International Journal of Engineering Science 60 (2012) 53–58 Contents lists available at SciVerse ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci Formulas for the speed and slowness of Stoneley waves in bonded isotropic elastic half-spaces with the same bulk wave velocities Pham Chi Vinh a,⇑, Peter G Malischewsky b, Pham Thi Ha Giang a a b Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam Institute for Geosciences, Friedrich-Schiller University Jena, Burgweg 11, 07749 Jena, Germany a r t i c l e i n f o Article history: Received August 2011 Received in revised form 20 March 2012 Accepted 12 May 2012 Available online 21 June 2012 Keywords: Stoneley waves The wave velocity The wave slowness Two bonded isotropic elastic half-spaces Holomorphic function a b s t r a c t This paper is concerned with the propagation of Stoneley waves in two bonded isotropic elastic half-spaces with the same bulk wave velocities Our main purpose is to find formulas for the wave velocity and the wave slowness By applying the complex function method, the exact formulas for the wave velocity and the wave slowness have been derived The derivation of these formulas also shows that there always exists a unique Stoneley wave for the case under consideration Ó 2012 Elsevier Ltd All rights reserved Introduction Interfacial waves traveling along the welded plane boundary of two different isotropic elastic half-spaces were first investigated by Stoneley (1924) He derived the secular equation of the wave, and showed by means of examples that such interfacial waves not always exist Subsequent studies by Sezawa and Kanai (1939) and Scholte (1942, 1947) focused on the range of existence of Stoneley waves Scholte (1947) found the equations expressing the boundaries of the existence domain and they are in complete agreement with the corresponding curves numerically obtained by Sezawa and Kanai (1939) for the case of Poisson solids (for which the corresponding Lame constants of the half-spaces are the same) Their studies showed that the restriction on material constants that permit the existence of Stoneley waves are rather severe However, Sezawa and Kanai (1939) and Scholte did not prove the uniqueness of Stoneley waves This question was settled by Barnett, Lothe, Gavazza, and Musgrave (1985) for general anisotropic half-spaces with bonded interface The propagation of Stoneley waves in anisotropic media was also studied by Stroh (1962) and Lim and Musgrave (1970) Much of the early attentions to Stoneley waves was directed toward geophysical applications Latter studies have indicated that interfacial waves may prove to be useful probes for the non-destructive evaluations (see Lee & Corbly, 1977, Rokhlin, Hefet, & Rosen, 1980) The propagation of Stoneley waves in two isotropic elastic half-spaces with the loosely bonded interface was studied by Murty (1975b, 1975a) The author has derived the secular equation of the wave and obtained a lot of numerical values of the Stoneley wave velocity by directly solving that equation for the case of Poisson solids The existence and uniqueness of Stoneley waves in two half-spaces in sliding contact was investigated by Barnett, Gavazza, and Lothe (1988) The authors showed that for the isotropic elastic half-spaces, if a Stoneley exists, then it is unique, while for the anisotropic half-spaces, the possibility of a new slip-wave mode, called the second slip-wave mode, arises ⇑ Corresponding author Tel.: +84 5532164; fax: +84 8588817 E-mail address: pcvinh@vnu.edu.vn (P.C Vinh) 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.ijengsci.2012.05.002 54 P.C Vinh et al / International Journal of Engineering Science 60 (2012) 53–58 For the Stoneley wave, its velocity is of great interest to researchers in various fields of science The formulas for the Stoneley wave velocity are 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 Recently, the exact formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces have been derived by