Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 4, pp. 1125-1135, Warsaw 2016 DOI: 10.15632/jtam-pl.54.4.1125 ON SOME PROBLEMS OF SH WAVE PROPAGATION IN INHOMOGENEOUS ELASTIC BODIES Sebastian Kowalczyk, Stanisław J. Matysiak University of Warsaw, Faculty of Geology, Institute of Hydrogeology and Engineering Geology, Warszawa, Poland e-mail: s.j.matysiak@uw.edu.pl Dariusz M. Perkowski Białystok University of Technology, Faculty of Mechanical Engineering, Białystok, Poland The paper deals with the propagation of shear horizontal (SH)waves in annonhomogeneous elastic half-space composed of a layer whose shear modulus andmass density have a power dependence on the distance from the lower plane and the periodically stratified half-space. The equation which relates the wave speed to the wave-number and functions of the shear modulus andmass density is derived. Thewave velocity is analyzed numerically. Especially, the influence of mechanical properties of the coating layer and the stratified foundation on the wave velocity is presented in the form of figures. Keywords: displacement, stresses, SH wave, shear modulus, stratified foundation 1. Introduction The phenomena of wave propagations through the Earth is useful in investigating the internal Earth structure, and it can be helpful in explorations of various materials beneath the Earth’s surface. It is well known that the Earth is not perfectly homogeneous and some forms of inho- mogeneity exist. Many rocks and soils are stratified and clearly pice-wise homogeneous. Some layers are characterized by mechanical parameters with continuous changing in spatial direc- tions (called as functionally gradedmaterials). The problems of modeling of wave propagations in inhomogeneous elastic bodies play a very important role in applied geophysics civil and me- chanical engineering (space structures, fusion reactors). The list of references connected with the problems of wave propagations in inhomogeneous elastic bodies is rather very large (for instance monographsby by Birykov et al. (1995), Brekhovskikh (1960), Kennet (1983), Nayfeh (1995); papers byAchenbach andBalogun (2010), Alenitsyn (1964), Alshits andMaugin (2005), Cerveny et al. (1982), Destrade (2007), Shuvalov et al. (2008), Vrettos (1990)). Achenbach and Balogun (2010) dealt with the propagation of anti-plane shear waves in an elastic half-space whose shear modulus and mass density had an arbitrary dependence on the distance from the boundary plane. Alenitsyn (1964) considered the problem of Rayleigh waves in a nonhomogeno- us elastic slab. Alshits and Maugin (2005) developed a theory for the elastic wave propagation in an arbitrary layered plane (piece-wise homogenous or continuously inhomogeneous). The de- scriptionwas based on the transfermatrix approach.TheGaussian beammethodof the solution of wave propagation problems in inhomogeneous bodies was applied by Cerveny et al. (1982). The seismic Rayleigh waves in an orthotropic elastic half-space with an exponentially graded propertieswere considered byDestrade (2007). Shuvalov et al. (2008) presented some analysis of the problem of shear horizontal waves in transversely inhomogeneous plates. Surface harmonic vibrations of soil deposits with variable shear modulus were analysed by Vrettos (1990). The problem of SH-wave propagation in anisotropic inhomogeneous layer with directional rigidities and density of mass changing as a power function was investigated by Upadhyay and 1126 S. Kowalczyk et al. Gupta (1972). The authors assumed that the outer layer was fixed to an isotropic homogeneous elastic