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

 Vietnam Journal of Mechanics, VAST, Vol.41, No. 1 (2019), pp. 51 – 62
 DOI: https://doi.org/10.15625/0866-7136/12710
 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
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 ... lation 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 + µbbIn
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 + µbbIn
 Similarly, for a horizontal load of the form
 A −ikct
 fx = Qδ(z − z0)δ(x − x0)e , (46)
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 + µI
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 + µbbIn
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 = µb/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
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.ÐT2017-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). https://doi.org/10.1063/1.3300474.
 [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. https://doi.org/10.1016/j.compbiomed.2017.06.001.
 [3] P. H. F. Nicholson, P. Moilanen, T. Kärkkäinen, 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. https://doi.org/10.1088/0967-3334/23/4/313.
 [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. https://doi.org/10.1016/0021-9290(96)00054-1.
 A theoretical study on propagation of guided waves in a fluid layer overlying a solid half-space 61
 [5] J. J. Kaufman, G. Luo, and R. S. Siffert. Ultrasound simulation in bone. IEEE Transac-
 tions on Ultrasonics, Ferroelectrics, and Frequency Control, 55, (6), (2008), pp. 1205–1218.
 https://doi.org/10.1109/tuffc.2008.784.
 [6] A. H. Nayfeh. Wave propagation in layered anisotropic media: With application to composites. El-
 sevier, (1995).
 [7] J. Achenbach. Wave propagation in elastic solids. North-Holland Publishing Company, (2012).
 [8] J. L. Rose. Ultrasonic guided waves in solid media. Cambridge University Press, (2014).
 [9] W. M. Ewing, W. S. Jardetzky, and F. Press. Elastic waves in layered media. McGraw-Hill, (1957).
[10] T. Kundu. Ultrasonic nondestructive evaluation: engineering and biological material characteriza-
 tion. CRC Press, (2003).
[11] N. T. K. Linh, P. C. Vinh, and L. T. Hue. An approximate formula for the H/V ra-
 tio of Rayleigh waves in compressible pre-stressed elastic half-spaces coated with a thin
 layer. Vietnam Journal of Mechanics, 40, (1), (2018), pp. 63–78. https://doi.org/10.15625/0866-
 7136/10417.
[12] P. C. Vinh, V. T. N. Anh, and V. P. Thanh. Rayleigh waves in an isotropic elastic half-space
 coated by a thin isotropic elastic layer with smooth contact. Wave Motion, 51, (3), (2014),
 pp. 496–504. https://doi.org/10.1016/j.wavemoti.2013.11.008.
[13] P. C. Vinh and V. T. N. Anh. Rayleigh waves in an orthotropic half-space coated by a thin
 orthotropic layer with sliding contact. International Journal of Engineering Science, 75, (2014),
 pp. 154–164. https://doi.org/10.1016/j.ijengsci.2013.11.004.
[14] J. D. Achenbach and S. P. Keshava. Free waves in a plate supported by a
 semi-infinite continuum. Journal of Applied Mechanics, 34, (2), (1967), pp. 397–404.
 https://doi.org/10.1115/1.3607696.
[15] H. F. Tiersten. Elastic surface waves guided by thin films. Journal of Applied Physics, 40, (2),
 (1969), pp. 770–789. https://doi.org/10.1063/1.1657463.
[16] T. T. Tuan and T. N. Trung. The dispersion of Rayleigh waves in orthotropic layered
 half-space using matrix method. Vietnam Journal of Mechanics, 38, (1), (2016), pp. 27–38.
 https://doi.org/10.15625/0866-7136/38/1/6191.
[17] C. L. Yapura and V. K. Kinra. Guided waves in a fluid-solid bilayer. Wave Motion, 21, (1),
 (1995), pp. 35–46. https://doi.org/10.1016/0165-2125(94)00043-5.
[18] H. Phan, Y. Cho, and J. D. Achenbach. Verification of surface wave solutions
 obtained by the reciprocity theorem. Ultrasonics, 54, (7), (2014), pp. 1891–1894.
 https://doi.org/10.1016/j.ultras.2014.05.003.
[19] H. Phan, Y. Cho, and J. D. Achenbach. Validity of the reciprocity approach for
 determination of surface wave motion. Ultrasonics, 53, (3), (2013), pp. 665–671.
 https://doi.org/10.1016/j.ultras.2012.09.007.
