Wave Motion 48 (2011) 647–657 Contents lists available at ScienceDirect Wave Motion j o u r n a l h o m e p a g e : w w w e l s ev i e r c o m / l o c a t e / wave m o t i On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces Pham Chi Vinh ⁎, Pham Thi Ha Giang 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 Article history: Received 30 December 2010 Received in revised form May 2011 Accepted May 2011 Available online 13 May 2011 a b s t r a c t Formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces are derived using the complex function method The derivation also shows that if a Stoneley wave exists, then it is unique By using the obtained formulas, we can easily reproduce the numerical results previously obtained by Murty [G S Murty, Phys Earth Planet Interiors 11 (1975), 65–79.] by directly solving the secular equation © 2011 Elsevier B.V All rights reserved Keywords: Stoneley waves Stoneley wave velocity Loosely bonded interface Holomorphic function Introduction Rayleigh surface waves [1] and Stoneley interfacial waves [2] in isotropic elastic solids discovered many years ago, in 1885 by Rayleigh and 1924 by Stoneley, respectively, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecommunications industry and materials science, for example The velocities of Rayleigh waves and Stoneley waves are of great interest to researchers in various fields of science The formulas for them provide powerful tools 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 The formulas for the velocity of Rayleigh waves and Stoneley waves are thus of theoretical as well as practical interest For Rayleigh waves, some formulas for the velocity have been obtained recently, see for instance [3–13], while no formulas have been derived for Stoneley waves so far, to the best of the authors' knowledge The main aim of this paper is to establish formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces by using the complex function method It is shown from the derivation of these formulas that, if a Stoneley wave exists, then it is unique, as proved by Barnett et al [14] by another technique Using the obtained formulas, it is easy to recover the numerical results obtained previously by Murty [15] by directly solving the secular equation Secular equation In this section we present briefly the derivation of the secular equation of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces For details, the reader is referred to the papers [15,16] by Murty Let us consider two isotropic elastic solids Ω and Ω⁎ occupying the half-spaces x2 ≥ and x2 ≤ 0, respectively Suppose that these two elastic half-spaces are not in welded contact with each other at the plane x2 = (see [15,16]) In particular, the normal ⁎ Corresponding author Tel.: + 84 5532164; fax: + 84 8588817 E-mail address: pcvinh@vnu.edu.vn (P.C Vinh) 0165-2125/$ – see front matter © 2011 Elsevier B.V All rights reserved doi:10.1016/j.wavemoti.2011.05.002 648 P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657 component of the particle displacement vector and the normal component of the stress tensor are continuous, while the shearing stress vanishes across the interface x2 = The plane x2 = is then called the loosely bonded interface (see [15]) of the half-spaces Ω and Ω⁎ Note that same