Modares Mechanical Engineering

Modares Mechanical Engineering

Application of multi-scale methods for Lamb wave propagation analysis in a steel beam

Document Type : Original Article

Authors
fatemeh imani 1
mahnaz shamahirsaz 2
mohammad mohammadi aghdam 2
1 New Technologies Research Center, Amirkabir University of Technology, Tehran, Iran
2 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran
10.48311/mme.2026.118877.82954
Abstract
This study investigates the propagation behavior of Lamb waves in a steel beam at varying inter-sensor distances, employing CWT, FFT, and STFT to evaluate their effectiveness for time–frequency and energy-based signal characterization. Results show that the Lamb-wave time-of-flight exhibits minimal variation despite changes in sensor spacing, yielding a nearly constant group velocity of approximately 4540 m/s, which confirms the stable propagation of the S₀ mode. Multiscale wavelet analysis reveals that nearly 90% of the total energy is concentrated in Scale‑1, while the combined contribution of Scales 2–5 remains below 10%, indicating a dominant, low-dispersion single-mode response.

Quantitative CWT-based indicators also show consistent trends with increasing distance. Wavelet energy increases by about 20%, whereas entropy decreases from 0.4908 to 0.4651, reflecting stronger energy localization in Scale‑1 and improved separation of the S₀ mode from background noise and secondary modes. The RMS value increases from 0.2199 to 0.239, suggesting reduced attenuation and lower dispersion along the propagation path. The reduction in kurtosis further indicates diminished impulsive peaks, increased waveform smoothness, and an enhanced signal-to-noise ratio. Overall, the findings demonstrate that CWT provides superior capability over FFT and STFT in analyzing energy evolution, attenuation characteristics, and time–frequency dynamics of guided waves. These advantages establish CWT as a robust quantitative tool for structural health monitoring and guided-wave analysis.
Keywords
Method of multi scale
Lamb wave
Time-frequency analysis
short- time fourier transform
Subjects
[1] J. L. Rose, Ultrasonic Guided Waves in Solid Media. Cambridge, UK: Cambridge University Press, 2014.
[2] J. D. Achenbach, Wave Propagation in Elastic Solids. Amsterdam, The Netherlands: North-Holland, 1973.
[3] H. Lamb, “On waves in an elastic plate,” Proc. Roy. Soc. London, vol. A93, pp. 114–128, 1917.DOI:10.1098/rspa.1917.0008
 
[4] I. A. Viktorov, Rayleigh and Lamb Waves. New York, NY, USA: Plenum Press, 1967. DOI: 10.1007/978-1-4899-5681-1
 
[5] M. J. S. Lowe, “Matrix techniques for modeling ultrasonic waves in multilayered media,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 42, no. 4, pp. 525–542, Jul. 1995.DOI: 10.1109/58.393096
 
[6] Ghodrati, Behnam, Amin Yaghootian, Afshin Ghanbarzadeh, and Hamid Mohammad Sedighi. "Extraction of dispersion curves for Lamb waves in an aluminium nitride (AlN) microplate using consistent couple stress theory." (2016): 248-256. DOI: 20.1001.1.10275940.1395.16.1.33.6.
 
[7] Ziaiefar, Hamidreza, Milad Amiryan, Mojtaba Ghodsi, Farhang Honarvar, and Yousef Hojjat. "Ultrasonic damage classification in pipes and plates using wavelet transform and SVM." Modares Mechanical Engineering 15, no. 5 (2015): 41-48. DOI: 20.1001.1.10275940.1394.15.5.19.3
 
[8] Liu, Zenghua, Xiaoyu Liu, Yanping Zhu, Zhaojing Lu, Long Chen, Bin Wu, and Cunfu He. "Multi-time Lamb waves space wavenumber imaging method based on ultrasonic-guided wavefield." Structural Health Monitoring 24, no. 4 (2025): 2521-2535. DOI: 10.1177/14759217241261091
 
[9] A. H. Nayfeh, “Stability of three-dimensional boundary layers,” AIAA Journal, vol. 18, no. 4, pp. 406–416, Apr. 1980. DOI:10.2514/3.50773
 
[10] J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods. New York, NY, USA: Springer, 1996.
 
[11] K. Kanda and T. Maruyama, “Forced Lamb waves analyzed using the method of multiple scales,” Ultrasonics, vol. 137, p. 107144, 2023. DOI:10.1007/s00707-023-03573-8
 
[12] K. Kanda, K. Kosuke, and T. Maruyama, “Theoretical analysis of the dispersion of Lamb waves forming a wave packet of finite bandwidth using the method of multiple scales,” International Journal of Solids and Structures, vol. 234, pp. 111268, Jan. 2022. DOI:10.1016/j.ijsolstr.2021.111268
 
[13] C. Potel and J.-F. Chaix, “Leaky Lamb waves for NDT applications,” Ultrasonics, vol. 40, nos. 1–8, pp. 191–197, 2002. DOI: 10.1016/S0041-624X(02)00124-7
 
[14] P. Kauffmann, M. A. Ploix, and J.-F. Chaix, “Multi-modal leaky Lamb waves in immersed plates,” J. August. Soc. Am., Vol. 146, no. 3, pp. 1825–1836, Sept. 2019. DOI:10.1121/1.5091689
 
[15] Kanda, Kosuke, and Toshihiko Sugiura. "Analysis of damped guided waves using the method of multiple scales." Wave Motion, vol. 82, pp. 86–95, Nov. 2018. DOI: 10.1016/j.wavemoti.2018.07.007
 
[16] Nayfeh, Ali H. Linear and nonlinear structural mechanics. John Wiley & Sons, 2024.
 
[17] Sugita, Naohiro, and Toshihiko Sugiura. "Nonlinear normal modes and localization in two bubble oscillators." Ultrasonics 74 (2017): 174-185. DOI: 10.1016/j.ultras.2016.10.008
 
[18] Nayfeh, A. H., And S. A. Nayfeh. "Nonlinear normal modes of a continuous system with quadratic nonlinearities." (1995): 199-205. DOI: 10.1115/1.2873898
Modares Mechanical Engineering
Volume 26, Issue 9
August 2026
Pages 733-742