[20] H. Phan, Y. Cho, and J. D. Achenbach. A theoretical study on scattering of surface waves by
 a cavity using the reciprocity theorem. In Nondestructive Testing of Materials and Structures,
 Springer Netherlands, (2013), pp. 739–744.
[21] H. Phan, Y. Cho, T. Ju, and J. D. Achenbach. Multiple scattering of surface waves by cav-
 ities in a half-space. In AIP Conference Proceedings, AIP, AIP, Vol. 1581, (2014), pp. 537–541.
 https://doi.org/10.1063/1.4864866.
[22] O. Balogun and J. D. Achenbach. Surface waves generated by a line load on a half-
 space with depth-dependent properties. Wave Motion, 50, (7), (2013), pp. 1063–1072.
 https://doi.org/10.1016/j.wavemoti.2013.03.001.
62 Phuong-Thuy Nguyen, Haidang Phan
[23] S. S. Kulkarni and J. D. Achenbach. Application of the reciprocity theorem to determine line-
 load-generated surface waves on an inhomogeneous transversely isotropic half-space. Wave
 Motion, 45, (3), (2008), pp. 350–360. https://doi.org/10.1016/j.wavemoti.2007.07.001.
[24] J. D. Achenbach. Reciprocity in elastodynamics. Cambridge University Press, (2003).
[25] J. D. Achenbach, A. K. Gautesen, and H. McMaken. Ray methods for waves in elastic solids: with
 applications to scattering by cracks. Pitman Advanced Publishing Program, (1982).
[26] A. T. de Hoop. Handbook of radiation and scattering of waves: Acoustic waves in fluids, elastic
 waves in solids, electromagnetic waves. Academic Press, (1995).
[27] L. R. F. Rose, W. K. Chiu, N. Nadarajah, and B. S. Vien. Using reciprocity to derive the far
 field displacements due to buried sources and scatterers. The Journal of the Acoustical Society
 of America, 142, (5), (2017), pp. 2979–2987. https://doi.org/10.1121/1.5009666.
[28] H. Phan, T. Q. Bui, H. T. L. Nguyen, and C. V. Pham. Computation of inter-
 face wave motions by reciprocity considerations. Wave Motion, 79, (2018), pp. 10–22.
 https://doi.org/10.1016/j.wavemoti.2018.02.008.
[29] H. Phan, Y. Cho, Q. H. Le, P. C. Vinh, H. T. L. Nguyen, P. T. Nguyen, and T. Q. Bui.
 A closed-form solution to propagation of guided waves in a layered half-space under
 a time-harmonic load: An application of elastodynamic reciprocity. Ultrasonics, (2019).
 https://doi.org/10.1016/j.ultras.2019.03.015.
[30] H. Phan, Y. Cho, and J. D. Achenbach. Application of the reciprocity theorem to scattering
 of surface waves by a cavity. International Journal of Solids and Structures, 50, (24), (2013),
 pp. 4080–4088. https://doi.org/10.1016/j.ijsolstr.2013.08.020.
[31] H. Phan, Y. Cho, and W. Li. A theoretical approach to multiple scattering of sur-
 face waves by shallow cavities in a half-space. Ultrasonics, 88, (2018), pp. 16–25.
 https://doi.org/10.1016/j.ultras.2018.02.018.
[32] H. Phan, Y. Cho, T. Ju, J. D. Achenbach, S. Krishnaswamy, and B. Strom. A novel approach of
 using the elastodynamic reciprocity for guided wave problems. In AIP Conference Proceedings,
 AIP, AIP, Vol. 1511, (2013), pp. 107–112. https://doi.org/10.1063/1.4789037.
[33] W. Liu, Y. Cho, H. Phan, and J. D. Achenbach. Study on the scattering of 2-D Rayleigh waves
 by a cavity based on BEM simulation. Journal of Mechanical Science and Technology, 25, (3),
 (2011), pp. 797–802. https://doi.org/10.1007/s12206-011-0133-5.
[34] S. Hao, B. W. Strom, G. Gordon, S. Krishnaswamy, and J. D. Achenbach. Scattering of the
 lowest Lamb wave modes by a corrosion pit. Research in Nondestructive Evaluation, 22, (4),
 (2011), pp. 208–230. https://doi.org/10.1117/12.880534.