Abstract

We report a Brillouin-based fully distributed and dynamic monitoring of the strain induced by a propagating mechanical wave along a 20m long composite strip, to which surface a single-mode optical fiber was glued. Employing a simplified version of the Slope-Assisted Brillouin Optical Time Domain Analysis (SA-BOTDA) technique, the whole length of the strip was interrogated every 10ms (strip sampling rate of 100Hz) with a spatial resolution of the order of 1m. A dynamic spatially and temporally continuous map of the strain was obtained, whose temporal behavior at four discrete locations was verified against co-located fiber Bragg gratings. With a trade-off among sampling rate, range and signal to noise ratio, kHz sampling rates and hundreds of meters of range can be obtained with resolution down to a few centimeters.

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References

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  1. A. Othonos and K. Kalli, Fiber Bragg Gratings (Artech House, 1999).
  2. B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
    [CrossRef]
  3. M. Froggatt and J. Moore, “High resolution strain measurement in optical fiber with Rayleigh scatter,” Appl. Opt.37, 1735–1740 (1998).
    [CrossRef] [PubMed]
  4. D. P. Zhou, Z. Qin, W. Li, L. Chen, and X. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express20(12), 13138–13145 (2012).
    [CrossRef] [PubMed]
  5. T. Kurashima, T. Horiguchi, and M. Tateda, “Distributed-temperature sensing using stimulated Brillouin scattering in optical silica fibers,” Opt. Lett.15(18), 1038–1040 (1990).
    [CrossRef] [PubMed]
  6. L. Thévenaz, “Inelastic Scatterings and Applications to Distributed Sensing” in Advanced Fiber Optics - Concepts and Technology, Thévenaz L. ed, (Switzerland: EPFL University, 2011).
  7. X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel)12, 8601–8639 (2012).
    [CrossRef] [PubMed]
  8. W. Li, X. Bao, Y. Li, and L. Chen, “Differential pulse-width pair BOTDA for high spatial resolution sensing,” Opt. Express16(26), 21616–21625 (2008).
    [CrossRef] [PubMed]
  9. S. M. Foaleng, M. Tur, J. C. Beugnot, and L. Thevenaz, “High spatial and spectral resolution long-range sensing using Brillouin echoes,” J. Lightwave Technol.28(20), 2993–3003 (2010).
    [CrossRef]
  10. T. Sperber, A. Eyal, M. Tur, and L. Thévenaz, “High spatial resolution distributed sensing in optical fibers by Brillouin gain-profile tracing,” Opt. Express18(8), 8671–8679 (2010).
    [CrossRef] [PubMed]
  11. A. W. Brown, B. G. Colpitts, and K. Brown, “Dark-pulse Brillouin optical time-domain sensor with 20-mm spatial resolution,” J. Lightwave Technol.25(1), 381–386 (2007).
    [CrossRef]
  12. K. Hotate and S. S. L. Ong, “Distributed fiber Brillouin strain sensing by correlation-based continuous-wave technique ~cm-order spatial resolution and dynamic strain measurement,” Proc. SPIE4920, 299–310 (2002).
    [CrossRef]
  13. R. Bernini, A. Minardo, and L. Zeni, “Dynamic strain measurement in optical fibers by stimulated Brillouin scattering,” Opt. Lett.34(17), 2613–2615 (2009).
    [CrossRef] [PubMed]
  14. Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express19(21), 19845–19854 (2011).
    [CrossRef] [PubMed]
  15. K. Y. Song, M. Kishi, Z. He, and K. Hotate, “High-repetition-rate distributed Brillouin sensor based on optical correlation-domain analysis with differential frequency modulation,” Opt. Lett.36(11), 2062–2064 (2011).
    [CrossRef] [PubMed]
  16. Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express20(8), 8584–8591 (2012).
    [CrossRef] [PubMed]
  17. A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
    [CrossRef]
  18. T. Yari, K. Nagai, and K. Hotate, “Monitoring aircraft structural health using optical fiber sensors,” Mitsubishi Heavy Ind. Tech. Rev.45, 5–8 (2008).
  19. A. Motil, Y. Peled, L. Yaron, and M. Tur, “BOTDA measurements in the presence of fiber vibrations,” Proc. SPIE8351, 835132, 835132-6 (2012).
    [CrossRef]
  20. J. Urricelqui, A. Zornoza, M. Sagues, and A. Loayssa, “Dynamic BOTDA measurements based on Brillouin phase-shift and RF demodulation,” Opt. Express20(24), 26942–26949 (2012).
    [CrossRef] [PubMed]
  21. J. C. Beugnot, M. Tur, S. F. Mafang, and L. Thévenaz, “Distributed Brillouin sensing with sub-meter spatial resolution: modeling and processing,” Opt. Express19(8), 7381–7397 (2011).
    [CrossRef] [PubMed]
  22. S. Timoshenko and D. N. Young, Vibration Problems in Engineering (V. Nostrand, N.J., 1955).

