Series Expanding of the Ultrasound Transmission Coefficient Through a Multilayered Structure
Abstract
To calculate the transmission coefficient of ultrasonic waves through a multi-layered medium, a new approach is proposed by expanding it into Debye’s series. Using this formalism, the transmission coefficient can be put in the form of resonance terms series. From this point of view, the relative amplitude of the transmitted wave can be considered as an infinite summation of terms taking into account all possible reflections and refractions on each interface. Our model is then used to investigate interaction between the ultrasonic plane wave and the N-plane-layer structure. Obviously, the resulting infinite summation has to be reduced to a finite one, according to some level of accuracy. The numerical estimation of the transmission coefficient using the exact expression (Eq. (1)) is then compared to the one of our method in the case of two or three plane-layer structure. The effect of the order of the finite summation on the calculated value of the transmission coefficient is, as well, studied. Finally, our proposed method may be used, with the decomposition into Gaussian beams of a pressure field created by a circular source, to draw a 3D image of the pressure field transmitted through a multilayered structure.Keywords:
multilayered structure, Debye’s series, resonance formalism, ultrasonic NDTReferences
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24. Maréchal P. et al. (2008), Modeling of a high frequency ultrasonic transducer using periodic structures, Ultrasonics, 48(2): 141–149, https://doi.org/10.1016/j.ultras.2007.11.007
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27. Messineo M.G., Rus G., Eliçabe G.E., Frontini G.L. (2016), Layered material characterization using ultrasonic transmission: An inverse estimation methodology, Ultrasonics, 65: 315–328, https://doi.org/10.1016/j.ultras.2015.09.010
28. Nayfeh A.H. (1991), The general problem of elastic wave propagation in multilayered anisotropic media, The Journal of the Acoustical Society of America, 89(4): 1521–1531, https://doi.org/10.1121/1.400988
29. Potel C., de Belleval J.-F. (1993), Propagation in an anisotropic periodically multilayered medium, The Journal of the Acoustical Society of America, 93(5): 2669–2677, https://doi.org/10.1121/1.405842
30. Rokhlin S.I., Wang Y.J. (1992), Equivalent boundary conditions for thin orthotropic layer between two solids: Reflection, refraction, and interface waves, The Journal of the Acoustical Society of America, 91(4): 1875–1887, https://doi.org/10.1121/1.403717
31. Scott W.R., Gordon P.F. (1977), Ultrasonic spectrum analysis for nondestructive testing of layered composite materials, The Journal of the Acoustical Society of America, 62(1): 108–116, https://doi.org/10.1121/1.381491
32. Shenand M., Cao W. (2000), Acoustic band gap formation in a periodic structure with multilayer unit cells, Journal of Physics D: Applied Physics, 33(10): 1150–1154, https://doi.org/10.1088/00223727/33/10/303
33. Solyanik F.I. (1977), Transmission of plane waves through a layered medium of anisotropic materials, Soviet Physics Acoustics-USSR, 23: 533–536.
34. Soucrati H., Chitnalah A., Aouzale N., Jakjoud H. (2018), Analytical model of three-dimensional ultrasonic beam interaction with an immersed plate, Archives of Acoustics, 43(4): 669–679, https://doi.org/10.24425/aoa.2018.125160
35. Storheim E., Lohne K.D., Hergum T. (2015), Transmission and reflection from a layered medium in water. Simulations and measurements, [in:] Proceedings of the 38th Scandinavian Symposium on Physical Acoustics, Norway.
36. Stovas A., Arntsen B. (2006), Vertical propagation of low-frequency waves in finely layered media, Geophysics, 71(3): T87–T94, https://doi.org/10.1190/1.2197488
37. Yang X., Zhang C., Wang C., Sun A., Ju B.-F., Shen Q. (2019), Simultaneous ultrasonic parameter estimation of a multi-layered material by the PSO-based least squares algorithm using the reflection spectrum, Ultrasonics, 91: 231–236, doi: 1016/j.ultras.2018.08.003.
2. Bakhtiari-Nejad M., Hajj M.R., Shahab S. (2020), Dynamics of acoustic impedance matching layers in contactless ultrasonic power transfer systems, Smart Materials and Structure, 29: 035037, https://doi.org/10.1088/1361-665X/ab6fe5
3. Chern E.J., Nielsen H.T.C. (1989), Generalized formulas for reflected pulse response of multilayered structures, Journal of Applied Physics, 66(7): 2833–2837, https://doi.org/10.1063/1.344212
4. Chern E.J., Nielsen H.T.C. (1990), Generalized pulse equations for through-transmission evaluation of arbitrary multilayered structures, Research in Nondestructive Evaluation, 2: 1–9.
