Reconstruction of Rayleigh-Lamb dispersion spectrum based on noise obtained from an air-jet forcing
aa r X i v : . [ phy s i c s . c l a ss - ph ] O c t 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: [email protected] (Dated: October 22, 2018)
The time-domain cross-correlation of incoherent and random noise recorded by a series ofpassive sensors contains the impulse response of the medium between these sensors. By usingnoise generated by a can of compressed air sprayed on the surface of a plexiglass plate, weare able to reconstruct not only the time of flight but the whole waveforms between thesensors. From the reconstruction of the direct A and S waves, we derive the dispersioncurves of the flexural waves, thus estimating the mechanical properties of the material withouta conventional electromechanical source. The dense array of receivers employed here allow aprecise frequency-wavenumber study of flexural waves, along with a thorough evaluation of therate of convergence of the correlation with respect to the record length, the frequency, and thedistance between the receivers. The reconstruction of the actual amplitude and attenuation ofthe 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 specific propagation modes, denotedLamb waves [Royer and Dieulesaint (2000); Viktorov (1967)], which are related to the traction free condition on bothsides of the medium. Depending on the purpose of the experiment, different measurement configurations have beenproposed. From the dispersion of Lamb waves, obtained from pitch-catch measures repeated for different ranges, onehas access to the velocities of bulk waves and to the thickness of the plate ( see for instance Gao et al. (2003)). Otherpulse-echo (or impact-echo) techniques have been proposed to assess the mechanical properties of the plate, based onthe simple and multiple reflection of bulk waves within the plate [Krautkr¨amer and Krautkr¨amer (1990)] or on theresonance of high order Lamb modes [Clorennec et al. (2007)]. Some studies also concern the dynamic evaluation offatigue 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 acquired at different dates and are eventually compared to each othersto monitor the medium.All these techniques require the use of controlled sources and receivers. In the following, they are referredto as ”active” experiments. Another idea has undergone a large development after the seminal experiments ofWeaver and Lobkis (2001) (see for instance the review of Wapenaar and Fokkema (2006), Weaver and Lobkis (2006)or Larose (2006)). By cross-correlating the incoherent noise recorded by two passive sensors, Weaver and Lobkisdemonstrated that one could reconstruct the impulse response of the medium as if a source was placed at one sensor.This noise correlation technique (also referred to as ”passive imaging”, or ”seismic interferometry” [Schuster et al. (2004)]) requires the use of synchronized sensors, and has the advantage of eliminating any controlled source.In section II of our paper, we compare the dispersion curves of flexural waves (Lamb waves in the low frequencyregime) in a plexiglass plate obtained by an active (pitch-catch) experiment to the ones obtained in a passive exper-iment. During the passive acquisition, we deployed synchronised receivers only, and used a random noise source: a1 mm thick high pressure air jet produced by a can of compressed air [McBride and Hutchison (1976)]. The experimentconducted in a plexiglass plate show at the same time: dispersive ( A ) and non dispersive ( S ) waves, reverberations,and absorption. Since the earth’s crust has similar properties for seismic waves, we believe that a plexiglass plate atkHz frequencies is a good candidate to built small scale seismic analogous experiments. The question of reconstruct-ing not only the phase, but also the amplitude of the wave is addressed at the end of part II. This is of interest inseismology, where the role of absorption and attenuation in the correlation has not yet been subject to experimentalinvestigation. In section III, we analyze the rate of convergence of the correlations to the real impulse response. Therole of the record length, the central frequency, and the distance between the receivers is investigated. II. EXPERIMENTAL SETUP AND DISPERSION CURVES
To study actively and passively the dispersion of flexural waves, we built a laboratory experiment using a 1 . m × . m large, 0.6 cm thick plexiglass plate. The plate is laid on an open steel frame that supports the edges of the plate butReconstruction of Rayleigh-Lamb dispersion from noise 1 IG. 1. Experimental setup. (a) In the active experiment (pitch-catch configuration), a broadband piezoelectric source S emitsa chirp that is sensed by a vertical accelerometer R placed at a distance d . (b) In the passive experiment, a turbulent jetproduced by a can of compressed air generates a white random noise recorded simultaneously at all sensors R i . The jet israndomly sprayed over the area in gray. leaves free the upper and lower sides. The traction-free condition is therefore achieved on both horizontal sides. Theresulting dispersion relation that connects the pulsation ω and the wave-vector k reads: ω c s = 4 k q (cid:18) − p tan( ph/ γ ) q tan( qh/ γ ) (cid:19) (1)where p = ω c p − k and q = ω c s − k ; c s (resp. c p ) is the shear (resp. compressional) velocity, and h is the thicknessof the plate. The parameter γ equals 0 for symmetric ( S ) modes, and π/ A ) modes. In the lowfrequency regime, only two modes are solutions: they are labeled A and S .For both acquisitions we used broad-band miniature (3 mm radius) accelerometers (ref. ◦ C).