2012

2011

2010

S. M. Foaleng, M. Tur, J. C. Beugnot, and L. Thevenaz, “High spatial and spectral resolution long-range sensing using Brillouin echoes,” J. Lightwave Technol.28(20), 2993–3003 (2010).
[CrossRef]

T. Sperber, A. Eyal, M. Tur, and L. Thévenaz, “High spatial resolution distributed sensing in optical fibers by Brillouin gain-profile tracing,” Opt. Express18(8), 8671–8679 (2010).
[CrossRef] [PubMed]

2009

2008

W. Li, X. Bao, Y. Li, and L. Chen, “Differential pulse-width pair BOTDA for high spatial resolution sensing,” Opt. Express16(26), 21616–21625 (2008).
[CrossRef] [PubMed]

T. Yari, K. Nagai, and K. Hotate, “Monitoring aircraft structural health using optical fiber sensors,” Mitsubishi Heavy Ind. Tech. Rev.45, 5–8 (2008).

2007

2002

K. Hotate and S. S. L. Ong, “Distributed fiber Brillouin strain sensing by correlation-based continuous-wave technique ~cm-order spatial resolution and dynamic strain measurement,” Proc. SPIE4920, 299–310 (2002).
[CrossRef]

2001

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
[CrossRef]

1998

1990

T. Kurashima, T. Horiguchi, and M. Tateda, “Distributed-temperature sensing using stimulated Brillouin scattering in optical silica fibers,” Opt. Lett.15(18), 1038–1040 (1990).
[CrossRef] [PubMed]

Allison, S. G.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
[CrossRef]

Bao, X.

Batten, C. F.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
[CrossRef]

Bernini, R.

Beugnot, J. C.

Brown, A. W.

Brown, K.

Chen, L.

Childers, B. A.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
[CrossRef]

Chitgarha, M.

A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
[CrossRef]

Colpitts, B. G.

Eyal, A.

T. Sperber, A. Eyal, M. Tur, and L. Thévenaz, “High spatial resolution distributed sensing in optical fibers by Brillouin gain-profile tracing,” Opt. Express18(8), 8671–8679 (2010).
[CrossRef] [PubMed]

Foaleng, S. M.

Froggatt, M.

Froggatt, M. E.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
[CrossRef]

Hare, D. A.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
[CrossRef]

He, Z.

Horiguchi, T.

T. Kurashima, T. Horiguchi, and M. Tateda, “Distributed-temperature sensing using stimulated Brillouin scattering in optical silica fibers,” Opt. Lett.15(18), 1038–1040 (1990).
[CrossRef] [PubMed]

Hotate, K.

K. Y. Song, M. Kishi, Z. He, and K. Hotate, “High-repetition-rate distributed Brillouin sensor based on optical correlation-domain analysis with differential frequency modulation,” Opt. Lett.36(11), 2062–2064 (2011).
[CrossRef] [PubMed]

T. Yari, K. Nagai, and K. Hotate, “Monitoring aircraft structural health using optical fiber sensors,” Mitsubishi Heavy Ind. Tech. Rev.45, 5–8 (2008).

K. Hotate and S. S. L. Ong, “Distributed fiber Brillouin strain sensing by correlation-based continuous-wave technique ~cm-order spatial resolution and dynamic strain measurement,” Proc. SPIE4920, 299–310 (2002).
[CrossRef]

Jegley, D. C.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
[CrossRef]

Kishi, M.

Kurashima, T.

T. Kurashima, T. Horiguchi, and M. Tateda, “Distributed-temperature sensing using stimulated Brillouin scattering in optical silica fibers,” Opt. Lett.15(18), 1038–1040 (1990).
[CrossRef] [PubMed]

Li, W.

Li, Y.

Loayssa, A.

Mafang, S. F.

Minardo, A.

Moore, J.

Moore, T. C.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
[CrossRef]

Motil, A.

Nagai, K.

T. Yari, K. Nagai, and K. Hotate, “Monitoring aircraft structural health using optical fiber sensors,” Mitsubishi Heavy Ind. Tech. Rev.45, 5–8 (2008).

Nuccio, S. R.

A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
[CrossRef]

Ong, S. S. L.

K. Hotate and S. S. L. Ong, “Distributed fiber Brillouin strain sensing by correlation-based continuous-wave technique ~cm-order spatial resolution and dynamic strain measurement,” Proc. SPIE4920, 299–310 (2002).
[CrossRef]

Peled, Y.

Qin, Z.