Vinh and Giang (2011) The main aim of this paper is to find exact formulas for the velocity and the slowness of Stoneley waves propagating in two bonded isotropic elastic half-spaces which have the same bulk wave velocities By employing the complex function method that is based on the properties of Cauchy integrals and the generalized Liouville theorem, the authors derive the exact formulas for the velocity and for the slowness The derivation of these formulas shows that: if a Stoneley wave exists then it is unique; and there always exits a Stoneley wave propagating along the bonded interface of two bonded isotropic elastic halfspaces which have the same bulk wave velocities The former was proved by Barnett et al (1985) by another method, and the latter was shown by Stoneley (1924) expanding the corresponding secular equation into Taylor series Exact formula for the velocity Let us consider two isotropic elastic solids X and X⁄ occupying the half-space x2 P and x2 0, respectively Suppose that these two elastic half-spaces are in welded contact with each other at the plane x2 = Then, the component of the particle displacement vector and the component of the stress tensor are continuous across the interface x2 = Note that same quantities related to X and X⁄ have the same symbol but are systematically distinguished by an asterisk if pertaining to X⁄ Suppose that the two half-spaces have the same bulk wave velocities, i.e ck ẳ ck k ẳ 1; 2ị, where c1, cÃ1 are the longitudinal wave velocities and c2, cÃ2 are the transverse wave velocities of the half-spaces These conditions appear to be satisfied at the Wiechert surface of discontinuity within the Earth, as indicated by Stoneley (1924) Since these two half-spaces become the same if q = q⁄, we therefore assume that the mass densities of the half-spaces are different from each other, i.e q – q⁄ According to Stoneley (1924) the secular equation of Stoneley waves for the case under consideration is (see Eq (3) in Stoneley, 1924): h h i pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffii x2 ðq À qà Þ2 q ỵ q ị2 x cx ỵ 4q q ị2 xị2 cxị ỵ x cxx 2ị ; 1ị where c ẳ c22 =c21 < c < 1ị and x ẳ c2 =c22 , c is the velocity of Stoneley waves Since < c < c2, it follows that x (0, 1) The secular Eq (1) can be rewritten as follows: pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðxÞ  q1 ðxÞ À q2 ðxÞ À x cx ẳ 0; < x < 1; 2ị where q1 xị ẳ b ỵ 4cịx2 42 ỵ cịx ỵ ; 2 q2 xị ẳ a2 x2 4b x ỵ 8b ; 3ị in which a ẳ ỵ r; b ẳ r; r ẳ q=q 1ị: 4ị Note that, since b < a it follows: q2 ðxÞ ¼ a2  2   ! 2 x 2b =a2 ỵ 4b b =a2 =a2 > x; ð5Þ and according to Stoneley (1924) we have: lim x!0 f xị ẳ a2 þ b c < 0: x2 ð6Þ Now, in the complex plane C we consider the equation: pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðzÞ  q1 zị ỵ q2 zị z cz ẳ 0; 7ị p p where qk(z) are given by (3), and z À 1; cz À are chosen as the principal branches of the corresponding square roots With the fact < c < it is easy to see that for z (0, 1), Eq (7) becomes Eq (2) We will prove the following theorem: Theorem If a Stoneley wave exists, then it is unique, and its squared dimensionless velocity xs ¼ c2 =c22 is defined by xs ¼ À  2 p b c ỵ I0 ; À ffiffiffi p a2 c ð8Þ where I0 ¼ p Z ( 1=c hðtÞdt; hðtÞ ¼ atan ) q1 ðtÞ pffiffiffiffiffiffiffiffiffiffi ffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi : q2 ðtÞ t À 1 À ct ð9Þ P.C Vinh et al / International Journal of Engineering Science 60 (2012) 53–58 55 Proof Denote L = [1,1/c], S = {z C, z R L}, N(z0) = {z S: < jz À z0j < e}, e is a sufficient small positive number, z0 is some point of the complex plane C If a function /(z) is holomorphic in X & C we write /(z) H(X) In order to solve Eq (2) in the interval (0, 1) we will find zeros of f(z) in the domain S ('(0, 1)) Theorem is proved as follows: Step 1: To express f(z) by product of two functions, the first function is non-zero in S, and the second one is a polynomial of third-order (see Eq (16)) Sine the first factor is non-zero in S, the