A theoretical study on propagation of guided waves in a fluid layer overlying a solid half-space

Nội dung text: A theoretical study on propagation of guided waves in a fluid layer overlying a solid half-space

1. Vietnam Journal of Mechanics, VAST, Vol.41, No. 1 (2019), pp. 51 – 62 DOI: A THEORETICAL STUDY ON PROPAGATION OF GUIDED WAVES IN A FLUID LAYER OVERLYING A SOLID HALF-SPACE Phuong-Thuy Nguyen1, Haidang Phan2,3,∗ 1Institute of Physics, VAST, Hanoi, Vietnam 2Graduate University of Science and Technology, VAST, Hanoi, Vietnam 3Institute of Mechanics, VAST, Hanoi, Vietnam ∗E-mail: haidangphan.vn@gmail.com Received: 02 July 2018 / Published online: 29 October 2018 Abstract. Ultrasonic guided waves propagating in a non-viscous fluid layer of uniform thickness bonded to an elastic solid half-space is theoretically investigated in this article. Based on the boundary conditions set for the joined configuration, a characteristic disper- sion equation is found and new expressions for free guided waves are introduced. Closed- form solutions of guided waves generated by a time-harmonic load are derived by the use of elastodynamics reciprocity theorems. Through calculation examples, it is shown that the obtained computation of the lowest wave mode approaches the result of the Rayleigh wave in the solid half-space as the layer thickness approaches zero. The aim of the present work is to improve the understanding of wave motions in layered half-spaces for potential applications in the area of bone quantitative ultrasound. Keywords: layered half-space; guided waves; reciprocity theorem; quantitative ultrasound. 1. INTRODUCTION Quantitative ultrasound (QUS) has shown a great potential in the assessment of bone characteristics in the recent research. Compared to X-ray method, QUS is more sensitive in the determinants of bone strength, non-ionizing and able to give some information about the elastic properties and defects of bones [1–3]. Various studies have been carried out to understand ultrasound interaction with the bone structure. Lowet and Van der Perre [4] studied the simulation of ultrasound wave propagation and the method to mea- sure velocity in long bones. Numerical simulations of wave propagation and experiment measurement were used to gain insights into the expected behaviour of guided waves in bone [3]. Simulation results have made significant steps to improve our understanding of ultrasound interaction with bone [5]. Our knowledge of wave interaction with bone is, however, still far from complete because of the lack of analytic solutions. c 2019 Vietnam Academy of Science and Technology
2. 52 Phuong-Thuy Nguyen, Haidang Phan Bones are normally composed of layers of different materials including cortical bone, cancellous bone and marrow. Propagation of guided waves in bone is largely influenced by the presence of overlying soft-tissue layer which is usually mimicked by a fluid layer of finite thickness. When the soft-tissue is relatively thin compared with the cortical bone, a fluid-solid layered half-space model can be used to study wave propagation in bone structures. In the current investigation, the soft tissue layer is considered as a non- viscous fluid while the cortical bone is assumed to be an isotropic solid half-space. This work aims to expand our understanding of guided wave propagation in a fluid-solid lay- ered half-space to explore the potential of using ultrasound-based methods for long-bone characterization. Wave motion in layered structures is indisputably one of the most fundamental prob- lems of elastodynamics that has been widely considered for applications in geophysics, acoustics, and medicines. Theory of free ultrasonic waves propagating in layered struc- tures can be found, for example, in the textbooks [6–11]. This classical topic is also ad- dressed in a large number of research articles available in the literature. Approximate for- mula for guided wave velocity in an elastic half-space coated by a thin elastic layer with a smooth contact was considered in [12]. In a similar manner, approximate secular equa- tions of the waves in an orthotropic half-space coated by a thin orthotropic layer with sliding contact were also derived and reported in [13] by the same authors. Achenbach and Keshava [14] analyzed dispersion curves for free waves in a layered half-space while Tiersten [15] investigated the influence of thin film on the propagation of guided waves in the film-halfspace structure with comparison to experiment data. Matrix method is used to investigate the dispersion of Rayleigh waves in orthotropic layered half-space [16]. Dispersion equations for a fluid-solid bilayered plate were derived and a discussion on the shapes of the wave modes was addressed in [17]. Wave motion generated by a loading is conventionally solved by using integral transform techniques [7]. The integral transform approach, however, becomes more diffi- cult for anisotropic solids, and impossible for inhomogeneous solids, for example, solids whose elastic moduli depend on the depth coordinate, as in geophysical applications and functionally graded materials. In order to avoid these difficulties, another method has been proposed in recent years, based on the