1. Introduction
Guided waves have been extensively investigated in the recent years as a nondestructive testing (NDT) technique because of their advantages compared to ultrasonic testing (UT) inspection. The key benefit of this technology is the ability to interrogate the entire thickness of thin-walled structures over large areas from a single location.
During guided wave propagation, waves experience dispersion that distorts the wave shape as the wave propagates, due to a dependence of velocity on frequency [
1,
2]. Dispersion curves show the relation of phase and group velocities against frequency, for a particular geometry and material. Determination of dispersion curves is important for any guided wave application; accurate dispersion curves enable wave modes of received wavepackets to be distinguished, specific wave modes to be cancelled to clean the acquired signal, the propagation of wave modes in a particular direction using phased arrays [
3] and the spatial location of damage in the structure to be determined based on time-of-flight measurements [
4]. Guided waves are being used in commercial products, for instance to detect wall thickness losses due to corrosion for pipeline inspection in the oil and gas industry. These commercial products generate a unique wave mode which propagates along the structure avoiding the creation of multimodal excitations, as this would increase the complexity of the analysis of the signals [
5,
6]. Problems with the application of guided waves occur when the material properties or thickness of the structure to be inspected are unknown due to imprecise records, commercial confidentiality or manufacturing and maintenance uncertainty. This is a particular problem when evaluating composite structures, like wind turbine blades, where elastic constants are not often available. To inspect this kind of structure in an industrial environment, it becomes impracticable as currently there are no means of generating the dispersion curves for such situations. Therefore, there is a need to be able to calculate in situ the phase and group velocities at frequencies of interest in a quick and reliable way without requiring prior knowledge of material properties or thickness.
There has been much work published on the determination of dispersion curves describing many analytical, numerical and experimental methods. For relatively simple structures like plates or pipes, dispersion curves can be predicted using the commercially available software Disperse
®, which uses the global matrix method [
7]. Other methods, such as the transfer matrix method [
8,
9] or pseudospectral collocation method [
10], have been utilized to determine dispersion curves. Finite element (FE) [
11] and semi-analytical finite element [
12,
13] methods have been also used. All these numerical and analytical methods require knowledge of material properties and thickness of the structure to determine the dispersion curves.
Experimental methods do not suffer this limitation, since guided wave data is directly acquired from the inspected structure. The most widely used experimental technique to measure dispersion curves is the 2D fast Fourier transform (FFT) [
14]. This signal processing technique requires the acquisition of many signals along the wave propagation direction in order to carry out a double FFT in time and space. The result is a wavenumber–frequency matrix, which can be rearranged to give the phase velocity dispersion map. The acquisition of the signals from all required locations is manually prohibitive; therefore, a laser scanning vibrometer (LSV) is commonly used to automatically obtain the signals from preselected points. This device is highly sensitive and bulky, being restricted to controlled areas like laboratories [
15] and difficult to use in situ.
Other experimental techniques have been proposed recently. Harb and Yuan [
16,
17] presented a noncontact technique using an air-coupled transducer (ACT) to generate the wave mode on the plate and an LSV to acquire the propagating mode. The technique requires precise control of the incidence angle of the ultrasonic pressure from the ACT upon the surface. By relating the frequency to the incidence angle at which the wave amplitude is maximized, it is possible to calculate the phase velocity using Snell’s law. However, the technique has limited applicability outside of the laboratory. In works by Mažeika et al. [
18,
19], a zero-crossing technique is used to calculate the phase velocity based on the measurement of the position of constant phase points of the pulse corresponding to zero amplitude as a function of time. A similar approach is taken in the phase velocity method in [
20], tracking the peaks of the pulse rather than the zero crossings. To produce convincing results, both techniques require the acquisition of a large number of signals at different distances, making these techniques time-consuming. For the sake of reducing time and sampling points in space, Adams et al. [
21] presented a method using an array probe from which a time–space matrix can be created. A 2D filter is applied to the matrix to extract the phase velocities. The technique was demonstrated to be valid in simulation using FE analysis. However, for realistic probe sizes, the technique has limited experimental applications. In [
22], sparse wavenumber analysis was used to experimentally recover the dispersion curves from an aluminum plate; however, seventeen sensors were needed to deploy the technique. An alternative approach to getting dispersion curves from experimental signals is to use time–frequency representations [
23], where group velocity can be directly determined. However, the calculation of phase velocity involves the integration of group velocity requiring precise values of ω and k at the lower limit, which are not easy to obtain experimentally. For medical purposes, two techniques, SVD and LRT, were evaluated for extracting the dispersion curves of cortical bones in [
24]. Both techniques require a large number of receivers and transmitters.
