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Reconstruction of Rayleigh-Lamb dispersion spectrum
based on noise obtained from an air-jet forcing
Éric Larose, Philippe Roux, Michel Campillo
To cite this version:
Éric Larose, Philippe Roux, Michel Campillo. Reconstruction of Rayleigh-Lamb dispersion spectrum
based on noise obtained from an air-jet forcing. Journal of the Acoustical Society of America, Acous-
tical Society of America, 2007, 122 (6), pp.3437. �hal-00175909�
hal-00175909, version 1 - 1 Oct 2007
Reconstruction of Rayleigh-Lamb dispersion spectrum based on noise obtained from an air-jet forcing
Eric LAROSE, Philippe ROUX, and Michel CAMPILLO
Lab. de G´eophysique Interne et Tectonophysique, Universit´e J. Fourier & CNRS, BP53, 38041 Grenoble,
France. Email: eric.larose@ujf-grenoble.fr
(Dated: October 1, 2007)
The time-domain cross-correlation of incoherent and random noise recorded by a series of
passive sensors contains the impulse response of the medium between these sensors. By using
noise generated by a can of compressed air sprayed on the surface of a plexiglass plate, we
are able to reconstruct not only the time of flight but the whole waveforms between the
sensors. From the reconstruction of the direct A
0
and S
0
waves, we derive the disper sion
curves of the flexural waves, thus estimating the mechanical properties of the material without
a conventional electromechanical source. The dense array of receivers employed here allow a
precise frequency-wavenumber study of flexural waves, along with a thorough evaluation of the
rate of convergence of the correlation with respect to the record length, the frequency, and the
distance between the receivers. The reconstruction of the actual amplitude and attenuation of
the impulse response is also addressed in this paper.
(accepted for publication in J. Acoust. Soc. Am. 2007)
I. INTRODUCTION
Elastic waves at kHz and MHz frequencies are widely used to evaluate the mechanical properties of structures,
material and tissues. In plates and shells, the elastic wave equation admits spec ific propagation modes, deno ted Lamb
waves [Royer and Dieulesaint (2000); Viktorov (1967)], which a re related to the traction free condition on both sides of
the medium. Depending on the purpose of the experiment, different measurement configurations have been proposed.
From the dispersion of Lamb waves, obta ined from pitch-catch measures repeated for different ranges, one has access
to the velo c ities of bulk waves and to the thickness of the plate ( see for instance Gao et al. (2003)). Other pulse-echo
(or impact-echo) techniques have been proposed to assess the mechanical properties of the plate, based on the simple
and multiple reflection of bulk waves within the plate [Krautkr¨amer and Krautkr¨amer (1990)] or on the r e sonance of
high order Lamb modes [Clorennec et al. (2007)]. Some studies also concern the dynamic evaluation of fatigue and/or
crack growth (see for instance Ihn and Chang (2004); Ing and Fink (1996)). In this last application, several impulse
responses of the medium are acquire d at different dates and are eventually compared to each others to monitor the
medium.
All these techniques require the use of controlled sources and receivers. In the following, they are referred to as
”active” experiments. Another idea has undergone a large development after the seminal experiments of Weaver and
Lobkis (2001) (see for instance the review of Wap e naar and Fokkema (2006), Weaver and Lobkis (2006) or Larose
(2006)). By cross-correlating the incoherent noise recorded by two passive sensors, Weaver and Lobkis demonstrated
that one could reconstruct the impulse respo ns e of the medium as if a source was plac e d at one sensor. This noise
correla tion technique (also referred to as ”passive imaging”, or ”seismic interferometry” [Schuster et al. (2004)])
requires the use of synchronized senso rs, and has the advantage of eliminating any controlled source.
In section II of our paper, we compare the dispersion cur ves of flexural waves (Lamb waves in the low frequency
regime) in a plexiglass plate obtained by an active (pitch-catch) experiment to the ones obtained in a passive exper-
iment. During the passive acquisitio n, we deployed synchronised receivers only, a nd used a random noise so urce: a
1 mm thick high pr e ssure air jet produced by a can of compressed air [McBride and Hutchison (1976)]. The experiment
conducted in a plexiglass pla te show at the same time: dispersive (A
0
) and non dispersive (S
0
) waves, reverberations,
and absorptio n. Since the earth’s crust has similar properties for seismic waves, we believe that a plexiglass plate at
kHz frequencies is a good candidate to built small scale s e ismic analogous expe riments. The question of reconstruct-
ing not only the phase, but also the amplitude of the wave is addressed a t the end of pa rt II. This is of interest in
seismology, where the role of absorption and attenuation in the correlation has not yet been subject to experimental
investigation. In section III, we analyze the rate of convergence of the correlations to the real impulse response. The
role of the record length, the central frequency, and the distance between the re ce ivers is investigated.