quantities related to Ω and Ω⁎ have the same symbol but are systematically distinguished by an asterisk if pertaining to Ω⁎ We are interested in planar motion in the (x1x2)-plane with the displacement components u1, u2, u3 such that: ui = ui ðx1 ; x2 ; t Þ; i = 1; 2; u3 ≡ 0; ð1Þ where t is the time Then the equations of motion are of the form [17]: 2 2 cL u1;11 + cT u1;22 + cL −cT u2;12 = u1̈ ; 2 2 cL −cT u1;12 + cL u2;22 + cT u2;11 = u2̈ ; ð2Þ in which a superposed dot denotes the differentiation pffiffiffiffiffiffiffiffiffiffi with respect to t, commas indicate the differentiation with respect to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spatial variables xi, cL = ðλ + 2μ Þ = ρ and cT = μ = ρ are speed of the longitudinal wave and the transverse wave, respectively, ρ is the mass density, λ and μ are usual the Lame constants The stress components on the planes x2 = const are related to the displacement gradients by: σ 12 = μ u1;2 + u2;1 ; σ 22 = λ u1;1 + u2;2 + 2μu2;2 : ð3Þ In addition to Eq (2), the decay condition is required, i.e.: ui = ði = 1; 2Þ; σ ij = ði; j = 1; 2Þ at x2 = + ∞: ð4Þ For Ω⁎ we have equations similar to Eqs (1)–(3) in which the quantities are asterisked, and the decay condition (4) are replaced by: ⁎ u⁎ i = ði = 1; 2Þ; σ ij = ði; j = 1; 2Þ at x2 = −∞: ð5Þ Since the half-spaces are not in welded contact in the meaning as mentioned above, we have: ⁎ ; σ = σ ⁎ = at x = 0: u2 = u2⁎ ; σ 22 = σ 22 12 12 ð6Þ Now we consider the propagation of a wave, travelling with velocity c and wave number k (N 0) in the x1-direction, being mostly confined to the neighbourhood of the interface x2 = Then the displacement components u1, u2 (of Ω) are found in the form: −kbx2 ikðx1 −ct Þ u1 = Ae e −kbx2 ikðx1 −ct Þ ; u2 = Be e ; ð7Þ where A, B, b are constants, and: Reb N 0; ð8Þ in order to ensure the decay condition (4) Substituting Eq (7) into Eq (2) leads to a system of two homogeneous linear equations for A, B, and vanishing its determinant yields the equation determining b, namely: h i 2 2 2 2 2 2 cL cT b − cL cT −c + cT cL −c b + cL −c cT −c = 0: ð9Þ Now we show that if Eq (8) holds, then we have (see also [18]): b c b cT : ð10Þ Indeed, as the coefficients of the quadratic Eq (9) for X = b are all real, and its determinant Δ = [cL2(cT2 − c 2) − cT2(cL2 − c 2)] ≥ 0, Eq (9) has always two real roots denoted by X1, X2, and they both must positive, otherwise Eq (8) doesn't hold This leads to X1X2 N 0, i.e.:: 2 2 cT −c N 0: cL −c ð11Þ As X1 + X2 = [cL2(cT2 − c 2) + cT2(cL2 − c 2)]/cL2cT2, if both (cL2 − c 2) and (cT2 − c 2) are all negative, then X1 + X2 b But this contradicts the fact that X1, X2 are both positive Therefore both (cL2 − c 2) and (cT2 − c 2) are all positive, i.e we have Eq (10) P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657 649 With the condition (10), Eq (9) has two roots satisfying Eq (8), namely: sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 b1 = 1− ; cL sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 b2 = 1− ; cT ð12Þ thus we have: −kb x −kb x ikðx −ct Þ ; u1 = A1 e + A2 e 2 e u2 = b i −kb x −kb x ikðx −ct Þ − A1 e + A2 e 2 e ; i b2 ð13Þ where A1, A2 are constants to be determined Similarly, the displacement components u1⁎, u2⁎ are expressed by: kb⁎ x kb⁎ x ikðx −ct Þ ; u1⁎ = A1⁎ e + A2⁎ e 2 e ! b1⁎ ⁎ kb⁎1 x2 i ⁎ kb⁎2 x2 ikðx1 −ct Þ e A e − A2 e ; i b⁎ u⁎ = ð14Þ where Rebk⁎ N (k = 1, 2), A1⁎, A2⁎ are constants to be determined: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u c2 ⁎ b1 = t1− ; c⁎ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u c2 ⁎ b2 = t1− ; c⁎ L ð15Þ T and if Reb⁎ N 0, then: b c b c⁎ T: ð16Þ From Eqs (10) and (16) we conclude that: Proposition If a Stoneley wave exists (this implies Reb N 0, Reb⁎ N 0), then its velocity is subject to: n o b c b cT ; c⁎ T : ð17Þ Making use of Eqs (3), (13), and (14) in the boundary condition (6) yields a system of four homogeneous linear equations for A1, A2, A1⁎, A2⁎, and making its determinant zero we have: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 u " sffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffi !2 # u 2 u u c2 c2 c2 44t1− c t1− c −@2− c A 1− 1− − 2− 2 2 c⁎ c⁎ c⁎ cT cL cT ρ⁎ c⁎4 L T T v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi = 0: cT + u ρ T u c2 c2 t1− 1− 2 cL c⁎ ð18Þ L Eq (18) is desired secular equation that coincides with the one derived by Murty [15,16] (see also, for example, [17,19]) From the above argument it is clear that Eqs (17) and (18) are the necessary condition for the existence of a Stoneley wave It is not difficult to show that they are also the sufficient condition Thus we have: Proposition For a Stoneley wave to exist, it is necessary and sufficient that Eqs (17) and (18) are both satisfied Formulas for the velocity of Stoneley waves Without loss of generality we can suppose that cT ≤ cT⁎ For the sake of simplicity we introduce the following notations: x= 2 c c ρ c c ; B = T2 ; D = ; F = T2 ; E = T2 : ⁎ ρ ⁎ ⁎ c2T cL c c T L ð19Þ 650 P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657 In terms of these notations Eq (18) is of the form: 1=2 ð2−xÞ ð1−FxÞ 1=2 −4ð1−xÞ 1=2 ð1−ExÞ 1=2 ð1−FxÞ + i h 1=2 1=2 1=2 1=2 ð2−BxÞ ð1−ExÞ −4ð1−BxÞ ð1−FxÞ ð1−ExÞ = 0: DB ð20Þ Due to Eq (17) we have: b x b 1: ð21Þ Now, in the complex plane C, we consider the equation: ð2−zÞ rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 1 F z− + EF z−1 z− z− + F E F " rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi# pffiffiffiffiffiffiffiffi pffiffiffi 1 1 ð Þ = 0; 2−Bz + BEF z− z− z− E z− E B F E DB2 ð22Þ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where z−1, z−1 = B, z−1 = E, z−1 = F are chosen as the principal branches of the corresponding square roots For the real values of z ∈ (01), Eq (22) is equivalent p toffiffiffiffiffiffiffiffiffiffi Eq (20) Note that Eq (22) always has a real root, namely z = Multiplying two sides of Eq (22) by z−1 (the key technique) gives: f ðzÞ =: ð2−zÞ + rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 1pffiffiffiffiffiffiffiffiffiffi 1 z−1 + EF ðz−1Þ z− z− F z− F E F pffiffiffi ð2−zBÞ E DB ð23Þ rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi 1pffiffiffiffiffiffiffiffiffiffi 1 1pffiffiffiffiffiffiffiffiffiffi BEF z−1 + z−1 = 0: z− z− z− z− E B F E DB From the assumption cT ≤ cT⁎(→ b ≤ 1) and the fact cT⁎ b c⁎ L (→ F b B), it follows that there are three basic possibilities: Case 1 N B N E N F N Case N B N F N E N Case N E N B N F N By replacing at least one inequality in a basic case by possible equalities we have a special case First, we consider the basic