Experimental calculation of bulk wave velocities in dispersive materials has been studied [
25]. Sachse and Pao used the method of phase spectral analysis, where the phase delay is calculated using the real and imaginary parts of the Fourier transform of the pulse, and then obtaining the group delay through the differentiation of the phase characteristics of the pulse with respect to frequency. Pialucha et al. presented the amplitude spectrum technique overcoming the requirement of the phase spectrum technique for the separation of successive pulses in the time domain for bulk waves [
26]. However, the phase spectral technique provided better results when pulses can be separated.
In this paper, an experimental technique for straightforward calculations of phase and group velocities of guided waves at frequencies of interest is presented. The technique requires just the acquisition of two signals spaced a few cm using conventional transducers, enabling its application on in-service structures located in poorly controlled environments. There is no requirement for prior knowledge of the material properties or thickness of the inspected structure. In
Section 2, the formulation on which the experimental technique relies is presented.
Section 3 describes in detail the methodology to extract the group delay and phase shift from experimental signals. Successive sections present validation tests using synthesized signals and simulated signals from FE analysis. Finally, experimental signals from a 3 mm-thick aluminium plate are used to validate the technique as a solution to the problem presented in the introduction.
2. Theoretical Basis
Assuming that the propagating pulse is sufficiently narrow-band that dispersion does not significantly distort the wavepacket over the propagation distance, the signal can be approximated as [
27],
where
is propagation distance,
is time,
is the angular frequency of the harmonic wave at frequency
and
is the wavenumber.
is a function defining the temporal pulse shape.
is defined such that the pulse has a peak at
.
is the group velocity at which the packet propagates, and so if the trajectory of the wavepacket peak is defined as the point whereby
, then
The wavepacket is shown schematically in
Figure 1. The harmonic part of the signal has a phase of
and propagates at the phase velocity [
6,
27].
The phase of the harmonic signal at the centre of the wavepacket
is
Rearranging and substituting for
and
gives
For nondispersive waves,
and Equation (7) gives the expected result that the phase at the wavepacket center is
for all propagation times. Where the system is dispersive there is a finite phase difference between the harmonic part of the signal and the wavepacket,
, which increases linearly with propagation time and is positive if
, negative if
. Assuming that this phase difference can be measured experimentally, Equations (4) and (5) further yield two useful equations for evaluating the phase velocity:
In
Figure 2, an illustrative example of a S
0-mode wavepacket at 150 kHz is shown. The solid line is the S
0 mode after 1.2-m propagation including the dispersion. The dashed line represents the S
0 mode after the propagation without dispersion, namely, the wavepacket was calculated using the same phase velocity for all frequencies. In this example, the phase shift is negative (–1.26 radians), which means the phase velocity is higher than the group velocity. Using (9), the calculation of the phase velocity would be 1.2 m divided by the sum of 223 µs, which is the time of the wavepacket to travel 1.2 m at group velocity, plus –1.34 µs which is the time that the signal at an angular frequency,
, takes to cover –1.26 radians. Group velocity and phase velocity are calculated using (2) and (9), respectively:
and
.
3. Methodology
In this section, the methodology for determining the phase shift and hence the phase velocity is presented. This general methodology is complemented by a description of a specific experiment setup described in
Section 6.1. The technique is based on the analysis of two signals acquired at different distances in the direction of wave propagation. One of them will be treated as the baseline and the other signal will be modified in time and phase in order to match with the baseline. The methodology is composed of different signal processing techniques which are presented in a block diagram in
Figure 3.