5. Conoir J.M., (1991), Interferences and periodic distribution of resonances in an elastic plate [in French: Interférences et périodicités sur les résonances dans une plaque élastique], Journal d’acoustique, 4: 377–412.
6. Derem A. (1982), Series of transmitted waves for a fluid and hollow cylinder: an exact solution [in French: Série des ondes transmises pour un cylindre fluide et creux: une solution exacte], Revue du CETHEDEC, 19(70): 1–27.
7. Derible S., Tinel A. (2011), Resonances of two elastic plates separated by a thickness of water. Study by means of transition terms [in French: Résonances de deux plaques élastiques séparées par une épaisseur d’eau. Etude au moyen des termes de transition], [in:] 20ème Congrès Français de Mécanique Besançon.
8. Deschamps M., Chengwei C. (1991), Reflection/refraction of a solid layer by Debye’s series expansion, Ultrasonics, 29(4): 288–293, https://doi.org/10.1016/0041-624X%2891%2990024-3
9. Fiorito R., Madigoskay W., Überall H. (1981), Acoustic resonances and the determination of the material parameters of a viscous fluid layer, The Journal of the Acoustical Society of America, 69(4): 897–903, https://doi.org/10.1121/1.385610
10. Fiorito R., Überall H. (1979), Resonance theory of acoustic reflection and transmission through a fluid layer, The Journal of the Acoustical Society of America, 65(1): 9–14, https://doi.org/10.1121/1.382275
11. Folds D.L., Loggins C.D. (1977), Transmission and reflection of ultrasonic waves in layered Media, The Journal of the Acoustical Society of America, 62(5): 1102–1109, https://doi.org/10.1121/1.381643
12. Gérard A. (1979), Field resulting from the incidence of P or SV waves on an elastic sphere [in French: Champ résultant de l’incidence d’ondes P ou SV sur une sphere élastique], Comptes rendus de l’Académie des Sciences. Série IIb, Mécanique, Elsevier, 289: 237–240.
13. Gérard A. (1980), Field resulting from the incidence of P and SV waves on a stratified medium with spherical symmetry [in French: Champ résultant de l’incidence d’ondes P et SV sur un milieu stratifié à symétrie sphérique], Comptes rendus de l’Académie des Sciences. Série IIb, Mécanique, Elsevier, 290(3): 43–46.
14. Gérard A. (1982), Factorization of the characteristic equation of a multi-layered elastic sphere: interpretation of resonances [in French: Factorisation de l’équation caractéristique d’une sphère élastique multicouches: interprétation des résonances], Comptes rendus de l’Académie des Sciences. Série IIb, Mécanique, Elsevier, 297: 17–19.
15. Gérard A. (1987), Modal formalism: interpretation [in French: Formalisme modal: interprétation], [in:] Diffusion Acoustique [in French: La Diffusion Acoustique], Gespa N., Poirée B. [Eds.], Revue du CETHEDEC, pp. 165–287.
16. Gérard A. (2022), Generalized Debye series theory for acoustic scattering: Some applications, [in:] Generalized Models and Non-classical Approaches in Complex Materials 1. Advanced Structured Materials, Visakh P.M. [Ed.], Vol. 89, Springer Singapore, pp. 349–374.
17. Gudra T., Banasiak D. (2020), Optimal selection of multicomponent matching layers for piezoelectric transducers using genetic algorithm, Archives Acoustics, 45(4): 699–707, https://doi.org/10.24425/aoa.2020.135276
18. Haskell N.A. (1953), The dispersion of surface waves on multilayered media, Bulletin of the Seismological Society of America, 43(1): 17–34, https://doi.org/10.1785/BSSA0430010017
19. Hsu D.K. (2009), Nondestructive evaluation of sandwich structures: A review of some inspection techniques, Journal of Sandwich Structures and Materials, 11(4): 275–291, https://doi.org/10.1177/1099636209105377