elastodynamic reciprocity theorem, strictly to determine the guided waves. Compared to the integral transform, the reciprocity ap- proach is simpler [18–21] and more general that is able to use for anisotropic and inho- mogeneous materials [22, 23]. Generally, reciprocity theorem is a relation between displacements, tractions and body forces for two different loading states of the same body. One of the states is re- ferred as the actual state, guided waves radiated from a time-harmonic load and the other is called the virtual state, an appropriately chosen free wave traveling in the struc- ture. Statements of elastodynamic reciprocity theorems have already presented, and curious readers can refer to, e.g., [24–26]. Reciprocity relations have been successfully used in direct applications to calculate wave motions generated by a time-harmonic load, see [18, 19, 22, 23, 27, 28]. The material to be studied may be inhomogeneous, anisotropic or viscoelastic. Balogun and Achenbach [22] examined surface waves generated by a line load on a half-space with depth-dependent properties. The applications of reciprocity
3. A theoretical study on propagation of guided waves in a fluid layer overlying a solid half-space 53 to surface waves on an inhomogeneous transversely isotropic half-space was discussed in [23]. Recently, Phan et al. [29] considered the computation of guided waves in structure of a solid layer joined to a solid half-space. The reciprocity approach was also applied to study scattering of surface waves by cavities on the surface of a half-space [30–33] and scattering of Lamb waves by a partial spherical corrosion pit in a plate [34]. In this article, we first find the characteristic dispersion equation and propose new explicit expressions for free guided waves. The expressions are essential to obtain closed- form solutions of wave fields generated by a time-harmonic load in the fluid-solid lay- ered half-space by reciprocity consideration. The next step is choosing an appropriate virtual state which is a single guided wave mode propagating in the joined structure. The two loading states are substituted into a reciprocity relation for a two-material body. The relation is largely simplified due to the characteristics of guided waves in the fluid layer overlying the solid half-space. After some manipulation, exact solutions of the guided waves due to the time-harmonic load are derived. The examples of calculation show that the obtained result of the lowest wave mode approaches the computation of the Rayleigh wave in the solid half-space as the layer thickness approaches zero. 2. FREE GUIDED WAVES IN FLUID LAYER BONDED TO SOLID HALF-SPACE Consider a fluid layer Ω of uniform thickness h and a solid half-space Ωb which are bonded together along the plane z = 0. The layered half-space relative to the Cartesian coordinate system (x, z) is shown in Fig.1. Free guided waves propagating in the layered half-space are discussed in this section. For a homogeneous isotropic elastic solid, the governing equations are the displacement equations of motion [7]   àbui,jj + bλ + àb uj,ji = ρbuăi , (1) where bλ, àb are the Lame constants and ρb is the mass density. For a non-viscous fluid, which does not sustain shear stresses, the equations of wave motion given in Eq. (1) can Instructions for Authors 3 be used by assuming àb = 0. Fig. 1. Coordinate system for fluid layer joined to solid half-space Fig. 1. Coordinate system for fluid layer joined to solid half-space where ˆ, ˆ are the Lame constants and ˆ is the mass density. For a non-viscous fluid, which does not sustain shear stresses, the equations of wave motion given in Eq. (1) can be used by assuming ˆ = 0 . The trial solutions in the layered half-space can be expressed as a combination of partial waves based on the partial wave theory discussed in detail, for example, in [8]. In this case, there are two partial waves in the fluid layer and another two in the solid half-space. Besides the amplitudes, guided waves in this joined structure are defined by an angular frequency  and a wavenumber k , where kc=  / , c being the phase velocity, as well as material properties  , of  and ˆ,, ˆ ˆ of ˆ . For the fluid layer, displacement components may be written as ik z−− ik z ik() x ct ux =+( Ae12 A e) e (2) ik z−− ik z ik() x ct uz =− ( Ae12 A e) e (3) and for the half-space, they are of the form ˆˆ ˆ ˆˆk 12 z k z ik() x− ct ux =+( A12 e A e) e (4) 1 ˆˆ ˆˆk 12 z k z ik() x− ct uˆz = − i A1 e + ˆ 2 A 2 e e (5) ˆ1 ˆ where A j and Ajj (= 1,2) are constants to be determined. In Eqs. (2) – (5), 22 = −1/ + ccL (6) 2 2 2 2 ˆˆ12=1 −c / cˆˆTL , = 1 − c / c (7) ˆ where cL =  / is longitudinal wave velocity of  while cˆˆT =  / ˆ and cˆˆL =+( 2  ) / ˆ are ˆ the transverse and longitudinal wave velocities, respectively, of  . In Eqs. (6) - (7), ,, ˆˆ12 are dimensionless quantities and they are generally complex. It is important to note that guided waves in the fluid-solid layered half-space may not have a real solution for phase velocity. Therefore, they may not exist for some material combination. A detailed study of the conditions of the material properties for the existence of guided waves is, however, beyond the scope of the current work.