cases 3.1 Case 1: N B N E N F N We will prove the following theorem: Theorem Let N B N E N F If a Stoneley wave exists, then it is unique, and its squared dimensionless velocity xs = c 2/cT2 is defined by: xs = − A2 − −I ̂ ; A3 E ð24Þ where A2, A3 given by Eq (55) and: 1=B 1=E 1=F 1@ ̂ I0 = ∫ θ1 ðt Þdt + ∫ θ2 ðt Þdt + ∫ θ3 ðt Þdt A; π 1=B 1=E ð25Þ in which θk(t) are given by Eqs (45)–(48) Proof Denote L = L1 ∪ L2 ∪ L3 with L1 = [1, 1/B], L2 = [1/B, 1/E], L3 = [1/E, 1/F], S = {z ∈ C, z ∉ L}, N(z0) = {z ∈ S : b |z − z0| b ε}, ε is a sufficient small positive number, z0 is some point of the complex plane C If a function ϕ(z) is holomorphic in Ω ⊂ C we write ϕ(z) ∈ H(Ω) From Eq (23) it is not difficult to show that the function f(z) has the properties: ( f1) f(z) ∈ H(S) ( f2) f(z) is bounded in N(1/F) and N(1) ( f3) f(z) = O(z 3) as |z| → ∞ P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657 651 ( f4) f(z) is continuous on L from the left and from the right (see [20]) with the boundary values f +(t) (the right boundary value of f(z)), f −(t) (the left boundary value of f(z)) defined as follows: F > > < f1 ðt Þ; t ∈ L1 F f ðt Þ = f2F ðt Þ; t ∈ L2 > > : F f3 ðt Þ; t ∈ L3 ð26Þ where: " f1ỵ t ị # r r p p rrr p pffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 E BEF = i ð2−t Þ F ð2−Bt Þ −t t−1 + −t t−1−4 −t −t −t t−1 F E B F E DB2 DB2 rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 1 −4 EF ðt−1Þ −t −t ; E F # rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffi E = i ð2−t Þ F ð2−Bt Þ −t t−1 + −t t−1 F E DB2 rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 BEF −4 −t −t t−1 −4 EF ð t−1 Þ −t −t ; t− B F E E F DB2 " rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 1pffiffiffiffiffiffiffiffiffiffi 1 BEF −t t−1 + −t t− t−1 + EF ð t1 ị t t t f3ỵ t ị = i ð2−t Þ2 F F B F E E F DB2 rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1pffiffiffiffiffiffiffiffiffiffi E + ð 2−Bt Þ t−1; t E DB 27ị " f2ỵ t ị fk t ị = fkỵ t ị; k = 1; 2; 3; the bar indicates the complex conjugate Note that fk+(t) (fk−(t)) is the right (left) boundary value of f(z) on Lk and i = as follows: ỵ f1 ðt Þ > > > ; t ∈ L1 > − > f > ðt Þ > > > < ỵ f2 t ị g t ị = ; t ∈ L2 > f2− ðt Þ > > > > > > f3ỵ t ị > > : − ; t ∈ L3 f3 ðt Þ ð28Þ ð29Þ ð30Þ pffiffiffiffiffiffiffiffi −1 Now we introduce the function g(t) (t ∈ L) ð31Þ From Eqs (26) and (31) it is clear that ỵ f t ị = g t Þf ðt Þ; t∈L: ð32Þ Consider the function Γ(z) defined by: ΓðzÞ = logg ðt Þ ∫ dt: 2πi L t−z ð33Þ It is not difficult to verify that: (γ1) Γ(z) ∈ H(S) (γ2) Γ(∞) = (γ3) Γ(z) = − (1/2)log(z − 1) + Ω0(z), z ∈ N(1), Γ(z) = Ω1(z), z ∈ N(1/F) where Ω0(z) (Ω1(z)) bounded in N(1) (N(1/F)) and takes a defined value at z = 1(z = 1/F) It is noted that (γ3) comes from the fact (see [20]): logg ð1Þ = iπ; logg ð1 = F Þ = 0: ð34Þ Introduce the function Φ(z) given by: ΦðzÞ = exp ΓðzÞ: ð35Þ 652 P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657 It is implied from (γ1) − (γ3) that: (ϕ1) (ϕ2) (ϕ3) (ϕ4) Φ(z) ∈ H(S) Φ(z) ≠ ∀ z ∈ S Φ(z) = O(1) as |z| → ∞ ΦðzÞ = ðz−1Þ−2 expΩ0 ðzÞ for z ∈ Nð1Þ; ΦðzÞ = expΩ1 ðzÞ; z ∈ Nð1 = F Þ: From the Plemelj formula [20], the function Φ(z) is seen directly to satisfy the boundary condition: ỵ t ị = g ðt ÞΦ ðt Þ; t ∈ L: ð36Þ We now consider the function Y(z) defined