zeros of f(z) are identical with those of the third-order polynomial in S Due to (6), z1 = 0, z2 = are zeros of the third-order polynomial in S Step 2: Finding the coefficients of z3 and z2 of the third-order polynomial (see (26) and (27)) by expanding f(z) and the inverse of the first factor into Laurent series at infinity Knowing these coefficients we obtain immediately the expression of the third zero, given by (8), of the third-order polynomial Step 3: Showing that: (i) if a Stoneley wave exists then it is unique (ii) if a Stoneley wave exists then its squared dimensionless velocity is the third zero of the third-order polynomial Step From (3) and (7) 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) and N(1/c) (f3) f(z) = O(z3) as jzj ? (f4) f(z) is continuous on L from the left and from the right (see Muskhelishvili, 1953) with the boundary values f+(t) (the right boundary value of f(z)), fÀ(t) (the left boundary value of f(z)) dened as follows: pp f ỵ tị ẳ q1 tị þ iq2 ðtÞ t À 1 À ct; f tị ẳ f ỵ tị; 10ị where the bar indicates the complex conjugate Now we define the function g(t) (t L) as follows: gtị ẳ f ỵ tị ; f À ðtÞ ð11Þ then it is obvious that: f ỵ tị ẳ gtịf tị; t L: 12ị Consider the function C(z) dened as: Czị ẳ 2pi Z L log gðtÞ dt: tÀz ð13Þ The function C(z) is an integral of the Cauchy type whose properties are examined in detail in Muskhelishvili (1953) (chaps 2–4) It is not difficult to verify that: (c1) C(z) H(S) (c2) C(1) = (c3) For c – 1/2: C(z) = X0(z), z N(1), C(z) = X1(z), z N(1/c), where X0(z) (X1(z)) bounded in N(1) (N(1/c)) and takes a defined value at z = (z = 1/c) (c4) For c = 1/2: C(z) = X2(z), z N(1), C(z) = (1/2) log (z À 2) + X3(z), z N(2), where X2(z) (X3(z)) bounded in N(1) (N(2)) and takes a defined value at z = (z = 2) It is noted that (c3) and (c4) come respectively from the facts (see Muskhelishvili, 1953, chap 4, Section 29): log g(1) = log g(1/c) = for c – 1/2 and logg(1) = 0, log g(2) = À1 for c = 1/2 We now consider the function Y(z) dened by Yzị ẳ f zịeCzị : 14ị From (f1) À (f3), (c1) À (c4), (12), (14) and the Plemelj formula (see Muskhelishvili, 1953, Chapter 2, Section 17), it is not difficult to assert that (see also Vinh & Giang, 2011, 2012): (y1) (y2) (y3) (y4) Y(z) H(S) Y(z) = O(z3) as jzj ? Y(z) is bounded in N(1) and N(1/c) Y+(t) = YÀ(t),t L 56 P.C Vinh et al / International Journal of Engineering Science 60 (2012) 53–58 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/c 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 Muskhelishvili, 1963) Thus, by the generalized Liouville theorem (Muskhelishvili, 1963) and taking account into (y2) we have: Yzị ẳ Pzị; 15ị where P(z) is a third-order polynomial From (14) and (15) we have: f zị ẳ eCzị Pzị: C(z) Since e ð16Þ – "z S (by (c1)), from (16) it follows that: f zị ẳ $ Pzị ẳ in S: ð17Þ From (6), (16) and eC(z) – "z S, it follows that z = is a double root of Eq P(z) = Step Also from (16) we have: Pzị ẳ f zịeCzị : ð18Þ From (5), (10) and (11) it implies: log gtị ẳ i/tị; /tị ẳ p 2htị; 19ị where h(t) is given by (9)2 From (13) and (19) it follows (see also Nkemzi, 1997): Czị ẳ X In ; nỵ1 z nẳ0 20ị in which: In ẳ 2p Z 1=c t n /tịdt; n ẳ 0; 1; 2; 3; : ð21Þ On use of (20) one can express eC(z) as follows: eCzị ẳ ỵ a1 a2 a3 ỵ ỵ ỵ OðzÀ4 Þ; z z z ð22Þ where a1, a2, a3 are constants to be determined Employing the identity: À ÀCðzÞ e ẳ Czịị0 eCzị ; 23ị and substituting (20), (22) into (23) yield: a1 ¼ I0 ; a2 ¼ By expanding I20 ỵ I1 ; a3 ẳ I30 þ I1 I0 þ I : ð24Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À 1=z; À 1=ðczÞ into Laurent series at infinity, it is not difficult to see that: À f zị ẳ A3 z3 ỵ A2 z2 ỵ A1 z ỵ A0 ỵ O z1 ; 25ị where A2, A3 are given by   pffiffiffi pffiffiffi a2 A2 ẳ b cị c ỵ p ; c A3 ẳ p ca2 : ð26Þ Substituting (22) and (25) into (18) yields: Pzị ẳ A3 z3 ỵ z2 A2 ỵ A3 a1 ị ỵ zA1 ỵ A3 a2 ỵ A2 a1 ị þ A3 a3 þ A2 a2 þ A1 a1 þ A0 : ð27Þ Since z = is a double root of Eq P(z) = 0, as mentioned above., from (19), (21), (24)1 and (27), the third root of Eq P(z) = 0, denoted by xs, is given by xs ¼ À   A2 1 À À þ I0 ; A3 c ð28Þ where A2, A3 are given by (26) and I0 is calculated by (9) Step Now we suppose that there exist two different Stoneley waves with the 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 < x(1), x(2) < From (17) it follows P(x(1)) = P(x(2)) = P(0) = But this is impossible because P(z) is a third-order polynomial and z = is a double root of Eq P(z) = Thus, if a Stoneley wave exists, it 57 P.C Vinh et al / International Journal of Engineering Science 60 (2012) 53–58 c 0.98 0.96 0.94 c2 10 0.1 r 0.2 γ 0.3 Fig Dependence of 0.4 0.5 pffiffiffi x ¼ c=c2 on c and r Table Values of the squared dimensionless velocity of Stoneley waves calculated by directly solving the secular Eq (2) (x⁄), by the exact formulas (8) (xs) and (32) (ys) for some given values of r Here c = 1/3 r x⁄ xs ys 1/ys 0.9901 0.9901 1.0100 0.9901 0.9686 0.9686 1.0324 0.9686 0.9501 0.9501 1.0525 0.9501 0.9357 0.9357 1.0687 0.9357 0.9246 0.9246 1.0815 0.9246 0.9158 0.9158 1.0919 0.9158 0.9087 0.9087 1.10043 0.9087 0.9029 0.9029 1.1075 0.9029 must be unique This fact was also proved by Barnett et al (1985) for the generally anisotropic case using the surface impedance method Suppose that there exists a (unique) Stoneley wave Then by above arguments, its squared dimensionless velocity x is a zero of the third-order polynomial P(z), and x – This implies that x must be xs given by (28), i.e by (8) The proof of Theorem is finished h Fig demonstrates the dependence of the Stoneley-wave velocity on c and r as a 3D-picture Exact formula for the slowness In this section we derive the formula for the squared dimensionless slowness y = 1/x of Stoneley waves that satisfies the equation: pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffi FðyÞ  p1 ðyÞ p2 yị y y c ẳ 0; y > 1; p1 yị ẳ b 8y2 42 ỵ cịy ỵ ỵ 4c y; p2 yị ẳ 8b y2 4b y ỵ a2 : ð29Þ where 2 ð30Þ Note that p2(y) > "y Eq (29) is derived from Eq (2) replacing x by 1/y In the complex plane C, Eq (29) takes the form: pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffi FðzÞ  p1 ðzÞ À p2 zị z z c ẳ 0; 31ị pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi where pk(z) are given by (30), and z À 1; z À c are chosen as the principal branches of the corresponding square roots For z (1 + 1) Eq (31) becomes Eq (29) Following the same procedure carried out in the previous section we have: Theorem If a Stoneley wave exists, then it is unique, and its squared dimensionless slowness ys ¼ c22 =c2 is given by ys ẳ 2a2 ỵ b c1 cÞð1 À 2cÞ   À I0 ; 2 a2 b c2 32ị where I0 ẳ p Z ( htịdt; c htị ẳ atan ) p1 ðtÞ p ffiffiffiffiffiffiffiffiffiffi ffi : pffiffiffiffiffiffiffiffiffiffiffi p2 ðtÞ t À c À t ð33Þ Table shows the numerical values of the squared dimensionless velocity of Stoneley waves which are calculated by directly solving the secular Eq (2) (for x⁄), by using the exact formulas (8) (for xs) and (32) (for ys) for some given values of r with c = 1/3 It is seen from Table that the corresponding values of x⁄, xs and 1/ys totally coincide with each other 58 P.C Vinh et al / International Journal of Engineering Science 60 (2012) 53–58 On the existence of Stoneley waves We will prove the following theorem: Theorem There always exists a (unique) Stoneley wave propagating along the bonded interface of two isotropic elastic halfspaces with the same bulk wave velocities Proof It is clear that in order a Stoneley wave to exist it is sufficient that ys > 1, where ys is given by (32) Since Eq F(z) = has no solution in the interval (c, 1] due to the discontinuity of F(z) in (0, 1), it follows ys R (c, 1] Therefore if ys > c then ys > This yields if ys > c, then a Stoneley wave can be propagate along the bonded interface of two isotropic elastic half-spaces with the same bulk wave velocities Now we prove that ys > c Indeed, from (33) it implies that ÀI0 P À(1 À c)/2 Taking into account this fact, from (32) we have: h i a2 cị ỵ b c cị2 ỵ 