A. Active experiment
In the active experiment, a source S (a piezoelectric polymer) is in a corner of the plate, approximately 30 cmfrom 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 repeated thisoperation 100 times to cover 100 cm of the plate. The dynamic impulse response h d ( t ) of the plate is reconstructedfor 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 , ,
80 cm are displayed in Fig. 2.The dispersive A mode is dominating the record (a), but the non-dispersive S wave (b) and reverberations fromthe edges of the plate (c) are also visible. The transit time in the plate is of the order of a few milliseconds. Theabsorption 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 aretherefore dominated by ballistic waves, but waves reverberated from the plate boundaries are also visible.From the set of 100 impulse responses h d ( t ), a spatio-temporal Fourier (f-k) transform was applied. The resultingdispersion curves are displayed in Fig. 3, showing the dispersive A mode and the weaker non-dispersive S mode.Theoretical dispersion curves are numerically obtained after Eq. 1 (dots), they perfectly fit the experimental datafor c p = 3130 m/s and c s = 1310 m/s. The dispersion curves are widely used to evaluate the mechanical propertiesof the plate, including thickness, presence of flaws... Nevertheless, the source-receiver configuration is sometimes ata disadvantage. As mentioned in the introduction, instead of using the conventional source-receiver configuration, itis possible to take advantage of random elastic noise to reconstruct the impulse response h d ( t ) between two sensors.Reconstruction of Rayleigh-Lamb dispersion from noise 2
101 d=10 cm(a)−101 (a)(b) d=40 cm N o r m a li z ed A cc e l e r a t i on FIG. 2. Source-receiver impulse response h d ( t ) for different distances d . (a) Dispersive anti-symmetric A mode propagating atvelocities approximately ranging from 150 to 900 m/s. (b) Non-dispersive S mode propagating at ≈ h/ λ F r equen cy ( k H z ) A modeS mode 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 receivers separated by 7 cm from each other onthe graduated line. At the former position of the source, we placed the accelerometer R that was kept fixed allthrough the experiment. As proposed by Sabra et al. (2007), we used the noise generated by a turbulent flow. Asa 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 theplate and the former active source. Note that the precise knowledge of the noise source is not necessary in the passiveexperiment. We also deployed an array of 16 receivers that allow for precise frequency-wave number analysis, andworked at higher frequencies, meaning a much thiner spatial resolution. We focused our attention on the A and S 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, sprayingReconstruction of Rayleigh-Lamb dispersion from noise 3 ensor position d (cm) T i m e τ ( m s )
20 40 60 80 1001234 −1−0.8−0.6−0.4−0.200.20.40.60.81
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 receivers recordedsynchronously this 10 s noise sequence, each record is labeled after its distance d from R . Then receivers R to R were translated one centimeter down the graduated line and the 10 s acquisition performed again. This operation wasrepeated 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 .. ( t )records. The time domain cross-correlation between the receiver R and the other receivers is processed afterward: C i ( τ ) = Z T r ( t ) r i ( t + τ ) dt. (3)As mentioned in the introduction, this cross-correlation 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 thecross-correlation convolved by the source spectrum (which is almost flat here). Nevertheless, the time-derivativeoperation was not performed here since it does not change the spatio-temporal Fourier transform of the data. Theseries of 100 correlations is displayed in Fig. 4 as a time-distance plot. The dispersive A mode is clearly visible,including reverberations at the edges. The symmetric S mode is very weak and almost invisible in this figure. In thepassive experiment, the size of the noise source is ≈ A and S wavelength. In that case, and taking into account the large difference in phase velocities, the energy ratio is clearlyin favour of the A mode, the S mode is much weaker (and additionally less