroh, Dislocations and cracks in anisotropic elasticity, Philos Mag (1958) 625–646 [11] A.N Stroh, Steady state problems in anisotropic elasticity, J Math Phys 41 (1962) 77–103 [12] Pham Chi Vinh, Geza Seriani, Comments to Stoneley waves and Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity by Abd-Alla and Ahmed, Appl Math Comput 135 (2003) 187–200 (Submitted to Applied Mathematics and Computations (2009)) [13] M Destrade, Seismic Rayleigh waves on an exponentially graded orthotropic elastic half-space, Proc Roy Soc A 463 (2007) 495–502 [14] M Destrade, Surface waves in orthotropic incompressible materials, J Acoust Soc Am 110 (2001) 837 [15] M Destrade, Rayleigh waves in symmetry planes of crystals: explicit secular equations and some explicit wave speeds, Mech Mater 35 (2003) 931 [16] J.D Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1973 [17] A.A Kaufman, A.L Levshin, Acoustic and Elastic Wave Fields in Geophysics, III, Elsevier, 2005 ... paper, we have derived the explicit secular equation of Stoneley waves in a non-homogeneous orthotropic elastic medium, under the in uence of gravity, using an appropriate representation of the. .. inhomogeneity on surface waves in anisotropic media, Sadhana 23 (1998) 247–258 [7] T.C.T Ting, An explicit secular equation for surface waves in an elastic material of general anisotropy, Quart... S.M Ahmed, Stoneley waves and Rayleigh waves in a non-homogeneous orthotropic elastic medium under the in uence of gravity, Appl Math Comput 135 (2003) 187–200 [6] P.K Pal, D Acharya, Effects of