Comparative Study of the Spectral Method, DISPERSE and Other ‎Classical Methods for Plotting the Dispersion Curves in ‎Anisotropic Plates

Document Type : Research Paper


Laboratory of Mechanics, Engineering and Innovation, National High School of Electricity and Mechanics, Hassan II University, Casablanca, Morocco‎


This paper presents a comparative study of different methods for obtaining the dispersion curves of ultrasonic guided waves in anisotropic media. First, we present the classical algorithms used to find zeros and propose some improvements. Next, the spectral method is explained for modeling the guided waves in anisotropic materials while presenting a technique that can distinguish the modes present in the structure. The dispersion curves are plotted using a Matlab program and the results are compared with those of the DISPERSE software. In addition, a comparison with the results obtained by Nayfeh’s works in the field of Nondestructive testing by ultrasonic guided waves is included. Then a discussion is developed to highlight the strengths of the spectral method. For proper non-destructive testing, we need reliable information about the modes that propagate in our waveguide. Both analytical and spectral approaches have limitations in obtaining the exact displacement and stress profiles in a plate media. To remedy this, normalization by the acoustic power is essential. Next, the displacement and stress fields obtained from the spectral method of the modes that can propagate in the plate are compared to those obtained analytically. A very good concordance is then noticed. Based on the results obtained, the spectral method presents a very good alternative for obtaining dispersion curves. It is a convergent method, stable, easy to implement with a very low calculation time.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Stazizar, A.J., Investigation of Flow Phenomena in a transonic Fan Rotor Using Laser Anemometry, ASME Journal of Engineering for Gas Turbines and Power, 107(2), 1985, 427-435.
[1] Moser, F., Jacobs, L.J., Qu, J., Modeling elastic wave propagation in waveguides with the finite element method, NDT & E International, 32(4), 1999, 225–234.
[2] Valle, C., Niethammer, M., Qu, J., Jacobs, L.J., Crack characterization using guided circumferential waves, The Journal of the Acoustical Society of America, 110(3), 2001, 1282–1290.
[3] Li, Y., Thompson, B.R., Influence of anisotropy on the dispersion characteristics of guided ultrasonic plate modes, The Journal of the Acoustical Society of America, 87(5), 1990, 1911–1931.
[4] Dayal, V., Kinra, V.K., Leaky lamb waves in an anisotropic plate. I: An exact solution and experiments, The Journal of the Acoustical Society of America, 85(6), 1989, 2268–2276.
[5] Ciampa, F., Meo, M., Acoustic emission source localization and velocity determination of the fundamental mode A0 using wavelet analysis and a Newton-based optimization technique, Smart Materials and Structures, 19(4), 2010, 045027.
[6] Ndiaye, E.B., Duflo, H., Non-destructive testing of sandwich composites: adhesion defects evaluation; experimental and finite element method simulation comparison, In Acoustics, 2012.
[7] Guo, S., Rebillat, M., Mechbal, N., Dichotomy property of dispersion equation of guided waves propagating in anisotropic composite plates, Mechanical Systems and Signal Processing, 164, 2022, 108212.
[8] Schwab, F., Knopoff, L., Surface-wave dispersion computations, Bulletin of the Seismological Society of America, 60(2), 1970, 321–344.
[9] Mal, A.K., Guided waves in layered solids with interface zones, International Journal of Engineering Science, 26(8), 1988, 873–881.
[10] Honarvar, F., Enjilela, E., Sinclair, A.N., An alternative method for plotting dispersion curves, Ultrasonics, 49(1), 2009, 15–18.
[11] Waas, G., Earth vibration effects and abatement for military facilities: report 3: analysis method for footing vibrations through layered media, Waterways Experiment Station, 1972.
[12] Ahmad, Z., Vivar-Perez, J.M., Gabbert, U., Semi-analytical finite element method for modeling of lamb wave propagation, CEAS Aeronautical Journal, 4(1), 2013, 21–33.
[13] Nissabouri, S., Allami, M.E., Boutyour, E.H., Quantitative evaluation of semi-analytical finite element method for modeling lamb waves in orthotropic plates, Comptes Rendus Mécanique, 348(5), 2020, 335–350.
[14] Barazanchy, D., Giurgiutiu, V., A comparative convergence and accuracy study of composite guided-ultrasonic wave solution methods: Comparing the unified analytic method, safe method and disperse, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 231(16), 2017, 2961–2973.
[15] Pavlakovic, B., Lowe, M., Alleyne, D., Cawley, P., Disperse: A general purpose program for creating dispersion curves, In Review of progress in quantitative nondestructive evaluation, Springer, 1997.