converged). In the active experimentthe source was 6 mm large, thus exciting the S mode more strongly and the S 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 mode is clearly reconstructed, the agreement betweenthe active, passive, and theoretical dispersion curves is perfect. Ghosts are also visible in this figure, and are due to1) small errors in positioning the sensors 2) the presence of reflections from the edges. As to Fig. 4, the S mode isnot visible. We therefore zoomed into the early part of the correlations (Fig.6-left) and muted all the data but theweak non-dispersive arrival. We then computed again the f-k transform of the data, as plotted in Fig. 6-right. Thestraight line that emerges from the noise is exactly the S dispersion curve, thus demonstrating that the weak arrivalin Fig. 6-left is indeed the S mode. It is important to emphasize that the weakness of the S mode is not a limitationof our correlation technique, but is connected to the noise generation. C. Reconstruction of the amplitudes
Since now, the reconstruction of the phase (the arrival time of the wave) of the Green function by correla-tion of noise has been widely studied. Feeble attention was paid to the information carried by the amplitudeReconstruction of Rayleigh-Lamb dispersion from noise 4 / λ F r equen cy ( k H z ) FIG. 5. Spatio-temporal (f-k) Fourier transform applied to the passive data (linear color bar, normalized amplitude). Thedispersion curve of the A mode is perfectly matching the active experiment. S mode is very weak, probably feebly excitedby the point-like noise source. The dots show the theoretical dispersion curves as obtained in the active experiment. Distance d (cm) T i m e τ ( m s )
20 40 60 80 10000.20.4 h/ λ F r equen cy ( k H z ) FIG. 6. Left: zoom into the early part of the passive data (saturated color bar). The non-dispersive S wave is pointed by thearrows. Right: f-k transform applied to the passive data after muting the A mode arrivals (set to zero). The S non-dispersivemode is now visible. of the reconstructed GF. We can report the first experiments of Weaver and Lobkis (2001) who noted thatboth the phase and amplitude of the signals were passively reconstructed, and also Larose et al. (2006) whoused this amplitude information to study weak localisation without a source. In Sec. II.B, we have seen that thephase information in the passive experiment perfectly matches the active data, what about the amplitude of the wave?To see if the amplitude decay is recovered using correlations, we plot on Fig. 7 the amplitude decay obtainedwith an active experiment (stars). In a homogeneous and lossless 2D plate, one would expect a decrease of theamplitude as ∝ /d . . Because of dispersion and the bandwidth used here (10 kHz), the exponent is slightlygreater ( ∝ /d . ). To be more rigorous, absorption adds another exponential decay. From the active experiments,we found that the absorption time was about 260 ms at 60 kHz and 2000 ms at 1 kHz (similar to Safaeinili et al. (1996)).On Fig. 7, we also plot the maximum of the amplitude of the correlations versus the distance d between the sensors(circles). Active and passive amplitudes are comparable: the geometrical spreading of the wave is well recovered.Nevertheless, small discrepancies are visible and are due the imperfect or irregular coupling between the sensorsand the plate. Estimating the absorption in the plate from the passive data is therefore more speculative with thisReconstruction of Rayleigh-Lamb dispersion from noise 5 −1 −1 −1 −1 −1 −1 FIG. 7. Amplitude decay obtained in an active experiment (stars) and by correlation (circles) without additional processing,and with whitening (triangles). Attenuation includes geometrical spreading and absorption. experimental setup and will be subject to further investigations.Nevertheless, in several applications [Roux et al. (2004); Sabra et al. (2005a); Shapiro et al. (2005)], an operationis performed prior to the correlation in order to balance the contribution of all frequencies. The whitening operation,performed here in the 1-60 kHz frequency range, reads:˜ r d ( t ) = IF F T (cid:18)