by: Y ðzÞ = f ðzÞ = ΦðzÞ: ð37Þ From ( f1)–( f3), Eq (32), (ϕ1) − (ϕ4) and Eq (37), it follows that: (y1) (y2) (y3) (y4) Y(z) ∈ H(S) Y(z) = O(z 3) as | z| → ∞ Y(z) is bounded in N(1) and N(1/F) Y +(t) = Y −(t), t ∈ L Properties (y1) and (y4) of the function Y(z) show that Y(z) is holomorphic in entire complex plane C, with the possible exception of points: z = and z = 1/F By (y3) these points are removable singularity points and it may be assumed that the function Y(z) is holomorphic in the entire complex plane C (see [21]) Thus, by the generalised Liouville theorem [21] and taking account into (y2) we have: Y ðzÞ = P ðzÞ: ð38Þ where P(z) is a third-order polynomial From Eqs (37) and (38) we have: f ðzÞ = ΦðzÞP ðzÞ: ð39Þ Since Φ(z) ≠ ∀ z ∈ S (by (ϕ2)), and Φ(z) → ∞ as z → 1, Φ(1/F) ≠ (by (ϕ4)), from Eq (39) we deduce: f ðzÞ = X P ðzÞ = in S ∪ f1g ∪ f1 = F g: ð40Þ In view of Eq (40), instead of finding zeros of f(z) we now look for three zeros of the third-order polynomial P(z) In order to that, first we have to determine P(z) From Eqs (35) and(39) we have: −ΓðzÞ P ðzÞ = f ðzÞe : ð41Þ From Eqs (27)–(31) it follows: log g ðt Þ = iϕðt ÞÞ; ϕðt Þ =: Arg g ðt Þ; ð42Þ where: < ϕ1 ðt Þ; ϕðt Þ = ϕ2 ðt Þ; : ϕ3 ðt Þ; t ∈ L1 t ∈ L2 : t ∈ L3 ð43Þ It is not difficult to verify that: ϕ1 ðt Þ = 2π + 2θ1 ðt Þ; ϕ2 ðt Þ = 2π + 2θ2 ðt Þ; ϕ3 ðt Þ = 2θ3 ðt Þ; ð44Þ where: θk ðt Þ = atanφk ðt Þ; k = 1; 2; 3; ð45Þ and: " φ1 ðt Þ = − ð2−t Þ rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # pffiffiffi 1 1 E BEF ð 2−Bt Þ −t + −t −4 −t −t −t F F E B F E DB2 DB2 = " rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 1 EF t−1 −t −t ; ð46Þ E F P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657 653 rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffi E ð2−t Þ2 F 1F −t + ð 2−Bt Þ −t E ffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi ; r φ2 ðt Þ = − pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi DB pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 1 1 BEF −t −t + EF t−1 −t −t t− B F E E F DB2 " rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 1 1 BEF φ3 ðt Þ = ð2−t Þ F −t + −t t− + EF t−1 t− −t t− F B F E E F DB2 ð47Þ = " pffiffiffi rffiffiffiffiffiffiffiffiffiffiffi # E ð2−Bt Þ t− : E DB2 ð48Þ From Eqs (33) and (42) it follows (see also [4]): ∞ −ΓðzÞ = ∑ n=0 In ; zn + ð49Þ in which: 1=F n ∫ t ϕðt Þdt; n = 0; 1; 2; 3… 2π In = ð50Þ On use of Eq (49) we can express e − Γ(z) as follows: −ΓðzÞ e =1+ a1 a a −4 ; + 22 + 33 + O z z z z ð51Þ where a1, a2, a3 are constants to be determined Employing the identity: −ΓðzÞ ′ −ΓðzÞ = ð−ΓðzÞÞ′ e e ; ð52Þ and substituting Eqs (49) and(51) into Eq (52) yields: I02 I3 + I ; a = + I I0 + I2 : ð53Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi By expanding 1−1 = z, 1−1 = ðBzÞ, 1−1 = ðEzÞ, 1−1 = ðFzÞ into Laurent series at infinity, it is not difficult to verify that: a1 = I0 ; a2 = −1 ; f ðzÞ = A3 z + A2 z + A1 z + A0 + O z ð54Þ where: pffiffiffi pffiffiffi E F+ D pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi F 1 E BEF A2 = − − pffiffiffi + EF − : + + +4 D 2E B DB2 F A3 = ð55Þ Substituting Eqs (51) and (54) into Eq (41) yields: P ðzÞ = A3 z + z ðA2 + A3 a1 Þ + zðA1 + A3 a2 + A2 a1 Þ + A3 a3 + A2 a2 + A1 a1 + A0 : ð56Þ It is clear from Eq (23) that f(0) = and f(1) = hence by Eq (40): P ð0Þ = 0; P ð1Þ = 0: ð57Þ On the use of Eqs (56) and(57) and taking into account the first of Eq (53), it is easy to see that the third root of the equation P(z) = 0, denoted by xs, is: xs = − A2 −1−I0 : A3 ð58Þ It follows from Eqs (44) and (50) that: I0 = −1 + I 0̂ : E thus xs is given by Eq (24) ð59Þ 654 P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657 Now we suppose that there exist two different Stoneley waves with corresponding velocities x (1), x (2) (x (1) ≠ x (2)) Then x (1), x (2) are two different roots of Eq f(z) = 0, and b x (1), x (2) b 1, according to Proposition From Eq (40) it follows P(x (1)) = P(x (2)) = This fact and Eq (57) imply that the third-order polynomial P(z) has four different roots But this is impossible Thus, if a Stoneley wave exists, it must be unique Suppose that there exists a (unique) Stoneley wave Then by above arguments, its squared dimensionless velocity x is a root of the third-order polynomial P(z), and x ≠ 0, x ≠ By Eq (57), x must be xs given by Eq (24) The proof of Theorem is completed 3.2 Case 2: N B N F N E N Following the same procedure carried out for the Case 1, and noting that the relations (44) are still valid for this case, one can see that: Theorem Let N B N F N E If a Stoneley wave exists, then it is unique, and its squared dimensionless velocity xs = c 2/cT2 is defined by: xs = − A2 − −I 0̂ ; A3 F ð60Þ where A2, A3 defined by Eq (55) and: 1=B 1=F 1=E 1@ ̂ I0 = ∫ θ1 ðt Þdt + ∫ θ2 ðt Þdt + ∫ θ3 ðt Þdt A; π 1=B 1=F ð61Þ θk(t) are given by Eq (45) in which: " rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # pffiffiffi 1 1 E BEF 2 ð Þ φ1 ðt Þ = ð2−t Þ F 2−Bt −t + −t −4 −t −t −t F E B F E DB2 DB2 " = rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 1 −4 EF t−1 −t −t ; E F rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffi E ð2−t Þ2 F 1F −t + ð 2−Bt Þ −t E ffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi ; ffi r φ2 ðt Þ = pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiDB pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 1 1 BEF −t −t −4 EF t−1 −t −t −4 t− B F E E F DB2 ð62Þ ð63Þ " pffiffiffi rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 1 1 1 E BEF ð2−Bt Þ −t + EF t−1 −t t− + t− −t t− φ3 ðt Þ = E E F B F E DB2 DB2 " = rffiffiffiffiffiffiffiffiffiffiffi # pffiffiffi ð2−t Þ F t− : F ð64Þ 3.3 Case 3: N E N B N F N Analogously, we have: Theorem Suppose that N E N B N F If a Stoneley wave exists, then it is unique, and its squared dimensionless velocity xs = c 2/cT2 is defined by: xs = − A2 1 − + −I 0̂ ; A3 B E ð65Þ where A2, A3 given by Eq (55) and: 1=E 1=B 1=F 1@ ̂ I0 = ∫ θ1 ðt Þdt− ∫ θ2 ðt Þdt + ∫ θ3 ðt Þdt A; π 1=E 1=B ð66Þ θk(t) are determined by Eq (45) in which: " rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # pffiffiffi 1 1 E BEF φ1 ðt Þ = ð2−t Þ F ð2−Bt Þ −t + −t −4 −t −t −t F E B F E DB2 DB2 " = rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 1 −4 EF t−1 −t −t ; E F ð67Þ P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657 655 rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi 1 E BEF ð 2−Bt Þ −4 −t −t t− t− 2 E B F E; DB DB q ffiffiffiffiffiffiffiffiffi ffi q ffiffiffiffiffiffiffiffiffi ffi q ffiffiffiffiffiffiffiffiffi ffi φ2 ðt Þ = pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2−t Þ2 