2c2   ys À c P > 0; 2 a2 À b c2 ð34Þ due to c 3/4 < and a2 > b2 The proof of Theorem is finished Note that the existence of Stoneley waves for the case under consideration was also asserted by Stoneley (1924) by another technique h Conclusions In this paper, the exact formulas for the velocity and the slowness of Sroneley waves propagating along the bonded interface of two isotropic elastic half-spaces having the same bulk wave velocities are derived using the complex function method By using the obtained exact formulas, it is shown that there always exists a unique Stoneley wave propagating in two bonded isotropic elastic half-spaces with the same bulk wave velocities Since the obtained formulas are explicit they are useful in practical applications Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), and by the DAAD References Barnett, D M., Gavazza, S D., & Lothe, J (1988) Slip waves along the interface between two anisotropic elastic half-spaces in sliding contact Proceedings of the Royal Society of London Series A – Mathematical and Physical Sciences, 415, 389–419 Barnett, D M., Lothe, J., Gavazza, S D., & Musgrave, M J P (1985) Consideration of the existence of interfacial (Stoneley) waves in bonded anisotropic elastic half-spaces Proceedings of the Royal Society of London Series A – Mathematical and Physical Sciences, 412, 153–166 Lee, D A., & Corbly, D M (1977) Use of interface waves for nondestructive inspection IEEE Transactions on Sonics and Ultrasonics, 24, 206–212 Lim, T C., & Musgrave, M J P (1970) Stoneley waves in anisotropic media Nature, 225, 372 Murty, G S (1975a) A theoretical model for the attenuation and dispersion of stoneley waves at the loosely bonded interface of elastic half spaces Physics of the Earth and Planetary Interiors, 11, 65–79 Murty, G S (1975b) Wave propagation at an unbounded interface between two elastic half-spaces Journal of the Acoustical Society of America, 58, 1094–1095 Muskhelishvili, N I (1953) Singular integral equations Noordhoff-Groningen Muskhelishvili, N I (1963) Some Basuc problems of mathematical theory of elasticity Netherlands: Noordhoff Nkemzi, D (1997) A new formula for the velocity of Rayleigh waves Wave Motion, 26, 199–205 Rokhlin, S., Hefet, M., & Rosen, M (1980) An elastic interface wave guided by a thin film between two solids Journal of Applied Physics, 51, 3579–3582 Scholte, J G (1942) On the Stoneley wave equation Proceedings/ Koninklijke Nederlandsche Akademie van Weten-schappen, 45, 159–164 Scholte, J G (1947) The range of existence of Rayleigh and Stoneley waves Monthly Notices of the Royal Astronomical Society Geophysical Supplement, 5, 120–126 Sezawa, K., & Kanai, K (1939) The range of possible existence of Stoneley waves, and some related problems Bulletin of the Earthquake Research Institute Tokyo University, 17, 1–8 Stoneley, R (1924) Elastic waves at the surface of separation of two solids Proceedings of the Royal Society of London Series A – Mathematical and Physical Sciences, 106, 416–428 Stroh, A N (1962) Steady state problems in anisotropic elasticity Journal of Mathematical Physics, 41, 77–103 Vinh, Pham Chi, & Giang, Pham Thi Ha (2011) On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces Wave Motion, 48, 646–656 Vinh, Pham Chi, & Giang, Pham Thi Ha (2012) Uniqueness of Stoneley waves in pre-stressed incompressible elastic media International Journal of NonLinear Mechanics, 47, 128–134  ... Conclusions In this paper, the exact formulas for the velocity and the slowness of Sroneley waves propagating along the bonded interface of two isotropic elastic half-spaces having the same bulk wave velocities. .. determining material parameters from the measured values of the wave speed Recently, the exact formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic. .. Liouville theorem, the authors derive the exact formulas for the velocity and for the slowness The derivation of these formulas shows that: if a Stoneley wave exists then it is unique; and there