Two PZ27-piezoelectric transducers of 27~mm radius and 0.43~mm thickness have been glued on the rail foot at positions A and B. A function generator is used to generate the excitation signal at transducer A and the signal received at transducer B is acquired using an oscilloscope synchronized with the function generator and linked to a computer through a GPIB interface (Figure 2(a)).

The transmitter (A) is excited with an electric signal corresponding to a one-cycle sine pulse train at 50 kHz and with an amplitude of 10 V. The measured signals are averaged over 128 acquisitions in order to improve the signal to noise ratio (SNR). Figure 3 shows a signal measured in this configuration. It is a typical reverberated signal consisting of the direct propagation from A to B and a large number of multiple reflexions, scattering, mode conversions on the domain boundaries (edges of the rail) and possible inhomogeneities. After 80 ms, this signal is completely attenuated.

For the noise-correlation experiment, the function generator is removed and both transducers A and B are used as receivers. In order to generate acoustic noise in the rail, ten piezoelectric transducers of the same type as in the previous experiment and glued on the rail foot at positions Si are successively fed by an electrical noise generator (Figure 2(b)).

The noise correlation function is experimentally estimated from finite-time correlations averaged over a number M of acquisitions. More precisely, an estimator of the correlation function corresponding to each noise source is defined as

where T is the recorded signal duration and is the signal recorded at X in the acquisition of the measurements performed when the noise source is active.

Then, the global correlation estimator for the case with the N noise sources is simply defined as

The study will focus on the influence of the number of acquisitions M and the number of noise sources N on the convergence of the correlation estimations towards the active response.

Considering that the signal duration in the considered frequency range is about 80 ms, as seen on Figure 3, a record duration T = 100 ms is chosen.

Figure 4(a) shows the first wave packets of the positive-time part of the estimated cross-correlation functions obtained with different values of the number of

acquisitions M, and a single noise-source position (N = 1). Since the absolute amplitudes of the presented waveforms are useless here, they have been systematically normalized with respect to their maximum values. For better visualization, a zoom on an arbitrary part of the waveforms is presented on Figure 4(b). It appears that the estimated correlation begins to converge when averaging over 20 acquisitions or more. This is regarding to the Equation (1). The lager the M, the better the convergence. Therefore, a value of M = 20 will be retained as a satisfying compromise between estimation accuracy and the required time for measurement and computation.

The next step aims to determine the number of source positions to converge towards the active response. Indeed, the diffuse field assumption is a necessary condition for this convergence. Multiplicating the number of sources distributed on the medium will help to get closer to this theoretical condition and subsequently ensure a better convergence. It is known that the temporal symmetry of the noise correlation functions is a good indicator of the quality of this convergence. The reason for that is the acoustic reciprocity principle [22] , which implies that the Green’s function from A to B is the same as the one from B to A. Therefore the symmetry of the estimated correlation functions for different values of the number N of noise sources will be checked. To that end, the positive-time and negative-time parts of are compared on Figures 5.

For N = 2, Figure 5(a), it is clear that the positive-time and negative-time parts of the estimated cross-correlation differ significantly. For N = 5, Figure 5(b), a better agreement between both parts is observed and, finally for N = 10, Figure 5(c), a clear similarity is obtained. Thus, in that case, the estimated cross-correlation function satisfies the reciprocity principle, as does the active response. It will be verified now that this case corresponds to a satisfying comparison between both.

The aim here is to compare this passively obtained response to the active signal. We define the quantity

where is the excitation signal used in the active experiment, and is the positive-time part of the correlation estimator.

In Figure 6(a), obtained with the 10 noise sources is compared to the active response. Here again, the presented waveforms have been normalized with respect to their maximum values. Wave-packets appear correctly reconstructed but a clear phase shift is observed. This phase shift can be interpreted as the effect of the (unknown) transduction impulse response (electro-mechanical and reciprocal conversions of the transducers). An empirical compensation of this phase shift, as shown in Figure 6(a) confirms that the noise-correlation response constitutes indeed a very good estimation (or “reconstruction”) of the active response. The slight differences between both responses can be interpreted as the reconstruction error, also referred to as “correlation residue”.

In the remainder of this paper, the cross-correlation functions will thus be estimated using N = 10 noise source positions and M = 20 acquisitions for each position.

The experimental setup used to perform the measurements presented here is the same as in the previous section (Figure 2). In order to allow repeatable and non-destructive experiments, we used a local change that did not damage irreversibly the rail sample. Therefore what will be referred to as the “defect” in the remaining of this paper is a local pressure applied to the rail foot by a locking plier as shown in Figure 7. Such static mechanical action will modify locally the guided wave propagation sufficiently to introduce reflection and transmission phenomena comparable to the effect of an actual damage such as a hole or a crack. This defect is placed at a distance d = 5 cm from the measurement point B as shown in Figure 8.