F F T ( r d ( t )) | F F T ( r d ( t )) | (cid:19) , (4)then the correlation ˜ r ( t ) × ˜ r d ( t ) is evaluated. We compared the active amplitude with the amplitude of thecorrelations using whitened data (triangles in Fig. 7). This time, though a decay in the correlation is visible, we donot recover the decay of the actual GF. Note that 1-bit processing (correlate the sign of the records, see for instanceLarose et al. (2004)) give similar disappointing results. Since whitening (or 1-bit) does not preserve the amplitudeof the records, we stress that correlations will unlikely give the actual attenuation. This is of practical interest forapplications like seismology. To summarize, in order to recover the amplitude by correlating the incoherent noise, wesuggest that each record be filtered in a narrow frequency band, and then correlated without additional processing.The amplitude decay would therefore be estimated for each frequency, and attenuation derived. On the other hand,the whitening (or 1-bit) processing is valuable when the only phase information is targeted (tomography for instance). III. RATE OF CONVERGENCE OF CORRELATIONS
Several authors [Campillo and Paul (2003); Derode et al. (2003)] have experimentally noticed that the correlations C d ( τ ) do not only contain the impulse response (the ”signal”) between the receivers but also show residual fluctua-tions that blur the weaker part of the signal. As soon as residual fluctuations are negligible compared to the referenceimpulse response, we consider that the correlations have converged. The convergence of correlations has been theoret-ically studied by Weaver and Lobkis (2005) and Sabra et al. (2005b). We propose here to check the validity of theirpredictions with our experimental data. Of course, theoretical works apply to a perfectly diffuse wavefield. Here, theReconstruction of Rayleigh-Lamb dispersion from noise 6 S i gna l − t o − N o i s e R a t i o d=10cm d=40cm d=80cmExperimenttheoretical fit FIG. 8. Signal-to-Noise Ratio of the correlation versus the duration T of the records. The theoretical fit for each distance d isof the form α √ T with α the fit parameter. noise is generated over a delimited area. Nevertheless, as developed in section III.D, reverberations compensate theuneven distribution of source, and we believe that previous theories should apply to our experiment. A. Convergence with time T To evaluate the degree of convergence of the correlations toward the real impulse response between the sensors, wesplit each noise record into N=1000 sub-records. Each sub-record lasts δt =10 ms. Correlations are processed foreach sub-record and each couple of receivers R − R d . We then compute the amplitude of the residual fluctuations: σ d ( N, τ ) = s h C d ( τ ) i − h C d ( τ ) i N − h . i represents an average over N sub-records. The fluctuations were found to vary weakly with time τ , wetherefore average σ d ( T = N δt, τ ) over τ to get a more robust estimation of the fluctuations σ d ( T ). To evaluate the”signal-to-noise ratio” (SNR) of the correlation, the maximum of each correlation is divided by the residual fluctuations σ d . Experimental results are plotted in Fig. 8 for three distances d and for the whole 1-60 kHz frequency range. Asnoticed theoretically and experimentally by several authors [Sabra et al. (2005b); Weaver and Lobkis (2005)], theSNR is found to increase like: SN R = α √ T (6)where α is the fit coefficient. This means that the longer the records, the better the reconstruction. The SNRincreases as the square root of the amount of ”information grain” contained in the records. This amount correspondsto the quantity T ∆ f (where ∆ f is the frequency bandwidth) and represents the number of uncorrelated pieces ofinformation transported by the waves [Derode et al. (1999); Larose et al. (2004)]. In our experiment, a few secondswas enough to get the direct arrival of the A mode.Another point visible in Fig. 8 is the decrease of the coefficient α with the distance d between the pair of receivers.This means that the passive reconstruction of the impulse response is harder when the receivers are far apart. Thiswas noticed in seismology for instance [Paul et al. (2005); Shapiro and Campillo (2004)]. A more precise estimationof the convergence with the distance d is developed in the following subsection. B. Convergence with distance d To study the dependence of the convergence rate on the distance d between the passive sensors, each coefficient α was evaluated for each SNR curves (as in Fig. 8) in the 1-60 kHz frequency range. The resulting α ( d ) is plotted inFig. 9 and is delimited by two theoretical curves of the form:Reconstruction of Rayleigh-Lamb dispersion from noise 7