half-space or to the rigid substrate. The present paper is concerned with the case of a shear horizontal (SH) wave in an inho- mogeneous elastic layer which is assumed to be ideally fixed to a periodically stratified elastic half-space, and the upper boundary plane is free of loadings. The considered layer is charac- terized by the shear modulus and mass density in the form of power functions of the distance from the lower boundary plane. The substrate is assumed to be composed of periodically re- peated two-layered laminae parallel to the boundary plane. Each component of the lamina is a homogeneous and isotropic body. The assumptions connected with the ideal bonding of the components on interfaces lead to a complicated boundary value problemwithin the framework of the classical theory of elasticity. For this reason, the classic idea is the use of the approximate procedure to replace the heterogeneous medium by an equivalent homogenized model, which gives the average behavior at the macroscopic scale. One of them is the homogenized model with microlocal parameters presented by Matysiak and Woźniak (1987, 1988). This model is derived by using the methods of the nonstandard analysis and taking into account the effects due to the periodic structure of the body. The governing equations of the model are formula- ted in terms of the unknown macro-displacements and certain extra unknowns being referred to as microlocal parameters. They are described by a relatively simple form of the equations satisfying the conditions of perfect interfacial bonding of constituents. The homogenized model has been successfully applied to a series of problems of the linear elasticity and thermoelasticity (problems of cracks, cavities, inclusions, contact problems, wave propagations), which it was partially resumed in (Matysiak, 1996; Woźniak and Woźniak, 1995). It should be underlined that the homogenization approach has been noticed to produce good physical results, at the same time being rather simple in mathematical aspects (Kulchytsky-Zhyhailo and Kołodziej- czyk, 2007; Kulchytsky-Zhyhailo and Matysiak, 2005, 2006; Kulchytsky-Zhyhailo et al., 2006). The wave problems in a periodically layered elastic half-space were investigated by Bielski and Matysiak (1992), Matysiak et al. (2009). The same dependence of the shear modulus is taken into account in many papers (see for instance Calladine and Greenwood, 1978; Wang et al., 2003)). The same dependence of the shearmodulus of the coating layer is considered in the pre- sent paper. The distributions of displacements and stresses in an inhomogeneous incompressible elastic half-space caused by line and point loads are considered in (Cerveny et al., 1982). The propagation of surface waves in a linear-elastic, isotropic, compressible half-space with constant mass density and Poisson’s ratio and shear modulus varying with depth is considered in (Vret- tos, 1990). The useful list of dependence forms for elastic modulus is presented by Wang et al. (2003). 2. Formulation and solution of the problem Consider the problem of shear waves propagation in an elastic nonhomogeneous layer and pe- riodically layered half-space. Let (x1,x2,x3) denote the Cartesian coordinate system such that the layer occupies the region x1 ∈ R, 0¬ x2 ¬ H, x3 ∈ R, where H > 0 is constant thickness of the FGMbody, Fig. 1. Let the upper boundaryplanex2 = H be free of loadings, and the layer is ideally fixed to the periodically two-layered half-space in the plane x2 =0. Let the stratified half-space be composed of periodically repeated fundamental laminae