4. 54 Phuong-Thuy Nguyen, Haidang Phan The trial solutions in the layered half-space can be expressed as a combination of partial waves based on the partial wave theory discussed in detail, for example, in [8]. In this case, there are two partial waves in the fluid layer and another two in the solid half-space. Besides the amplitudes, guided waves in this joined structure are defined by an angular frequency ω and a wavenumber k, where k = ω/c, c being the phase velocity, as well as material properties λ, ρ of Ω and bλ, àb, ρb of Ωb . For the fluid layer, displacement components may be written as  ikαz −ikαz ik(x−ct) ux = A1e + A2e e , (2)  ikαz −ikαz ik(x−ct) uz = α A1e − A2e e , (3) and for the half-space, they are of the form   kα1z kα2z ik(x−ct) ubx = Ab1e b + Ab2e b e , (4)   1 kα1z kα2z ik(x−ct) ubz = −i Ab1e b + bα2 Ab2e b e , (5) bα1 where Aj and Abj (j = 1, 2) are constants to be determined. In Eqs. (2)–(5), q 2 2 α = −1 + c /cL , (6) q q 2 2 2 2 bα1 = 1 − c /cbT , bα2 = 1 − c /cbL , (7) p q where cL = λ/ρ is longitudinal wave velocity of Ω while cbT = àb/ρb and cbL = q (bλ + 2àb)/ρb are the transverse and longitudinal wave velocities, respectively, of Ωb . In Eqs. (6)–(7), α, bα1, bα2 are dimensionless quantities and they are generally complex. It is important to note that guided waves in the fluid-solid layered half-space may not have a real solution for phase velocity. Therefore, they may not exist for some material combi- nation. A detailed study of the conditions of the material properties for the existence of guided waves is, however, beyond the scope of the current work. From Eqs. (2)–(5), stress components τxx, τzz of the layer Ω and τbxx, τbxz, τbzz of the half-space Ωb can be easily calculated by the use of Hooke’s law. For guided waves in the layered half-space, there are one free boundary condition at the free surface (z = h) and three conditions at the interface (z = 0) τzz = 0 at z = h, (8) uz = ubz , τbxz = 0, τzz = τbzz at z = 0. (9) Eqs. (8) and (9) result in  eikαh e−ikαh 0 0      A1 0 α −α i/α iα  b1 b2   A2   0    = . (10)  0 0 bα1 + 1/bα1 2bα2         Ab1   0  2 2 àb 2
   àb 1 + α 1 + α 2 α + 1 Ab2 0 λ b1 λ
5. 4 The Author’s names From Eqs. (2) – (5), stress components xx, zz of the layer  and ˆxx,,  ˆ xz  ˆ zz of the half-space ˆ can be easily calculated by the use of Hooke’s law. For guided waves in the layered half-space, there are one free boundary condition at the free surface ()zh= and three conditions at the interface (z = 0)  zz = 0 at zh= (8) uuz=ˆ z, ˆ xz = 0,  zz =  ˆ zz at z = 0 (9) Equations (8) and (9) result in ik h− ik h ee 00 A 1 0 ˆˆ − ii/ 12 A 2 0 = (10) A theoretical study0 on propagation 0 ˆ of1 guided+ 1/ ˆ waves 1 in a 2 fluid ˆ 2 layer overlyingˆ a solid half-space 55 A1 0 ˆˆ 2 2 2 ˆ 0 1+ 1 + 2( ˆ1 + 1) A In order to have nontrivial solutions, the determinant of2 the four-by-four matrix in Eq. (10) must be zero. It leads to which is referred to as the characteristic dispersion In order to have nontrivial solutions, the determinant of the four-by-four matrix in Eq. (10) must be zero. equation It leads to which is referred to as the characteristic dispersion equation 1 2
    2