20 40 60 80 10010 distance d (cm) c on v e r gen c e α ( d ) ( a r b i t r a r y un i t s ) experiment1/d β fit FIG. 9. The fit coefficient α of the SNR is plotted versus the distance d between the two correlated receivers. c on v e r gen c e α ( f ) ( a r b i t r a r y un i t s ) error barexperiment1/f .5 fit FIG. 10. The fit coefficient α of the SNR is plotted versus the central frequency f of the records. α ∝ /d β . (7)with 0 . < β < .
65. At first sight, our result is similar to previous theoretical works that predicted a SNR evolution as p /d in a 2D space. Nevertheless, the theory was developed for a non-dispersive medium, where the wave spreadingfactor is p /d . In our dispersive plate, the exponent is slightly greater ( ≈ .
6, see section II.A). The SNR decay seemsto be driven by the decay of the actual Green function. note that the role of the absorption can not be evaluated inour experiment, and will be subject to further investigations.
C. Convergence with frequency f It is now commonly acknowledged that high frequencies are harder to reconstruct by cross correlation than lowfrequencies. We now propose to quantitatively evaluate the role of the central frequency f of the record in the rateof convergence of the correlations. To that end, we performed several whitening operations in different frequencyband (every 5 kHz, ranging from 1 to 60 kHz). Fluctuations where then evaluated for each frequency band and eachsub-record. The experimental SNR was fitted by α ( f ) √ T , with α the fit parameter plotted in Fig. 10. The predictionthat SN R ∝ / √ f is well recovered. Reconstruction of Rayleigh-Lamb dispersion from noise 8 IG. 11. Rate of convergence of the correlation (SNR) with the record time T. The air-jet is randomly sprayed over the darkgray areas, doted line is the array of receivers. Upper left plot: sources located on the side of the array of receivers. Lowerright plot: sources located in others places.
D. Convergence with the noise sources position
Several theoretical approaches have been proposed to model the reconstruction of the GF between passivesensors . Some invoke a perfect distribution of sources surrounding the medium to image [Derode et al. (2003);Wapenaar (2004)]. Others refer to a perfect diffuse wavefield, obtained either with scatterers or with thermal noise[Weaver and Lobkis (2001)]. Our experiment does not correspond to the first case, since the noise is generated inlimited area. Does it correspond to the diffuse case? Because ballistic waves strongly dominate the records, this isnot guaranteed and should be checked thoroughly!To answer this point, we chose to spray over an area in the side of the array of receivers (see the dark gray areain Fig. 11). If only ballistic waves were present, the direct waves along the receiver line would not be reconstructed.This was not the case here: after an integration over a 35 s long record, we could reconstruct the same waveforms asin he active experiment. The only difference is the rate of convergence of the correlations: it was found to be muchslower (see SNR in Fig. 11 in log-log scale). This proves the elastic wavefield to be (at least slightly) diffuse in theplate employed in the experiment. To be more precise, the ballistic and the diffusive regime coexist in our plate.Right after a noise source occurs, the propagating wave paquet is in the ballistic regime. It turns into the diffusiveregime after a dozen of reverberation. The first regime dominates the record; the second is weaker but not completelynegligible. This last experiment is also a clear evidence that even with imperfectly diffusive wavefield, correlations ofincoherent noise yield the Green function. This is of importance in applications like seismology where the medium isnot always perfectly diffusive and the noise sources are not perfectly distributed. Figure 11 also demonstrates thatwhen the reconstruction of the direct waves is expected, spraying over an area located in the direction of the arrayrequires much less data than any other configuration.