II. EXPERIMENTAL SETUP AND DISPERSION CURVES
To study actively and passively the disper sion of flexural waves, we built a laboratory experiment using a 1.5m×1.5m
large, 0.6 cm thick plexiglass plate. The plate is laid on an open steel frame that supports the edges of the pla te but
Reconstruction of Rayleigh-Lamb dispersion from noise 1
FIG. 1. Experimental setup. (a) In the active experiment (pitch-catch configuration), a broadband piezoelectric source S emits
a chirp that is sensed by a vertical accelerometer R placed at a distance d. (b) In the passive experiment, a turb ulent jet
produced by a can of compressed air generates a white random noise recorded simultaneously at all sensors R
i
. The jet is
randomly sprayed over the area in gray.
leaves free the upper and lower sides. The tractio n-free condition is therefore achieved on both horizontal sides. The
resulting dispersion relation that connects the pulsation ω a nd the wave-vector k reads:
ω
4
c
4
s
= 4 k
2
q
2
1 −
p tan(ph/2 + γ)
q tan(qh/2 + γ)
(1)
where p
2
=
ω
2
c
2
p
−k
2
and q
2
=
ω
2
c
2
s
−k
2
; c
s
(resp. c
p
) is the shear (resp. compressio nal) velocity, a nd h is the thickness
of the plate. The parameter γ equals 0 for sy mmetric (S) modes, and π/2 for anti-symmetric (A) modes. In the low
frequency regime, only two modes are solutions: they are labeled A
0
and S
0
.
For both acquisitions we used broad-band miniature (3 mm radius) accelerometers (ref.# 4518 from Bruel&Kjaer).
They show a flat response in the 20 Hz-70 kHz fr e quency range. We fixed our accelerometers on our plate using a hot
chemical glue (phenyl-salicylic acid) that solidifies with cooling (below 43
◦
C).
A. Active experiment
In the ac tive experiment, a sour ce S (a piezoelectric polymer) is in a corner of the plate, approximately 30 cm
from each side and is emitting a 3 s chirp s(t) in a linear range of frequencies f from 1 kHz to 60 kHz. A receiver
R is initially placed at d =1 cm away from the source toward the center of the plate on a straight graduated line.
After each acquisition, we moved the receiver a centimeter away from the source down the diagonal; we repe ated this
operation 100 times to cover 100 cm of the plate. The dynamic impulse response h
d
(t) of the plate is reconstructed
for each distance d by correlating the record r
d
(t) by the source chirp:
h
d
(t) = r
d
(t) × s(t) (2)
As an example, impulse responses obtained for three different distances d = 10, 40, 80 cm are displayed in Fig. 2.
The dispersive A
0
mode is dominating the record (a), but the non-dispersive S
0
wave (b) and reverberations from
the edges of the plate (c) are also visible. The transit time in the plate is of the order of a few milliseconds. T he
absorption time of the plate that strongly depends on the frequency is of the order of a few hundreds of milliseconds.
The signal-to noise in the experiment do not allow to record more than a few tens of millisecond. Records are
therefore dominated by ballistic waves, but waves reverberated from the plate boundaries a re also visible.
From the set of 100 impulse responses h
d
(t), a spatio-temporal Fourier (f-k) trans form was applied. The resulting
dispersion curves are displayed in Fig. 3, showing the dispersive A
0
mode and the weaker non-dispersive S
0
mode.
Theoretical dispersion curves are numerically obtained after Eq. 1 (dots), they perfectly fit the experimental data
for c
p
= 3130 m/s and c
s
= 1310 m/s. The dispersion curves are widely use d to evaluate the mechanical properties
of the plate, including thickness, presence of flaws... Nevertheless, the source-receiver configuration is sometimes at
a disadvantage. As mentioned in the introduction, instead of using the conventional source-receiver configuration, it
is possible to take advantage of ra ndom elastic noise to recons truct the impulse response h
d
(t) between two sensors.
Reconstruction of Rayleigh-Lamb dispersion from noise 2
−1
0
1
d=10 cm(a)
−1
0
1
(a)(b) d=40 cm
Normalized Acceleration
0 0.5 1 1.5
−1
0
1
d=80 cm(a)
(b)
(c)
Time (ms)
FIG. 2. Source-receiver impulse response h
d
(t) for different distances d. (a) Disp ersive anti-symmetric A
0
mode propagating at
velocities approximately ranging from 150 to 900 m/s. (b) Non-dispersive S
0
mode propagating at ≈2400 m/s. (c) Reflex ions
from surrounding edges.
h/λ
Frequency (kHz)
A
0
mode
S
0
mode
0 0.1 0.2 0.3 0.4 0.5 0.6
0
10
20
30
40
50
60
0
0.2
0.4
0.6
0.8
FIG. 3. f-k transform of the source-receiver impulse responses. X-axis: dimensionless wavenumber (h is the slab thickness).