   2  àb + α 1 − bα1 bα2 tan kαh + bα1 + 1 − 4bα1bα2 = 0. (11) α 1 222 ˆ λ + (1 − ˆ1) ˆ 2 tankh +( ˆ 1 + 1) − 4 ˆ 1 ˆ 2 = 0 (11) As the thickness  of the layer approaches zero( in the limit, i.e.) tan kαh = 0, Eq. (11) becomes the famous equation of Rayleigh surface waves in a half-space. Unlike Rayleigh waves,As the guided thickness waves of the in layer the layered approaches half-space zero in arethe dispersivelimit, i.e. tan becausekh = there 0 , Eq. is ( a11 frequency) becomes the famousterm equation via k appearing of Rayleigh in Eq.surface (10 ).waves It also in meansa half-space. that there Unlike is anRayleigh infinite waves, number guided of wave waves in the modeslayered propagatinghalf-space are in dispersive the structure because of a there layer is joined a frequency to a half-space. term via k appearing in Eq. (10). It also means that there is an infinite number of wave modes propagating in the structure of a layer joined to a half-space. Table 1. Material properties of water and aluminum Table 1. Material properties of water and aluminum Material ρ (kg/cm3) λ (GPa) à (GPa) Water 1000 2.25 0 Aluminum 2700 55.25 25.94 Fig.Fig. 2 2 DispersionDispersion curves curves for for a awater water layer layer of of1 thickness thickness and and an an aluminum aluminum half-space half-space As an example of calculation, the dispersion curves of a water layer and an alu- minum half-space with the material properties tabulated in Tab.1 are shown in Fig.2. It can be seen that the phase velocity values are confined over a certain range. The upper bound of the phase velocity value is the shear wave velocity in the aluminum half-space while its lower limit is the longitudinal wave velocity in the water. The wave velocity approaches the Rayleigh surface wave in aluminum at the low frequency limit where the layer thickness is much smaller than the wavelength. When the phase velocity is larger
6. 56 Phuong-Thuy Nguyen, Haidang Phan than the shear wave velocity of the aluminum half-space, wave energy will leak into the half-space. The guided wave mode with complex phase velocity attenuates and is not considered here as we are only interested initially in non-leaky wave modes. Since the determinant of the matrix in Eq. (10) is zero, it has actually three indepen- dent equations with four unknowns. The process of solving Eq. (10) is straightforward but quite tedious. Therefore, we propose expressions for free guided waves in the lay- ered half-space without a detailed proof. The displacements and stress components of the layer are ik(x−ct) ux = AUx(z)e , (12) ik(x−ct) uz = AUz(z)e , (13) ik(x−ct) τxx = ikλATxx(z)e , (14) where ikαz −ikαz Ux(z) = d1e + d2e , (15)  ikαz −ikαz Uz(z) = α d1e − d2e , (16) 2
    ikαz −ikαz Txx(z) = 1 + α d1e + d2e . (17) For the half-space ik(x−ct) ubx = AUbx(z)e , (18) ik(x−ct) ubz = −iAUbz(z)e , (19) ik(x−ct) τbxx = ikàbATbxx(z)e , (20) ik(x−ct) τbxz = kàbATbxz(z)e , (21) where kα1z kα2z Ubx(z) = db1e b + db2e b , (22) 1 kα1z kα2z Ubz(z) = db1e b + bα2db2e b , (23) bα1 kbα1z 2 2
   kbα2z Tbxx(z) = 2db1e + 2bα2 − bα1 + 1 db2e , (24)   1 kα1z kα2z Tbxz(z) = bα1 + db1e b + 2bα2db2e b . (25) bα1 In Eqs. (15)–(17) and Eqs. (22)–(25), d1, d2, db1, db2 are dimensionless quantities defined as 2
   d1 = 1 − bα1 bα2 , (26) 2 2
   d2 = β bα1 − 1 bα2 , (27) 2
   db1 = 2iα 1 + β bα1bα2 , (28) 2
   2
   db2 = −iα 1 + β 1 + bα1 , (29) where β = eikαh. In Eqs. (12)–(14) and Eqs. (18)–(21), there is only one unknown con- stant A. The explicit expressions of guided waves are essential to direct application of