20. Ingard K.U., Morse P.M. (1968), Theoretical Acoustics, Princeton University Press, Princeton, New Jersey.
21. Khaled A., Maréchal P., Lenoir O., Ech-Cherif El-Kettani M., Chenouni D. (2013), Study of the resonances of periodic plane media immersed in water: Theory and experiment, Ultrasonics, 53(3): 642–647, https://doi.org/10.1016/j.ultras.2012.11.011
22. Lenoir O., Maréchal P. (2009), Study of plane periodic multilayered viscoelastic media: Experiment and simulation, [in:] 2009 IEEE International Ultrasonics Symposium Proceedings, pp. 1028–1011, https://doi.org/10.1109/ULTSYM.2009.5441518
23. Lowe M.J.S. (1995), Matrix techniques for modeling ultrasonic waves in multilayered media, [in:] IEEE Transaction of Ultrasonic, Ferroelectric, and Frequency Control, 42(4): 525–542, https://doi.org/10.1109/58.393096
24. Maréchal P. et al. (2008), Modeling of a high frequency ultrasonic transducer using periodic structures, Ultrasonics, 48(2): 141–149, https://doi.org/10.1016/j.ultras.2007.11.007
25. Maréchal P., Lenoir O., Khaled A., Ech-Cherif El-Kettani M., Chenouni D. (2014), Viscoelasticity effect on a periodic plane medium immersed in water, Acta Acustica united with Acustica, 100(6): 1036–1043, https://doi.org/10.3813/AAA.918783
26. Messineo M.G., Frontini G.L., Eliçabe G.E., Gaete-Garretón L. (2013), Equivalent ultrasonic impedance in multilayer media. A parameter estimation problem, Inverse Problems in Science and Engineering, 21(8): 1268–1287, https://doi.org/10.1080/17415977.2012.757312
27. Messineo M.G., Rus G., Eliçabe G.E., Frontini G.L. (2016), Layered material characterization using ultrasonic transmission: An inverse estimation methodology, Ultrasonics, 65: 315–328, https://doi.org/10.1016/j.ultras.2015.09.010
28. Nayfeh A.H. (1991), The general problem of elastic wave propagation in multilayered anisotropic media, The Journal of the Acoustical Society of America, 89(4): 1521–1531, https://doi.org/10.1121/1.400988
29. Potel C., de Belleval J.-F. (1993), Propagation in an anisotropic periodically multilayered medium, The Journal of the Acoustical Society of America, 93(5): 2669–2677, https://doi.org/10.1121/1.405842
30. Rokhlin S.I., Wang Y.J. (1992), Equivalent boundary conditions for thin orthotropic layer between two solids: Reflection, refraction, and interface waves, The Journal of the Acoustical Society of America, 91(4): 1875–1887, https://doi.org/10.1121/1.403717
31. Scott W.R., Gordon P.F. (1977), Ultrasonic spectrum analysis for nondestructive testing of layered composite materials, The Journal of the Acoustical Society of America, 62(1): 108–116, https://doi.org/10.1121/1.381491
32. Shenand M., Cao W. (2000), Acoustic band gap formation in a periodic structure with multilayer unit cells, Journal of Physics D: Applied Physics, 33(10): 1150–1154, https://doi.org/10.1088/00223727/33/10/303
33. Solyanik F.I. (1977), Transmission of plane waves through a layered medium of anisotropic materials, Soviet Physics Acoustics-USSR, 23: 533–536.
34. Soucrati H., Chitnalah A., Aouzale N., Jakjoud H. (2018), Analytical model of three-dimensional ultrasonic beam interaction with an immersed plate, Archives of Acoustics, 43(4): 669–679, https://doi.org/10.24425/aoa.2018.125160
35. Storheim E., Lohne K.D., Hergum T. (2015), Transmission and reflection from a layered medium in water. Simulations and measurements, [in:] Proceedings of the 38th Scandinavian Symposium on Physical Acoustics, Norway.
36. Stovas A., Arntsen B. (2006), Vertical propagation of low-frequency waves in finely layered media, Geophysics, 71(3): T87–T94, https://doi.org/10.1190/1.2197488
37. Yang X., Zhang C., Wang C., Sun A., Ju B.-F., Shen Q. (2019), Simultaneous ultrasonic parameter estimation of a multi-layered material by the PSO-based least squares algorithm using the reflection spectrum, Ultrasonics, 91: 231–236, doi: 1016/j.ultras.2018.08.003.