Dots are theoretical solutions of Eq. 1.
This is performed in the following section.
B. Passive experiment
In the passive experiment, we removed the source and placed 16 rece ivers separated by 7 cm from e ach other on
the graduated line. At the former position o f the source, we placed the accelerometer R
0
that was kept fixed all
through the exper iment. As proposed by Sabra et al. (2007), we used the noise generated by a turbulent flow. As
a noise source, we employed a dry air blower. Note that, contrary to Sabra’s work, our source was easy-to-handle.
During the experiment, it was randomly moved to cover a 30 cm×30 cm large area located between the corner of the
plate and the for mer active source. No te that the precise knowledge of the noise so urce is not neces sary in the passive
experiment. We also deployed an array of 16 receivers that allow fo r precise frequency-wave number analysis, and
worked at higher frequencies, meaning a much thiner spatial reso lutio n. We focused our attention on the A
0
and S
0
direct arrivals. Since we want to reconstruct direct waves, we chose to spray only at one end of the array of receivers
(end-fire lobes described by Roux et al. (2004) or coherent zones described by Larose (2006)). Nevertheless, spraying
Reconstruction of Rayleigh-Lamb dispersion from noise 3
Sensor position d (cm)
Time τ (ms)
20 40 60 80 100
1
2
3
4
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
FIG. 4. Passively reconstructed impulse responses (linear color bar, normalized amplitude).
elsewhere gives the same waveforms but requires much more data, thus much longer acquisitions (see section III).
The noise in the plate was created by spraying continuously for approximately T=10 s. The 16 re c e ivers recorded
synchronously this 10 s noise s equence, each record is labeled after its distance d from R
0
. Then receivers R
1
to R
15
were translated one centimeter down the graduated line and the 10 s a c quisition performed again. This operation was
repeated seven times to cover the 100 cm of the diagonal with a pitch of 1 cm. We ended with a set of 106 r
d=0..105
(t)
records. The time domain cross-correlation between the receiver R
0
and the other receivers is processed afterward:
C
i
(τ) =
Z
T
0
r
0
(t)r
i
(t + τ)dt. (3)
As mentioned in the introduction, this cross-c orrelation C
d
(τ) is very similar to the impulse response h
d
(t)
obtained in the pitch-catch experiment. Strictly speaking, the impulse response equals the time-derivative of the
cross-correlation convolved by the source spectrum (which is almost flat here). Nevertheless, the time-der ivative
operation was not performed here since it does not change the spatio-temporal Fourier transform of the data. The
series of 10 0 correlations is displayed in Fig. 4 as a time-distance plot. The dispersive A
0
mode is clearly visible,
including reverberatio ns at the edges . The symmetric S
0
mode is very weak and almost invisible in this figure. In the
passive experiment, the size of the noise source is ≈1 mm large, which is q uasi-punctual compared to the A
0
and S
0
wavelength. In that case, a nd taking into account the large difference in phase velocities, the energy ratio is clearly
in favour of the A
0
mode, the S
0
mode is much weaker (and additionally less co nverged). In the active experiment
the source was 6 mm large, thus ex c iting the S
0
mode more strongly and the S
0
mode was more visible in Fig.3.
Like in the active experiment, we computed the spatio-temporal Fourier transform of the set of 100 traces C
d
(τ).
The resulting dispersion curves are plotted in Fig. 5. The A
0
mode is clearly reconstructed, the agreement between
the active, passive, and theoretical dispersion curves is perfect. Ghosts are also visible in this figure, and are due to
1) small errors in positioning the sensors 2) the presence of reflections from the edges. As to Fig. 4, the S
0
mode is
not visible. We therefore zoomed into the early part o f the correlations (Fig.6-left) and muted all the data but the
weak non-dispersive arrival. We then computed again the f-k transfor m of the data, as plotted in Fig. 6-right. The
straight line that emerges from the noise is exac tly the S
0
dispersion curve, thus demonstrating that the weak arrival
in Fig. 6-left is indeed the S
0
mode. It is important to emphasize that the weakness of the S
0
mode is not a limitation
of our correlation technique, but is connected to the noise generation.
C. Reconstruction of the amplitudes
Since now, the reconstruction of the phase (the arr ival time of the wave) of the Green function by c orrela-
tion of noise ha s been widely studied. Feeble attention was paid to the information carried by the amplitude
Reconstruction of Rayleigh-Lamb dispersion from noise 4
