Theoretical model of guided waves in a bone-mimicking plate coupled with soft-tissue layers

Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan


Quantitative ultrasound has shown a significant promise in the assessment of bone characteristics in the recent reports. However, our understanding of wave interaction with bone tissues is still far from complete since the propagation of ultrasonic waves in bones is a very challenging topic due to their multilayer nature. The aim of the current study is to develop a theoretical model for guided waves in a bone-mimicking plate coupled with two soft-tissue layers. Here, the bone plate is modeled as an isotropic solid layer while the soft tissues are modeled as fluid layers. Based on the boundary conditions set for the three-layered structure, a characteristic equation is obtained which results in dispersion curves of the phase and group velocities. New expressions for free guided waves propagating in the trilayered plate are introduced. The amplitudes of wave modes generated by time-harmonic loads applied in the plate are theoretically computed by reciprocity consideration. As an example of calculation, the normalized amplitudes of the lowest wave modes are presented. The obtained results and equations discussed in this study could be, in general, useful for further applications in the area of bone quantitative ultrasound.


guided waves; bone plate; trilayered structures; reciprocity; quantitative ultrasound

Full Text:



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:/

V.-H. Nguyen and S. Naili. Ultrasonic wave propagation in viscoelastic cortical bone plate coupled with fluids: a spectral finite element study. Computer Methods in Biomechanics and Biomedical Engineering, 16, (9), (2013), pp. 963–974. https:/

P. Laugier and G. Haïat. Bone quantitative ultrasound. Springer, (2010).

P. Laugier. Quantitative ultrasound of bone: looking ahead. Joint Bone Spine, 73, (2), (2005), pp. 125–128. https:/

G. Lowet and G. Van der Perre. Ultrasound velocity measurement in long bones: measurement method and simulation of ultrasound wave propagation. Journal of Biomechanics, 29, (10), (1996), pp. 1255–1262. https:/

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 Measurement, 23, (4), (2002). https:/

V.-H. Nguyen, T. N. H. T. Tran, M. D. Sacchi, S. Naili, and L. H. Le. Computing dispersion 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:/

J. Achenbach. Wave propagation in elastic solids. North-Holland Publishing Company, (1973).

J. L. Rose. Ultrasonic guided waves in solid media. Cambridge University Press, (2014).

S. Banerjee and C. B. Pol. Theoretical modeling of guided wave propagation in a sandwich plate subjected to transient surface excitations. International Journal of Solids and Structures, 49, (23-24), (2012), pp. 3233–3241. https:/

A. Chattopadhyay, P. Singh, P. Kumar, and A. K. Singh. Study of Love-type wave propagation in an isotropic tri layers elastic medium overlying a semi-infinite elastic medium structure. Waves in Random and Complex Media, 28, (4), (2018), pp. 643–669. https:/

C. L. Yapura and V. K. Kinra. Guided waves in a fluid-solid bilayer. Wave Motion, 21, (1), (1995), pp. 35–46. https:/

E. Kausel, P. Malischewsky, and J. Barbosa. Osculations of spectral lines in a layered medium. Wave Motion, 56, (2015), pp. 22–42. https:/

E. Kausel. Fundamental solutions in elastodynamics: a compendium. Cambridge University Press, (2006).

R. K. N. D. Rajapakse and Y. Wang. Green’s functions for transversely isotropic elastic half space. Journal of Engineering Mechanics, 119, (9), (1993), pp. 1724–1746. https:/

W. M. Ewing, W. S. Zardetzky, and F. Press. Elastic waves in layered media. McGraw-Hill, (1957).

J. A. Achenbach and J. D. Achenbach. Reciprocity in elastodynamics. Cambridge University Press, (2003).

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:/

H. Phan, T. Q. Bui, H. T.-L. Nguyen, and C. V. Pham. Computation of interface wave motions by reciprocity considerations. Wave Motion, 79, (2018), pp. 10–22. https:/

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:/

P. T. Nguyen, H. Nguyen, D. Le, and H. Phan. A model for ultrasonic guided waves in a cortical bone plate coupled with a soft-tissue layer. In AIP Conference Proceedings, Vol. 2102, (2019). https:/

H. Phan, Y. Cho, C. V. Pham, H. Nguyen, and T. Q. Bui. A theoretical approach for guided waves in layered structures. In AIP Conference Proceedings, Vol. 2102, (2019). https:/

P.-T. Nguyen and H. Phan. A theoretical study on propagation of guided waves in a fluid layer overlying a solid half-space. Vietnam Journal of Mechanics, 41, (1), (2019), pp. 51–62. https:/

H. Phan, Y. Cho, Q. H. Le, C. V. Pham, 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, 96, (2019), pp. 40–47. https:/

H. Phan, Y. Cho, and W. Li. A theoretical approach to multiple scattering of surface waves by shallow cavities in a half-space. Ultrasonics, 88, (2018), pp. 16–25. https:/

J. Lee, V. Ngo, H. Phan, T. Nguyen, D. K. Dao, and Y. Cho. Scattering of surface waves by a three-dimensional cavity of arbitrary shape: analytical and experimental studies. Applied Sciences, 9, (24), (2019). https:/

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:/

D. K. Dao, V. Ngo, H. Phan, C. V. Pham, J. Lee, and T. Q. Bui. Rayleigh wave motions in an orthotropic half-space under time-harmonic loadings: A theoretical study. Applied Mathematical Modelling, 87, (2020), pp. 171–179. https:/

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:/



  • There are currently no refbacks.

Copyright (c) 2021 Vietnam Academy of Science and Technology


Editorial Office of Vietnam Journal of Mechanics

3rd Floor, A16 Building, 18B Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam
Tel: (+84) 24 3791 7103