7. A theoretical study on propagation of guided waves in a fluid layer overlying a solid half-space 57 reciprocity to obtain closed-form solutions of wave fields generated by a time-harmonic load in the next section. 3. COMPUTATION OF GUIDED WAVES DUE TO TIME-HARMONIC LOADING In this section, a reciprocity theorem is applied to obtain the amplitudes of guided waves due to a time-harmonic line load. We first consider a vertical load applied at (x0, z0) where x0, z0 are the x-coordinate and the z-coordinate, respectively, of the point of application. The load is of the form A −ikct fz = Pδ(z − z0)δ(x − x0)e . (30) The load will generate guided waves along the layered half-space in both the positive P+ x-direction and the negative x-direction with unknown relative scattered amplitudes Am P− and Am , respectively. Here, m = 0, 1, . . . , ∞ indicate wave mode. This is the actual state A whose amplitudes are to be determined by the use of reciprocity consideration. The expansions for the far-field displacements of state A in the positive direction may be written as ∞ ∞ m P+ m ikm(x−cmt) ux = ∑ ux = ∑ Am Ux (z)e , (31) m=0 m=0 ∞ ∞ m P+ m ikm(x−cmt) uz = ∑ uz = ∑ Am Uz (z)e , (32) m=0 m=0 ∞ ∞ m P+ m ikm(x−cmt) ubx = ∑ ubx = ∑ Am Ubx (z)e , (33) m=0 m=0 ∞ ∞ m P+ m ikm(x−cmt) ubz = ∑ ubz = −i ∑ Am Ubz (z)e . (34) m=0 m=0 Reciprocity theorem offers a relation between displacements, tractions and body forces of two different loading states. Based on the reciprocity relation of the two states, the scattered amplitudes of guided waves of the actual state are derived. The ideal was introduced in [24] for a half-space and a plate body and recently developed for layered structures [28, 29]. For a two-material body, the reciprocity follows from Eq. (38) of [28] Z Z  A B B A  A B B A fj uj − fj uj dΩ+ fbj ubj − fbj ubj dΩb = Ω Ωb Z Z (35)  B A A B  B A A B τij uj − τij uj nidS + τbij ubj − τbij ubj nbidSb, S Sb where S and Sbdefines contours around Ω and Ωb without the interface, respectively, while ni and nbi are normal vectors along S and Sb, respectively. Superscripts A and B denote two A elastodynamic states. State A, the actual state, is the field generated by fz while state B, the virtual state, is the field of a free guided wave in the layered half-space. The first step is choosing a virtual state, i.e., state B based on the explicit expressions of free guided waves given in Eqs. (12)–(14) and Eqs. (18)–(21). State B is set to include
8. 58 Phuong-Thuy Nguyen, Haidang Phan only a single wave mode represented by amplitude Bn. If state B is chosen in the negative x-direction, it is of the form n n −ikn(x+cnt) ux = −BnUx (z)e , (36) n n −ikn(x+cnt) uz = BnUz (z)e , (37) n n −ikn(x+cnt) ubx = −BnUbx (z)e , (38) n n −ikn(x+cnt) ubz = −iBnUbz (z)e . (39) We then replace the expressions of states A and B into the reciprocity relation given in Eq. (35). The left-hand side of Eq. (35) can be simplified since the loading is applied only at (x0, z0). If state Aand state B propagate in the same direction, the right-hand side of Eq. (35) vanishes. Thus, there is only contribution from the counter-propagating waves, see [24, 28] for detail. It can be easily seen that there is no contribution of the integration along to the top surface of the layered half-space because a free boundary condition is applied and along the line at z → ∞ since the waves vanish. Moreover, using the orthogonality condition in Eq. (9.4.23) of [24], the right-hand side of Eq. (35) cancels out for m 6= n. Note that the time-harmonic loading can be anywhere in the joined structure. Without loss of generality, the load is applied in the half-space Ωb . After some manipulation, we finally obtain the amplitude of guided waves in the positive x- direction n −ikn x0 P+ −iPUbz (z0)e An =   , (40) 2 λIn + àbbIn where Z h n n In = ikn [Txx(z)Ux (z)] dz, (41) 0 Z 0 h n n n n i bIn = ikn Tbxx(z)Ubx (z) + Tbxz(z)Ubz (z) dz. (42) −∞ The integrals in Eqs. (41) and (42) can be calculated as 1 + α2 h    i I = e2ikαh − 1 d2 − e−2ikαh − 1 d2 + 4ikαhd d , (43) n 2α 1 2 1 2 3α2 + 1 2α α − α2 + 3 4α2 − α2 + 1  = b1 2 + b1b2 b1 + b2 b1 2 bIn i 3 db1 db1db2 db2 . (44) 2bα1 bα1 2bα2 Note that In, bIn are connected to the guided wave of mode n. Therefore, k, α, bα1, bα2, d1, d2, db1, db2 are the quantities of mode n although we have ignored subscript n in the expres- sions of Eqs. (43) and (44). If a virtual wave of mode n in the positive x-direction is chosen, we find n ikn x0 P− −iPUz (z0)e An =   . (45) 2 λIn + àbbIn Similarly, for a horizontal load of the form A −ikct fx = Qδ(z − z0)δ(x − x0)e , (46)
9. 8 The Author’s names Similarly, for a horizontal load of the form A− ikct fx = Q()() z − z00 x − x e (46) we find ˆ n −ikn x0 Q+ −QUx () z0 e An = (47) ˆ ˆ 2(IInn+ ) ˆ n ikn x0 Q− QUx () z0 e An = (48) ˆ ˆ 2(IInn+ ) 4. RESULTS This section presents calculation of phase velocity and displacement amplitudes of guided waves due to time-harmonicA theoretical loading. study Consider on propagation a ofwater guided waveslayer in and a fluid an layer aluminum overlying a solid half half-space-space whose material 59 properties are given in Tab. 1. It is discussed in Section 2 that there is only the lowest wave mode propagatingwe find in the layered half-space as the thickness of the water layer is much smaller than the n −ikn x0 Q+ −QUbx (z0)e wavelength. As the layer thickness approachesAn = zero in the limit, , the phase velocity c approaches (47) the 2 λI + àI velocity of Rayleigh surface wave cˆR in the half-spacen, seeb bFig.n 2. n ikn x0 We are now interested in showing thatQ− the displacementQUb (z0)e amplitudes of the lowest wave mode will A = x . (48) approach the amplitudes of Rayleigh wavesn as the thickness of the layer h goes to zero in the limit for 2 λIn + àbbIn a fixed finite values of frequency f . The frequency is chosen as f=1 MHz and the thickness of the layer varies from h = 0 to h= 0.5 mm . The magnitude4. RESULTS of both vertical and horizontal loads is chosen as P Q PQ==ˆ /2This. The section loads presents generatecalculation the lowest of wave phase modes velocity with and scattered displacement amplitudes amplitudes A0 and of A0 , respectively.guided These waves amplitudes due to time-harmonic are compared with loading. the ones Consider of surface a water waves layer in an and aluminum an aluminum half-space, P half-spaceQ whose material properties are given in Tab.1. It is discussed in Section 2 AR and AR , obtained by Phan et al. [17]. The amplitude ratios of the lowest mode to the Rayleigh that there is only the lowest wave mode propagating in the layered half-space as the PP P QQ wave, AAthickness0 / R due of to the vertical water layerload is much and AA smaller0 / R thandue to the horizontal wavelength. load As Q the, are layer displayed thickness in Fig. 3. PP QQ Clearly,approaches AA0 /1R → zero and in AA0 the/1 limit,R → the as the phase thickness velocity of ctheapproaches layer approaches the velocity to zero. of RayleighThis shows the validationsurface of the wave reciprocitycbR in the approach half-space, discussed see Fig. in2 .the current study. Fig.Fig. 3. 