Sagues, M.

Shamee, B.

A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
[CrossRef]

Song, K. Y.

Sperber, T.

T. Sperber, A. Eyal, M. Tur, and L. Thévenaz, “High spatial resolution distributed sensing in optical fibers by Brillouin gain-profile tracing,” Opt. Express18(8), 8671–8679 (2010).
[CrossRef] [PubMed]

Tateda, M.

T. Kurashima, T. Horiguchi, and M. Tateda, “Distributed-temperature sensing using stimulated Brillouin scattering in optical silica fibers,” Opt. Lett.15(18), 1038–1040 (1990).
[CrossRef] [PubMed]

Thevenaz, L.

Thévenaz, L.

J. C. Beugnot, M. Tur, S. F. Mafang, and L. Thévenaz, “Distributed Brillouin sensing with sub-meter spatial resolution: modeling and processing,” Opt. Express19(8), 7381–7397 (2011).
[CrossRef] [PubMed]

T. Sperber, A. Eyal, M. Tur, and L. Thévenaz, “High spatial resolution distributed sensing in optical fibers by Brillouin gain-profile tracing,” Opt. Express18(8), 8671–8679 (2010).
[CrossRef] [PubMed]

Tur, M.

Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express20(8), 8584–8591 (2012).
[CrossRef] [PubMed]

A. Motil, Y. Peled, L. Yaron, and M. Tur, “BOTDA measurements in the presence of fiber vibrations,” Proc. SPIE8351, 835132, 835132-6 (2012).
[CrossRef]

J. C. Beugnot, M. Tur, S. F. Mafang, and L. Thévenaz, “Distributed Brillouin sensing with sub-meter spatial resolution: modeling and processing,” Opt. Express19(8), 7381–7397 (2011).
[CrossRef] [PubMed]

Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express19(21), 19845–19854 (2011).
[CrossRef] [PubMed]

A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
[CrossRef]

S. M. Foaleng, M. Tur, J. C. Beugnot, and L. Thevenaz, “High spatial and spectral resolution long-range sensing using Brillouin echoes,” J. Lightwave Technol.28(20), 2993–3003 (2010).
[CrossRef]

T. Sperber, A. Eyal, M. Tur, and L. Thévenaz, “High spatial resolution distributed sensing in optical fibers by Brillouin gain-profile tracing,” Opt. Express18(8), 8671–8679 (2010).
[CrossRef] [PubMed]

Urricelqui, J.

Voskoboinik, A.

A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
[CrossRef]

Wang, J.

A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
[CrossRef]

Willner, A. E.

A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
[CrossRef]

Yari, T.

T. Yari, K. Nagai, and K. Hotate, “Monitoring aircraft structural health using optical fiber sensors,” Mitsubishi Heavy Ind. Tech. Rev.45, 5–8 (2008).

Yaron, L.

A. Motil, Y. Peled, L. Yaron, and M. Tur, “BOTDA measurements in the presence of fiber vibrations,” Proc. SPIE8351, 835132, 835132-6 (2012).
[CrossRef]

Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express19(21), 19845–19854 (2011).
[CrossRef] [PubMed]

Zeni, L.

Zhang, L.

A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
[CrossRef]

Zhou, D. P.

Zornoza, A.

Appl. Opt.

J. Lightwave Technol.

A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS-based fiber optical sensing using frequency-domain simultaneous tone interrogation,” J. Lightwave Technol.29(11), 1729–1735 (2011).
[CrossRef]

J. Lightwave Technol.

Mitsubishi Heavy Ind. Tech. Rev.

T. Yari, K. Nagai, and K. Hotate, “Monitoring aircraft structural health using optical fiber sensors,” Mitsubishi Heavy Ind. Tech. Rev.45, 5–8 (2008).

Opt. Express

T. Sperber, A. Eyal, M. Tur, and L. Thévenaz, “High spatial resolution distributed sensing in optical fibers by Brillouin gain-profile tracing,” Opt. Express18(8), 8671–8679 (2010).
[CrossRef] [PubMed]

Opt. Lett.

T. Kurashima, T. Horiguchi, and M. Tateda, “Distributed-temperature sensing using stimulated Brillouin scattering in optical silica fibers,” Opt. Lett.15(18), 1038–1040 (1990).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Proc. SPIE

K. Hotate and S. S. L. Ong, “Distributed fiber Brillouin strain sensing by correlation-based continuous-wave technique ~cm-order spatial resolution and dynamic strain measurement,” Proc. SPIE4920, 299–310 (2002).
[CrossRef]

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four 8-m optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001).
[CrossRef]

A. Motil, Y. Peled, L. Yaron, and M. Tur, “BOTDA measurements in the presence of fiber vibrations,” Proc. SPIE8351, 835132, 835132-6 (2012).
[CrossRef]

Sensors (Basel)

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel)12, 8601–8639 (2012).
[CrossRef] [PubMed]

Other

S. Timoshenko and D. N. Young, Vibration Problems in Engineering (V. Nostrand, N.J., 1955).

A. Othonos and K. Kalli, Fiber Bragg Gratings (Artech House, 1999).

L. Thévenaz, “Inelastic Scatterings and Applications to Distributed Sensing” in Advanced Fiber Optics - Concepts and Technology, Thévenaz L. ed, (Switzerland: EPFL University, 2011).