F 1F −t + EF t−1 t− 1E 1F −t " rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 1 1 BEF φ3 ðt Þ = ð2−t Þ2 F −t + EF t−1 t− −t + −t t− t− F E F B F E DB2 ð68Þ = " pffiffiffi rffiffiffiffiffiffiffiffiffiffiffi # E ð 2−Bt Þ : t− E DB2 ð69Þ Note that for this case, instead of Eq (44) we have the following: ϕ1 ðt Þ = 2π + 2θ1 ðt Þ; ϕ2 ðt Þ = π−2θ2 ðt Þ; ϕ3 ðt Þ = 2θ3 ðt Þ: ð70Þ Special cases As B N F and b E ≤ 3/4 b 1, there are five special cases as follows: Case 1.1: = B N E N F N 0, Case 1.2: N B = E N F N 0, Case 1.3: N B N E = F N 0, Case 1.4: = B N E = F N 0, Case 2.1: = B N F N E N The first four cases originate in the basic Case 1, the last comes from the Case The results of all these cases are deduced directly from the corresponding basic cases In particular, for the Case 1.1, the squared dimensionless velocity xs = c 2/cT2 of the Stoneley wave is defined by Eq (24) in which: pffiffiffi pffiffiffi pffiffiffiffiffiffi 1 F E 9+ 9+ + EF + − ; 2D F D E pffiffiffi 1=E 1=F pffiffiffi E ; I 0̂ = @ ∫ θ2 ðt Þdt + ∫ θ3 ðt Þdt A; A3 = F + D π 1=E A2 = − ð71Þ θ2(t), θ3(t) are defined by Eqs (45), (47) and (48) For the Case 1.2, xs is defined by Eq (24) where: pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffiffi F E F 9+ 1+ ; + EF − +4 2D DE F E pffiffiffi 1=E 1=F pffiffiffi E ̂ ; I = @ ∫ θ1 ðt Þdt + ∫ θ3 ðt Þdt A; A3 = F + D π 1=E A2 = − ð72Þ θ1(t), θ3(t) are defined by Eqs (45), (46) and (48) For the Case 1.3, after some manipulations and taking into account Eq (28), we have: xs = − A1 1 + 1− −I 0̂ ; A2 B ð73Þ where: A1 = h pffiffiffi i pffiffiffiffiffiffi B2 D E−1 + BE−B I 0̂ = 1=B B2 D ; A2 = + 1=E ; D ð74Þ 1@ ∫ atanφ1 ðt Þdt + ∫ atanφ20 ðt Þdt A; π 1=B ð75Þ in which φ1(t) is given by Eq (46), and: φ20 rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffiffiffiffiffiffiffi BE 1 E t−1 −t + −1 t− E B E DB = : ð2−t Þ2 + ð Þ 2−Bt DB2 ð76Þ Putting B = in Eqs (73)–(76) provides the result for the Case 1.4 In particular, for this case xs is given by: pffiffiffi 1 = E xs = 1− E − ∫ atanφ0 ðt Þdt; π ð77Þ 656 P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657 where: φ0 ðt Þ = pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 1−Et t−1 : ð2−t Þ2 ð78Þ Since xs given by Eq (77) is the velocity of a Rayleigh wave propagating in an isotropic elastic half-space with the bulk wave velocities cL and cT (see [4,13]), it follows b xs b This means that, according to Proposition 2, a Stoneley wave is always possible for the case when two isotropic elastic half-spaces having the same bulk wave velocities This fact has also been proved by Barnett et al [14] Analogously, the result for the Case 2.1 is obtained directly from the one of the Case In particular, xs = c 2/cT2 is defined by Eq (60) in which: pffiffiffi pffiffiffi pffiffiffiffiffiffi 1 F E 9+ 9+ + EF + − ; 2D F D E pffiffiffi 1=F 1=E pffiffiffi E ̂ ; I = @ ∫ θ2 ðt Þdt + ∫ θ3 ðt Þdt A; A3 = F + D π 1=F A2 = − ð79Þ where θ2(t), θ3(t) are defined by Eqs (45), (63) and (64) Remark i) From Theorems 1, 2, 3, it is shown that there is at most one Stoneley wave can travel along the interface of two isotropic elastic half-spaces in sliding contact This fact was also proved by Barnett et al [14] by another method ii) From Proposition 2, the necessary and sufficient condition for the