Waveform differencing (also known as baseline subtraction) is used throughout the experimental measurements presented in this section. Therefore, active and noise-cor- relation measurements are first performed without defect. The obtained signals constitute the reference (or baseline) waveforms and (or equivalently), respectively. Second, the same measurements are performed with the presence of the defect and corresponding waveforms are noted and. Finally, the reference waveforms are subtracted from those obtained with the defect, yielding

the differential active response, and

the differential noise-correlation response (or again equivalently ).

These differential waveforms are representative of the defect signature. The early wave-packets of the measured, normalized with respect to the maximum value of, are shown in Figure 9(a). The first wave packet (beginning just before 0.1 ms) corresponds to the first reflection on the defect, whereas the other ones correspond to combinations of transmitted or reflected waveforms on the defect and reflections at the rail sample extremities.

The differential noise-correlation response, obtained with the experimental parameters previously determined (M = 20, N = 10) and normalized with respect to, is shown in Figure 9(b). Comparing Figure 9(a) and Figure 9(b), some similarities between both waveforms are visible. Yet, significant differences are also noticeable. In particular, a first wave-packet of significant amplitude is apparent between 0 and 0.08 ms in the differential noise-correlation response. This wave-packet makes the one associated to the reflection on the defect far less evident than it is in the active differential response. This can be explained by the fact that the reconstruction of the active responses by the noise correlation process relies on all reverberated wave-packets. Since

the introduction of any local change in the medium will both modify all wave-packets going through it and add reflected ones, it will entail changes in the reconstruction process itself. This means that in, the identifiable defect signature (temporally related to the defect position, as in) will be mixed with differences in the correlation residues that have no particular localization in time.

Though this shows obviously that noise correlation is clearly sensitive to a change in the medium, the difficulty to relate the differential noise-correlation response to the position might be a problem. Indeed, the need to discriminate a local defect from a change in the boundary conditions at the rail extremities or a global material variation such as induced by temperature changes or other effects might be crucial in a realistic situation.

Therefore, it is important to be able to extract some information coherently related to the defect position. To that end, we propose in the next section an experiment based on several acquisitions where multiple defect-sensor distances are considered.

A rather natural way to exhibit information related to the position of either a source or a scatterer is to interrogate several propagation distances. Therefore, the same kind of measurements as in the previous paragraph will be performed here, but with varying defect-transducer distances. For practical reasons, instead of moving the transducers or using several ones at different positions (as would be logical in a real SHM situation), we will move the local “defect” constituted by the locking plier pressure.

The transducers used are the same as previously (A and B) and the noise-correlation parameters are kept unchanged (N = 10 source positions, M = 20 acquisitions per source). A set of active signals and correlation responses for 24 defect positions are recorded. The positions are defined by, where is the distance between defect and transducer B (see Figure 10). The initial distance d is 5 cm and the displacement step is 1 cm.

By subtracting here again the reference signal and correlation responses without defect to the newly recorded waveforms, 24 differential active responses and 24 differential noise-correlation responses are obtained. The upper script indicates the defect position index. These waveforms are presented in Figure 11(a) and Figure 11(b), respectively, in the form of spatial-temporal information. Interpretation of Figure 11(a) is straightforward: the first wave-packet corresponding to the reflection on the defect has a propagation time proportional to the distance, which manifests itself by a neat slope (between approximately 0.1 ms at 5 cm and 0.25 ms at 29 cm) on the spatial-temporal image.

Figure 11(b) is visually noisier. The first wave-packet (before 0.05 ms) does not apparently propagate and, though it is evidently related to the defect’s presence, it is independent from its position. This confirms the remark made previously about Figure 9(b). A propagative component is also visible, though not very clearly, with the same slope as in the actively-obtained image. Note that we have represented here directly and not the time-derivate and convoluted version, which is only useful for precise quantitative comparisons with the active responses. For identification of the propagative component, is as suitable as.

In order to better emphasize the presence of this propagative component related to the first reflection on the defect, a two-dimensional (spatial-temporal) Fourier transform (2D-FT) has been applied. This leads to a representation in the frequency-wave- number domain that can be directly compared to the dispersion curves of elastic guided waves in the rail.

Figure 12(a) shows the 2D-FT of the differential active responses. The dispersion curves, computed using the Semi-Analytical Finite Element (SAFE) technique [23] , have been superimposed for easier interpretation of the frequency-wavenumber image. The strongest propagative component (corresponding to the slope visible on Figure 11(a) and commented above) appears to match with the guided mode of highest wavenumber. Logically, the mode shape of this mode happens to be essentially concentrated in the rail foot, where the transducers and the defect are located.

The same kind of interpretation can be made for Figure 12(b), where the 2D-FT image of the differential noise-correlation responses is compared to the same dispersion curves. First, it can be observed that high-amplitude spots are present around zero wavenumber. This corresponds to the non-propagative components resulting from the correlation residues as discussed above. However, the propagative component related to the same guided mode as in the active experiment is also clearly visible.

This confirms that the wave-defect interaction process, ordinarily exhibited by actively launching incident waves, manifests itself in the same way in noise-correlation measurements. This means in particular that information related to the defect-sensor

distances (and consequently to the defect position) is effectively present in, and eventually extractible from, an ambient noise.