3. Amplitude Amplitude ratio ofof lowest lowest mode mode to to Rayleigh Rayleigh wave wave mode mode due due to time-harmonic to time-harmonic loading loading We are now interested in showing that the displacement amplitudes of the lowest wave mode will approach the amplitudes of Rayleigh waves as the thickness of the layer h goes to zero in the limit for a fixed finite values of frequency f . The frequency is chosen as f = 1 MHz and the thickness of the layer varies from h = 0 to h = 0.5 mm. The magnitude of both vertical and horizontal loads is chosen as P = Q = àb/2. The loads P Q generate the lowest wave modes with scattered amplitudes A0 and A0 , respectively. These amplitudes are compared with the ones of surface waves in an aluminum half- P Q space, AR and AR , obtained by Phan et al. [19]. The amplitude ratios of the lowest mode P P Q Q to the Rayleigh wave, A0 /AR due to vertical load P and A0 /AR due to horizontal load
10. 60 Phuong-Thuy Nguyen, Haidang Phan P P Q Q Q, are displayed in Fig.3. Clearly, A0 /AR → 1 and A0 /AR → 1 as the thickness of the layer approaches to zero. This shows the validation of the reciprocity approach discussed in the current study. 5. CONCLUSION We have proposed a theoretical approach for ultrasonic guided waves propagating in a fluid layer overlying a solid half-space. Based on the boundary conditions, a char- acteristic dispersion equation has been found and explicit expressions for free guided waves in the structure have been obtained. One of the main contributions of the present work is the derivation of exact solutions of wave fields generated by a time-harmonic load in the fluid-solid layered half-space. It has been shown in calculation examples that as the layer thickness goes to zero, the computation of the lowest wave mode approaches the result of the Rayleigh surface wave. The theoretical solutions obtained in the present research will be useful to build mod- els for a cortical bone with overlying soft tissue as the cortical bone plate is relatively thick compared with the soft-tissue layer. The models allow us to discover the relation among transducer characteristics, frequencies, and geometry and material properties of bone tissues. They will definitely deliver a fast and computationally inexpensive calculation of generation, propagation, reflection, refraction, transmission and absorption as ultra- sound interacts with bone tissue. The analytical simulation could also provide a better understanding of the experiment signals, improve the data interpretation and acquisi- tion. The work will ultimately benefit physicians and scientists in developing ultrasonic methods for diagnosis and treatment of bone diseases and monitoring of bone healing after surgery. ACKNOWLEDGMENTS This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2016.23; and Graduate Univer- sity of Science and Technology under grant number GUST.STS.éT2017-CH01. REFERENCES [1] L. H. Le, Y. J. Gu, Y. Li, and C. Zhang. Probing long bones with ultrasonic body waves. Applied Physics Letters, 96, (11), (2010). [2] V. H. Nguyen, T. N. H. T. Tran, M. D. Sacchi, S. Naili, and L. H. Le. Computing disper- sion curves of elastic/viscoelastic transversely-isotropic bone plates coupled with soft tissue and marrow using semi-analytical finite element (SAFE) method. Computers in Biology and Medicine, 87, (2017), pp. 371–381. [3] P. H. F. Nicholson, P. Moilanen, T. Kọrkkọinen, J. Timonen, and S. Cheng. Guided ultrasonic waves in long bones: modelling, experiment and in vivo application. Physiological Measure- ment, 23, (4), (2002), pp. 755–768. [4] G. Lowet and G. Van der Perre. Ultrasound velocity measurement in long bones: measure- ment method and simulation of ultrasound wave propagation. Journal of Biomechanics, 29, (10), (1996), pp. 1255–1262.
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