existence of a Stoneley wave is b xs b 1, where xs is given by Eq (24) or (60) or (65) That means, for given material parameters, by simply calculating the quantity xs we can know whether a Stoneley wave exists or not, and its velocity in the case of existence This will make the obtained formulas useful in practical applications iii) When B N 1, i.e cT N cT⁎, the formulas (24), (60) and (65) are still applicable, but in which B, D, E, F, xs must be replaced by B⁎ = 1/B, D⁎ = 1/D, E⁎ = F/B, F⁎ = E/B, x⁎s = c 2/cT⁎ 2, respectively iv) By using the obtained formulas we easily obtain the numerical results obtained by Murty [15] by solving directly the secular Eq (20) For examples: - Taking B = 1/2 ; D = 3.4 (→ R = BD = 1.7) ; E = 1/3 ; F = 1/6, and using the formula (24) give c 2/cT2 = 0.9738 that coincides with the one corresponding to R = 1.7, D = 3.4 in Table of paper [15] - Taking B = 0.3 ; D = 3.0 (→ R = BD = 0.9) ; E = 1/3 ; F = 0.1, and using the formula (65) give c 2/cT2 = 0.9996 that coincides with the one corresponding to R = 0.9, D = 3.0 in Table of paper [15] - For B = 1/2 ; D = 0.2 (→ R = BD = 0.1) ; E = 1/3 ; F = 1/6, the formula (24) gives xs = The Stoneley wave therefore does not exist for this case This fact was also shown by Murty [15] - Taking B = 0.7 ; D = ; F = 0.45 ; E = 0.25, and using the formula (60) provide xs = 0.8735 that coincides with the one obtained directly from the secular Eq (20) - Consider the case of Poisson solids (i.e when λ = λ⁎, μ = μ⁎, see [15]) with D = ; R = 1.5 (E = 1/3, F = 1/2) As B = R/ D = 1.5 N we have to employ the above obtained formulas in which B, D, E, F, xs are replaced by B⁎ = 1/B = 1/1.5, D⁎ = 1/ D = 1, E⁎ = F/B = 1/3, F⁎ = E/B = 1/4.5, x⁎s = c 2/cT⁎ 2, respectively Applying the formula (24) with asterisked parameters yields c/cT⁎ = 0.9893 that coincides with the result shown in [15] Conclusions In this paper, by using the complex function method we have found formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces From the derivation of these formulas, it is shown that if a Stoneley wave exists, then it is unique Using the obtained formulas it is easy to know, for given material parameters, whether a Stoneley wave exists or not, and its velocity when it exists These formulas will therefore be useful in practical applications The method presented in this paper may be applicable to the case of perfectly bonded interfaces, as well as to Stoneley waves propagating along a material interface between two half spaces (see, for instance, [22,23]) Acknowledgments The work was supported by the Vietnam National Foundation For Science and Technology Development (NAFOSTED) under Grant No 107.02-2010.07 References [1] L Rayleigh, On waves propagating along the plane surface of an elastic solid, Proc R Soc Lond A17 (1885) 4–11 [2] R Stoneley, Elastic waves at the surface of separation of two solids, 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elastic half-spaces From the derivation of these formulas, it is shown that if a Stoneley wave exists, then it is unique Using the obtained formulas. .. sliding contact, Proc R Soc Lond A 415 (1988) 389–419 G.S Murty, A theoretical model for the attenuation and dispersion of Stoneley waves at the loosely bonded interface of elastic half spaces, Phys