with thickness δ, which include two homogeneous isotropic sub-layers denoted by 1 and 2with thicknesses δj, j =1,2, and δ = δ1+δ2. Let µj, ρj, j =1,2 be the shearmodulus andmass densities of the subsequent constituents of the composite half-space. Herein and in the sequel, all quantities (material components, stresses) pertaining to sub-layer 1 and 2 will be labeled by the index j taking values 1 and 2, respectively. The On some problems of SH wave propagation in inhomogeneous elastic bodies 1127 Fig. 1. Scheme of the considered SH problem considerations are limited to the anti-plane harmonicwave propagation in the 0x1 direction. Let u(x1,x2, t) = (0,0,u3(x1,x2, t)) be the displacement vector, where t denotes time. The shear modulus µ and mass density ρ of the upper layer are assumed the same as in (Upadhyay and Gupta, 1972), namely µ = µ0(1+αx2) p ρ = ρ0(1+αx2) p (2.1) where µ0, ρ0, α, p are given constants. The non-zero stress components σ13 and σ23 in the coating layer are expressed in the form σ13(x1,x2, t)= µ0(1+αx2) p∂u3(x1,x2, t) ∂x1 σ23(x1,x2, t)= µ0(1+αx2) p∂u3(x1,x2, t) ∂x2 (2.2) The anti-plane wave motion is governed by the following equation ∂σ13 ∂x1 + ∂σ23 ∂x2 = ρ0(1+αx2) p∂ 2u3 ∂t2 x1 ∈ R 0 < x2 < H (2.3) where the body forces are omitted. From equations (2.2) and (2.3), it follows that ∂2u3 ∂x21 + αp 1+αx2 ∂u3 ∂x2 + ∂2u3 ∂x22 = ρ0 µ0 ∂2u3 ∂t2 x1 ∈ R 0 < x2 < H t ∈ R (2.4) To determine the displacement and stresses in the periodically layered half-space x2 < 0, the ho- mogenizedmodel withmicrolocal parameters (Bielski andMatysiak, 1992; Kulchytsky-Zhyhailo and Kołodziejczyk, 2007; Kulchytsky-Zhyhailo and Matysiak, 2005, 2006; Kulchytsky-Zhyhailo et al., 2006; Matysiak et al., 2009; Matysiak and Woźniak, 1987, 1988) is applied. Here only a brief outline of the governing equations for the case of anti-plane state of strain will be presen- ted. The homogenized procedure presented by Matysiak andWoźniak (1987, 1988) is based on theorems of the nonstandard analysis and some physical assumptions, which leads, in the case of anti-plane state of strain, to the following approximations u3(x1,x2, t)= w3(x1,x2, t)+h(x2)q3(x1,x2, t)≈ w3(x1,x2, t) ∂u3(x1,x2, t) ∂x1 ≈ ∂w3(x1,x2, t) ∂x1 ∂u3(x1,x2, t) ∂t ≈ ∂w3(x1,x2, t) ∂t ∂u3(x1,x2, t) ∂x2 ≈ ∂w3(x1,x2, t) ∂x2 +h′(x2)q3(x1,x2, t) (2.5) where w3, q3 are unknowns called macro-displacement andmicrolocal parameters, respectively. The function h (called the shape function) is given in the form h(x2)=    x2− 1 2 δ1 for 0¬ x2 ¬ δ1 − ηx2 1−η − 1 2 δ1+ δ1 1−η for δ1 ¬ x2 ¬ δ h(x2+ δ)= h(x2) (2.6) 1128 S. Kowalczyk et al. and η = δ1 δ (2.7) Since |h(x2)| < δ for every x2 ∈ R, then for small δ the termswith h in equations (2.5) are small and are neglected. However, the derivative h′ is not small and the terms involving h′ cannot be neglected. The formof the shape functionh given in (2.6) secures the fulfilment of the conditions of ideal bonding on the composite interfaces. The homogenized model presented by Matysiak andWoźniak (1987, 1988) in the case of anti-plane state of strain leads to the following equations for the unknowns w3 and q3 µ̃ (∂2w3 ∂x21 + ∂2w3 ∂x22 ) +[µ] ∂q3 ∂x2 = ρ̃ ∂2w3 ∂t2 µ̂q3+[µ] ∂w3 ∂x2 =0 (2.8) where ρ̃ = ηρ1+(1−η)ρ2 µ̃ = ηµ1+(1−η)µ2 [µ] = η(µ1−µ2) µ̂ = ηµ1+ η2µ2 1−η (2.9) The non-zero stress components σ (j) 13 , σ (j) 23 , j =1,2 in the layer of j-th kind are expressed in the form σ (j) 13 = µj ∂w3 ∂x1 σ (j) 23 = µj (∂w3 ∂x2 +h′(x2)q3 ) (2.10) Eliminating themicrolocal parameter q3 from(2.8)1 (2.10) byusing (2.8)2, leads to the equations µ̃ ∂2w3 ∂x21 +C ∂2w3 ∂x22 = ρ̃ ∂2w3 ∂t2 (2.11) and σ (j) 13 = µj ∂w3 ∂x1 σ (j) 23 = C ∂w3 ∂x2 j =1,2 (2.12) where C = µ̃− [µ] 2 µ̂ = µ1µ2 (1−η)µ1+ηµ2 > 0 (2.13) The following boundary conditions are taken into consideration: a) on the upper boundary of the FGM layer σ23(x1,H,t)= 0 x1 ∈ R t ∈ R (2.14) b) on the interface x2 =0 between the FGM layer and the periodically stratified half-space u3(x1,0 +, t)= w3(x1,0 −, t) σ23(x1,0 +, t)= σ (1) 23 (x1,0 −, t) x1 ∈ R t ∈ R (2.15) c) the regularity condition at infinity lim x2→−∞ w3(x1,x2, t)= 0 (2.16) On some problems of SH wave propagation in inhomogeneous elastic bodies 1129 Let us consider a SHwave solution of the form u3(x1,x2, t)= U3(x2)e ik(x1−ct) w3(x1,x2, t)= W3(x2)e ik(x1−ct) (2.17) where i = √ −1, U3 and W3 are unknown amplitude of displacement in the outer layer and the periodically layered half-space, respectively, and k and c are the wave number and the phase velocity, respectively. By using equations (2.4) and (2.11) and (2.17), an ordinary differential equation are obtained d2U3(x2) dx22 + αp 1+αx2 dU3(x2) dx2 +k2 (c2 c20 −1 ) U3(x2)= 0 0 < x2 < H (2.18) and d2W3(x2) dx22 + k2 C (ρ̃c2− µ̃)W3(x2)= 0 x2 < 0 (2.19) where c20 = µ0 ρ0 (2.20) The ordinary differential equation of the second order with variable coefficients (2.18) belongs to well-known type (Kamke, 1976, p. 401). Its general solution has the form U3(x2)= (1+αx2) 1−p 2 [ A1J |1−p| 2 ( q (1 α +x2 )) +A2Y |1−p| 2 ( q (1 α +x2 ))] 0 < x2 < H (2.21) where q2 = k2 (c2 c20 −1 ) (2.22) on the assumption that c > c0, and A1, A2 are unknown constants, which should be determined fromboundary conditions (2.5), andJ|1−p|/2(·), Y|1−p|/2(·) areBessel functions. Equations (2.19) and (2.17) with condition (2.16) lead to the following solution W3(x2)= A3exp(βx2) x2 < 0 β 2 = k2µ̃ C ( 1− c2 c̃2 ) c̃2 = µ̃ ρ̃ (2.23) on the assumption that c < c̃ and A3 is an unknown constant. The constant A1, A2, A3 should be calculated from boundary conditions (2.14) and (2.15). The further analysis needs to take into consideration two cases: p ¬ 1 and p > 1. Case 1 Consider that p ¬ 1 so |1−p|=1−p (2.24) Todetermine the stress componentσ23, the followingdifferential relations for theBessel functions should be applied (Lebiediev, 1957) dzνJν(z) dz = zνJν−1(z) dzνYν(z) dz = zνYν−1(z) (2.25) 1130 S. Kowalczyk et al. Bearing inmindequations (2.2), (2.17), (2.21) and (2.24), it follows that the stress componentσ23 is expressed in the form σ23(x1,x2, t)= qµ0(1+αx2) 1+p 2 [ A1J−1−p 2 ( q (1 α +x2 )) +A2Y−1−p 2 ( q (1 α +x2 ))] eik(x1−ct) (2.26) where 0 < x2 < H. From boundary condition (2.14) and conditions of continuity (2.15) as well as equations (2.26), (2.17), (2.23), (2.12), (2.21), the following algebraic equations for the unknowns A1, A2, A3 are obtained A1J−1−p 2 ( q (1 α +H )) +A2Y−1−p 2 ( q (1 α +H )) =0 A1J1−p 2 (q α ) +A2Y1−p 2 (q α ) = A3 µ0q [ A1J−1−p 2 (q α ) +A2Y−1−p 2 (q α )] = CβA3 (2.27) Eliminating A3 from the system of equations (2.27), it follows that A1J−1−p 2 ( q (1 α +H )) +A2Y−1−p 2 ( q (1 α +H )) =0 A1 [ µ0qJ−1−p 2 (q α ) −CβJ1−p 2 (q α )] +A2 [ µ0qY−1−p 2 (q α ) −CβY1−p 2 (q α )] =0 (2.28) The system of algebraic equations (2.28) has a non-zero solution under the following condition J−1−p 2 ( q (1 α +H ))[ µ0qY−1−p 2 (q α ) −CβY1−p 2 (q α )] −Y−1−p 2 ( q (1 α +H ))[ µ0qJ−1−p 2 (q α ) −CβJ1−p 2 (q α )] =0 (2.29) Equation (2.29) will be solved numerically. Case 2 Consider now that p > 1 so |1−p|= p−1 (2.30) Todetermine the stress componentσ23, the followingdifferential relations for theBessel functions should be applied (Lebiediev, 1957) dz−νJν(z) dz =−z−νJν+1(z) dz−νYν(z) dz =−z−νYν+1(z) (2.31) Bearing in mind equations (2.2), (2.17), (2.21) and (2.31), it follows that the stress component σ23 is expressed in the form σ23(x1,x2, t)=−qµ0(1+αx2) 1+p 2 [ A1J1+p 2 ( q (1 α +x2 )) +A2Y1+p 2 ( q (1 α +x2 ))] eik(x1−ct) (2.32) where 0 < x2 < H. On some problems of SH wave propagation in inhomogeneous elastic bodies 1131 From boundary conditions (2.14) and (2.15) and equations (2.32), (2.17), (2.23), (2.12), (2.21), the following linear algebraic equations for the unknowns A1, A2, A3 are obtained A1Jp+1 2 (q α (1+αH) ) +A2Yp+1 2 (q α (1+αH) ) =0 A1Jp+1 2 (q α ) +A2Yp+1 2 (q α ) = A3 −µ0q [ A1J1+p 2 (q α ) +A2Y1+p 2 (q α )] = CβA3 (2.33) Eliminating A3 from the system of equations (2.33), it follows that A1Jp+1 2 (q α (1+αH) ) +A2Yp+1 2 (q α (1+αH) ) =0 A1 [ µ0qJp+1 2 (q α ) +CβJp−1 2 (q α )] +A2 [ µ0qYp+1 2 (q α ) +CβYp−1 2 (q α )] =0 (2.34) The system of algebraic equations (2.34) has non-zero solutions under the following condition Jp+1 2 (q α (1+αH) )[ µ0qYp+1 2 (q α ) +CβYp−1 2 (q α )] −Yp+1 2 (q α (1+αH) )[ µ0qJp+1 2 (q α ) +CβJp−1 2 (q α )] =0 (2.35) Equation (2.35) will be solved numerically. 3. Numerical results Equations (2.29) and (2.35) will be solved numerically applying the bisection method. For this aim, the following notations are introduced ψ = c2 c20 Ĉ = Cβ µ0 (3.1) Case 1 For p ¬ 1 from (2.28) and (3.1), it follows that J−1−p 2 (k α √ ψ−1(1+αH) )[ k √ ψ−1Y−1−p 2 (k α √ ψ−1 ) − ĈY1−p 2 (k α √ ψ−1 )] −Y−1−p 2 (k α √ ψ−1(1+αH) )[ k √ ψ−1J−1−p 2 (k α √ ψ−1 ) − ĈJ1−p 2 (k α √ ψ−1 )] =0 (3.2) Case 2 For p > 1 from (3.1) and (2.35), it follows that Jp+1 2 (k α √ ψ−1(1+αH) )[ k √ ψ−1Yp+1 2 (k α √ ψ−1 ) + ĈYp−1 2 (k α √ ψ−1 )] −Yp+1 2 (k α √ ψ−1(1+αH) )[ k √ ψ−1Jp+1 2 (k α √ ψ−1 ) + ĈJp−1 2 (k α √ ψ−1 )] =0 (3.3) The obtained numerical results for the dimensionless ratio ψ = c2/c20 are presented in the form of figures. Figure 2a presents the ratio ψ as a function of the parameter p for three cases of kH = 1, 2, 4, parameters η = 0.5, α = 0.05 and ratios µ1/µ2 = 4, µ1/µ0 = 2, ρ1/ρ0 = 2. For α =0.05 and a small values of H being the thickness of the FGM layer it follows form equation 1132 S. Kowalczyk et al. Fig. 2. The distribution of the parameter ψ = c2/c2 0 as a function of the parameter p for kH =1, 2, 4, η =0.5, µ1/µ2 =4, µ1/µ0 =2, ρ1/ρ2 =2, ρ1/ρ0 =2; (a) α =0.05, (b) α =0.5 (2.1) that the coating layer is almost homogeneous for all values of p. For this reason, the values of the ratio ψ are almost constant. A different case is presented in Fig. 2b, where the same values of the parameters as in Fig. 2a are taken into account without the parameter α =0.5. A weak influence of the nonhomogeneity of the coating layer on the wave speed ψ = c2/c20 can be noticed. Figure 3apresents thedistributionsofψ as functionsofkH forη =0.5,µ1/µ0 =2,ρ1/ρ0 =2, α = 0.05, p = 0.5 and three cases of values of the ratios: 1 – µ1/µ2 = 4, 2 – µ1/µ2 = 6, 3 – µ1/µ2 =8. This figure shows that the influence of different features of the sub-layers being components of the considered foundation on the wave speed ψ = c2/c20 is rather small. The distributions of ψ as a function of the ratio µ1/µ2 = ρ1/ρ2 for four cases of values p =0, 0.5, 1, 2 and η =0.5, µ1/µ0 =2, ρ1/ρ0 =2, α =0.05, kH =1 are presented in Fig. 3b. The curve numbered by 1 (Fig. 3b) shows the dependence of the ratio ψ for p = 0, so it is the homogenous coating layer and the periodically layered foundation. It can be observed that values of ψ decrease together with an increase in the parameter p. Fig. 3. The distributions of ψ = c2/c2 0 : (a) as a function of kH for 1: µ1/µ2 =4, 2: µ1/µ2 =6, 3: µ1/µ2 =8; (b) as a function of µ1/µ2 Figure 4a shows the distributions of the ratio ψ as a function of µ1/µ0 for four cases of the ratio µ1/µ2 = 4, 6, 8, 10 and η = 0.5, α = 0.05, ρ1/ρ0 = ρ1/rho2 = 2, p = 0.5, kH = 1. The curve numbered by 1 presents the smallest values of ψ for all the considered nonhomogenities of the periodically layered foundation. On some problems of SH wave propagation in inhomogeneous elastic bodies 1133 The distributions of the ratio ψ as a function of the parameter η for α = 0.05, µ1/µ0 = ρ1/ρ0 = ρ1/ρ2 =2, p =0.5, kH =1 and for cases of the ratio µ1/µ2 =4, 6, 8, 10 are given in Fig. 4b. It can be observed that for η → 1 all curves numbered by 1, 2, 3 and 4 tend to the same point. Fig. 4. The distributions of ψ = c2/c2 0 : (a) as a function of µ1/µ0, (b) as a function of η The limit case η → 1 leads to the homogeneous foundationwith the shearmodulusµ1 =2µ0, coated by the FGM layer with the shear modulus and the mass density dependent in the form given by (2.1) with respect to the distance from its lower boundary plane. In the case η → 0, the half-space being the foundation with the shear modulus µ2 is obtained. The values of µ2 depend on the taken into account value of the ratio µ1/µ2. From the assumptions in Fig. 4b, it follows that the curves are adequate for the cases: curve 1 for µ0 = 2µ2, curve 2 for µ0 = 3µ2, curve 3 for µ0 =4µ2 and curve 4 for µ0 =5µ2, respectively. From Fig. 4b it can seen that the values of ψ decrease with an increase in the ratio µ1/µ2 for fixed values of the parameter η. 4. Final remarks Theproblemof SHwave propagation in an elastic nonhomogeneous half-space is considered.The body is assumed to be composed of the FGM layer being a coating and periodically stratified two-layer half-space. The investigations are limited to the anti-plane shear harmonic waves in the nonhomoeneous body on the assumption that the boundary surface is free of loadings. The main aim is to determine the wave speed by using the wave number and the mechanical properties of the components of the half-space. The numerical results present the wave speed in the dimensionless form. The obtained figures show the influence of the nonhomogeneity of the coating layer aswell as thenonhomogenity of the foundationon thewave speed.Theassumptions of p =0, µ1 = µ2 = µ0, ρ1 = ρ2 = ρ0 lead to Love’s wave propagation in the homogenous half- space coated by the homogeneous layer well-known in the literature (see for exampleAchenbach, 1973; Nowacki, 1970), which is shown in Appendix. A. Appendix Taking into account p =0 µ1 = µ2 ρ1 = ρ2 (A.1) 1134 S. Kowalczyk et al. and using equation (2.22), (2,23), (2.9) and (2.13), it follows that C = µ1 q = k √ c2 c20 −1 β = k √ 1− c 2 c21 c21 = µ1 ρ1 c0 < c < c1 (A.2) Substituting (A.1) and (A.2) into (2.29) and using following relations (Lebiediev, 1957) J1 2 (z)= √ 2 πz sinz J−1 2 (z)= √ 2 πz cosz Y1 2 (z)=−J−1 2 (z) Y−1 2 (z)= J1 2 (z) (A.3) we obtain µ0q [ sin q α cos ( q (1 α +H )) − cos q α sin ( q (1 α +H ))] +µ1β [ cos q α cos ( q (1 α +H )) +sin q α sin ( q (1 α +H ))] =0 (A.4) From equation (A.4), it follows that µ1β = µ0q tan(qH) (A.5) Equation (A.5) agrees with the characteristic equation for the case of Love’s wave presented in the monograph by Nowacki (1970) (p. 612, eq. (13)). References 1. Achenbach J.D., 1973, Wave Propagation in Elastic Solids, North-Holland Publ. Co., Amsterdam-London 2. Achenbach J.D., Balogun O., 2010, Anti-plane surface waves on a half-space with depth- dependent properties,Wave Motion, 47, 59-65 3. 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