Supplementary Material (5)

» Media 1: MOV (576 KB)     
» Media 2: MOV (689 KB)     
» Media 3: MOV (821 KB)     
» Media 4: MOV (519 KB)     
» Media 5: MOV (855 KB)     

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Figures (10)

Fig. 1
Fig. 1

Brillouin gain spectra of a fiber segment under time average (black), high (blue) and low (red) strain values. A frequency working point is chosen near the center of the left slope; see the black point on the black Lorentzian. Positive (negative) strains introduced to the fiber, shift the BGS to higher (lower) frequencies, thereby modulating the Brillouin gain experienced by the propagating probe wave.

Fig. 2
Fig. 2

The composite strip with two glued fibers: the long one for the Brillouin sensing and the short one for measurement validation using fiber Bragg gratings.

Fig. 3
Fig. 3

Experimental setup: DFB-LD: Narrow linewidth (<10kHz) laser diode, EOM: electro-optic modulator, EDFA: Erbium-doped fiber amplifier, CIR: circulator, FBG: fiber Bragg grating, PS: polarization scrambler, IS: isolator, ATT: attenuator, PD: photodiode, FUT: fiber under test.

Fig. 4
Fig. 4

The BFS (red) along the strip, together with −3dB mid-slope contours, showing quite a small dependence on distance and a BGS linewidth of 80MHz. With such a small distance variation of the −3dB contours, there is no need for a tailored probe wave, as in [14], and a CW probe fixed frequency of 10.96MHz was used.

Fig. 5
Fig. 5

(a) A single-frame from the recorded movie (Media 1) showing the flexural bell-shaped wave. (b) A Gaussian-modeled flexural wave along the strip (blue). The sensing fiber (red) is glued to the strip top surface. (c) The corresponding strain wave at the strip top surface. While the real wave is not Gaussian, the model wave height (0.35m), as well as its FWHM spatial width (2m), were chosen to produce a strain wave having spatial scales similar to those observed in the experiment.

Fig. 6
Fig. 6

Theoretical predictions for the expected Brillouin-determined induced strain along the strip. (a) The modeled strain (green) and the Brillouin result (blue). Note the only slight reductions in the positive and negative peaks and also the delayed response. (b) Time aligned temporal signatures of the strain (assuming a velocity of 10m/s for the flexural wave): The Brillouin results (blue), strain readings from a short FBG (green) and temporally smoothed FBG results (red).

Fig. 7
Fig. 7

The Brillouin measured SA-BOTDA spatially and temporally continuous results as a function of both time and distance for the first 6 meters of the strip. Results are compared with the data from the four spatially discrete FBGs (black lines). In Figs. 7-10 time is measured from an arbitrary origin, where data collection commenced (not to be confused with the time origin of the simulation of Fig. 6, which coincided with the moment the pump pulse entered the strip).

Fig. 8
Fig. 8

(a) The time dependence of the strains at the locations of the four FBGs, as measured by: The SA-BOTDA (blue); 1cm spatial resolution FBGs (low-pass filtered to 100Hz and temporally aligned with the Brillouin data, green); A smoothed version of the FBG data (red). (b) A snapshot of the spatial distribution of the strain at t = 1.2sec, showing the continuous Brillouin results (blue) vs. the discrete nature of the FBG technique (red dots), Media 2.

Fig. 9
Fig. 9

Two 3D views of the strain accompanying the flexural wave as the latter propagates along the 20m tape with a velocity of ~9m/s. These figures clearly show that the wave slightly accelerates towards the end of the strip, while simultaneously decaying. The black curve in (b) represents a snapshot of the strain at t = 0.7s (Media 3).

Fig. 10
Fig. 10

Two counter-propagating flexural waves, simultaneously launched from the two ends of the strip. (a) A single frame from a movie (Media 4). (b) 3D and 2D snapshots from the measurement movie (Media 5) of the strain fields of the two counter-propagating flexural waves. The black and blue lines are 2D presentations of the strain distribution along the strip before (at t = 0.5 sec) and after (at t = 1.1 sec) their collision.

Equations (1)

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ε = w 2 2 y z 2 ,

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