IV. CONCLUSION AND DISCUSSION
In this article, we presented two experiments using Lamb waves detected by a series of accelerometers fixed on aplate. In the first experiment, a conventional source-receiver configuration was employed to construct the dispersioncurves of the A and S flexural waves. In the second experiment, we only used receivers: we recorded the noisegenerated by a can of compressed air whose turbulent stream generates random excitations. Contrarily to an activeexperiment, the advantage here was that we neither needed to know the position of the noise source, nor to employany electronics for the emission. By correlating the noise records, we reconstructed the impulse response between thesensors, and recovered the same dispersion curves as in the active experiment (except for the low frequency S modethat was not excited by the thin turbulent jet).The amplitude decay of the actual A mode with distance was successfully retrieved when correlating the rawReconstruction of Rayleigh-Lamb dispersion from noise 9ecords, but was not retrieved using either whitened or 1-bit records. This is of importance for applications wherepassive reconstruction of the attenuation of the media is envisioned. To summarize, whitening (or 1-bit) the data ishelpful to reconstruct the phase of the GF (for imaging applications or f-k analysis), but should not be employed toreconstruct the amplitude.Using our dense array of receivers, we also carefully studied the fluctuations of the correlations, which are con-nected to the rate of convergence of the correlations to the real impulse response. The Signal-to-Noise Ratio (orcorrelation-to-fluctuation ratio) obtained in our experiments is in agreement with previous studies [Sabra et al. (2005b); Weaver and Lobkis (2005)], both qualitatively and quantitatively. We showed that the reconstruction ofthe impulse response is better when: 1) we use long record (more data); 2) we employ close receivers; 3) we work atlow frequency. The resulting SNR curves were best fitted by: SN R = B s T ∆ f cd . f (8)The parameter B , as defined by Weaver and Lobkis (2005) was found to be [0 . ± .
02] (we take c=350m/s asa mean value for A mode), which is much greater than seismic experiments [Shapiro and Campillo (2004)]. Thisis in part due by the very uneven distribution of sources in our experiment, resulting in an anisotropic wave fluxthat accelerates the convergence of the correlation to the direct A wave. This parameter B was found to be tentimes smaller while spraying on one side of the array of receiver. Concerning the dependence with distance d , thedecay matches the amplitude decay of the actual Green function. We emphasize that it is not a trivial term whendispersive waves are considered. A proper prediction for the SNR should account for the dispersion and the absorption.Practically speaking, in the 1-60 kHz frequency range, we obtained very good reconstruction of the impulse re-sponses (SNR ≥
5) for distances up to 100 cm with less than ten seconds of noise. This estimation is fast enoughto be repeated continuously and provides a route for the passive monitoring of structures and materials. At seis-mic frequencies, correlations of ambient noise are already used to monitor active volcanoes [Brenguier et al. (2007);Sabra et al. (2006); Sens-Sch¨onfelder and Wegler (2006)]. As suggested by Sabra et al. (2007) in a recent paper, weforesee similar applications in the field of on board passive structural health monitoring, in noisy environments suchas aircrafts or ground vehicles, and also on civil engineering structures as bridges and buildings.
ACKNOWLEDGEMENTS
We are thankful to D. Anache-Menier, P. Gouedard, Stefan Hiemer, L. Margerin, Adam Naylor, B. Van Tiggelenand R. L. Weaver for fruitful discussions and experimental help. Jean-Paul Masson is strongly acknowledged fortechnical help in designing the experiment. This work was funded by a french ANR ”chaire d’excellence 2005” grant.
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