[16] Adamou, A., Craster, R.V., Spectral methods for modelling guided waves in elastic media, The Journal of the Acoustical Society of America, 116(3), 2004, 1524-1535.
[17] Karpfinger, F., Gurevich, B., Bakulin, A., Modeling of wave dispersion along cylindrical structures using the spectral method, The Journal of the Acoustical Society of America, 124(2), 2008, 859–865.
[18] Karpfinger, F., Valero, H.P., Gurevich, B., Bakulin, A., Sinha, B., Spectral-method algorithm for modeling dispersion of acoustic modes in elastic cylindrical structures, Geophysics, 75(3), 2010, 19–27.
[19] Yu, B., Yang, S., Gan, C., Lei, H., A new procedure for exploring the dispersion characteristics of longitudinal guided waves in a multilayered tube with a weak interface, Journal of Nondestructive Evaluation, 32(3), 2013, 263–276.
[20] Quintanilla, F.H., Lowe, M.J.S., Craster, R.V., Modeling guided elastic waves in generally anisotropic media using a spectral collocation method, The Journal of the Acoustical Society of America, 137(3), 2015, 1180-1194.
[21] Quintanilla, F.H., Lowe, M.J.S., Craster, R.V., Full 3D dispersion curve solutions for guided waves in generally anisotropic media, Journal of Sound and Vibration, 363, 2016, 545-559.
[22] Quintanilla, F.H., Lowe, M.J.S., Craster, R.V., The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides, The Journal of the Acoustical Society of America, 141(1), 2017, 406-418.
[23] Yu, J.G., Lefebvre, J.E., Guided waves in multilayered hollow cylinders: the improved Legendre polynomial method, Composite Structures, 95, 2013, 419-429.
[24] Othmani, C., Dahmen, S., Njeh, A., Ghozlen, M.H.B., Investigation of guided waves propagation in orthotropic viscoelastic carbon–epoxy plate by Legendre polynomial method, Mechanics Research Communications, 74, 2016, 27-33.
[25] Zheng, M., He, C., Lu, Y., Wu, B., State-vector formalism and the Legendre polynomial solution for modelling guided waves in anisotropic plates, Journal of Sound and Vibration, 412, 2018, 372-388.
[26] Kuznetsov, S.V., Similarity and discrepancy of Lamb wave propagation in functionally graded, stratified, and homogeneous media, International Journal of Dynamics and Control, 8(3), 2020, 717-722.
[27] Nayfeh, A.H., Chimenti, D.E., Free wave propagation in plates of general anisotropic media, Journal of Applied Mechanics, 56(4), 1989, 881-886.
[28] Eiger, A., Sikorski, K., Stenger, F., A bisection method for systems of nonlinear equations, ACM Transactions on Mathematical Software, 10(4), 1984, 367–377.
[29] Kelley, C.T., Solving nonlinear equations with Newton’s method, SIAM, 2003.
[30] Weideman, J.A., Reddy, S.C., A MATLAB differentiation matrix suite, ACM Transactions on Mathematical Software, 26(4), 2000, 465-519.
[31] Huber, A.M., Sause, M.G., Classification of solutions for guided waves in anisotropic composites with large numbers of layers, The Journal of the Acoustical Society of America, 144(6), 2018, 3236-3251.
[32] Sanderson, R., A closed form solution method for rapid calculation of guided wave dispersion curves for pipes, Wave Motion, 53, 2015, 40–50.
[33] Draudviliene, R., Raišutis, R., Žukauskas, E., Jankauskas, A., Validation of dispersion curve reconstruction techniques for the a0 and s0 modes of lamb waves, International Journal of Structural Stability and Dynamics, 14(07), 2014, 1450024.
[34] Crespo, B.H., Courtney, C.R.P., Engineer, B., Calculation of guided wave dispersion characteristics using a three-transducer measurement system, Applied Sciences, 8(8), 2018, 1253.
[35] Hernandez-Crespo, B., Engineer, B., Courtney, C., Empirical technique for dispersion curve creation for guided wave applications, In 8th European Workshop On Structural Health Monitoring, 2016.
[36] Pant, S., Laliberte, J., Martinez, M., Rocha, B., Derivation and experimental validation of Lamb wave equations for an n-layered anisotropic composite laminate, Composite Structures, 111, 2014, 566-579.
[37] Guo, S., Contribution to the study of guided waves propagation and attenuation in anisotropic composite laminates made up of viscoelastic composite materials: Application to A380 mounted nacelle parts, Doctoral dissertation, Ecole Nationale Supérieure d’Arts et Métiers Paris, 2021.
[38] Kausel, E., Malischewsky, P., Barbosa, J., Osculations of spectral lines in a layered medium, Wave Motion, 56, 2015, 22-42
[39] Li, Y., Thompson, R.B., Influence of anisotropy on the dispersion characteristics of guided ultrasonic plate modes, The Journal of the Acoustical Society of America, 87(5), 1990, 1911-1931.
[40] Royer, D., Dieulesaint, E., Elastic waves in solids I: Free and guided propagation, Springer Science & Business Media, 1999.
[41] Auld, B.A., Acoustic fields and waves in solids, Krieger Pub. Co., 1973.