The experimental setup is composed of one transmitter and two receivers along the propagation direction of the wave to be measured, as shown in
Figure 4. For isotropic systems the wave velocity is independent of direction. The technique analyses the effect of propagation between the first receiver and the second receiver. Spacing between receivers of a few centimeters is selected in order to avoid large phase shifts and waveform deformations due to dispersion.
The first step in analyzing the data is the isolation of the desired wavepacket by removing the rest of the signal. The boundaries of each wavepacket are determined by studying the slope of its envelope. The boundaries are set at the closest zero slopes before and after each peak, see example of real signals in
Figure 5. Once the region containing the wavepacket is identified, the remainder of the signal is discarded. Both signals are truncated with the same boundaries in order to maintain the time difference between the wavepackets,
and
. The boundaries used are the left limit of the signal of the closest receiver to the transmitter and the second limit of the signal of the further receiver, as can be seen in
Figure 6. Afterwards, the Hilbert Transform (H(…)) [
28] is applied to both wavepackets and the magnitude extracted in order to determine the envelope of each wavepacket:
These two envelopes are then cross-correlated to determine the time delay between them:
where
is the applied time delay and
the value that maximizes the correlation. Once this time delay is known, the phase delay to the harmonic part of
that maximizes the cross-correlation between the delayed phase-shifted wavepacket
and
is calculated. This process consists of the modification in phase of the signal from the furthest receiver by a phase angle,
, and finding the phase shift that maximizes the cross correlation. The phase shift,
, is applied to
; then, the cross correlation between
and
is calculated. The optimum phase shift is the phase angle when the correlation coefficient with time delay,
, is highest.
This methodology enables the calculation of the lag time and phase shift between the two signals acquired at different distances. By knowing these three variables (distance, time and phase) and also the excitation frequency, (2) and (9) can be used to determine the group and phase velocity, respectively.
The proposed methodology in this paper is based on the acquisition of two signals spaced a few centimeters apart, since for such short distances the distortion for any wave mode is relatively small, enabling a straightforward determination of the phase shift.
4. Synthetic Signal Analysis
The first validation was carried out using synthesized signals. The signal synthesis is based on the adjustment in frequency and wavenumber of the input signal in the frequency domain. Knowing the waveform at one point, in this case the input signal at the transmitting point, and the dispersion characteristics of each wave mode, the wave modes can be reconstructed in the time domain at any distance [
29]. If
is the reconstructed wave mode at a distance
from the transmitting point,
is the time,
is the Fourier transform of the input signal and
is the wavenumber as a function of the angular frequency, then
The relation between wavenumber and angular frequency is extracted directly from the dispersion curves provided by Disperse® and introduced in the integral.
As an initial test of the signal processing method, developed signals are synthesized that are highly pure, without noise and overlapping between modes. The higher the sampling frequency is set, the better accuracy the technique provides. The sampling frequency was set at 10 MHz, since it has been observed that it provides a good balance between computational time and accuracy.
A set of signals for symmetric (S0), antisymmetric (A0) and shear horizontal (SH0) wave modes were created by evaluating equation (14) in MATLAB for two propagation distances, 30 cm and 35 cm: the analysed propagated distance () in this case is 5 cm. The analysed frequencies are from 60 to 370 kHz with steps of 10 kHz.
The results after applying the signal processing algorithm described in
Section 3 to the synthesized signals are presented in
Figure 7. The results (black circles) accurately reconstruct the theoretical values from Disperse
® (grey lines). In
Figure 8, the extracted phase shift has been represented against the frequency for the three fundamental modes. For the S
0 mode, the phase values have negative values decaying from –2 degrees at 60 kHz to –52 degrees at 370 kHz. For the A
0 mode, conversely, the phase is positive, increasing along the frequency, since the phase velocity is lower than the group velocity; from 377 degrees at 60 kHz to 673 degrees at 370 kHz. In this case, the phase values are much higher than the S
0 and SH
0 ones, because the A
0 mode is highly dispersive at these frequencies. For the SH
0 mode, the extracted phase values are 0 degrees at 60 kHz and 1 degree at 370 kHz. The SH
0 is a nondispersive mode so the phase value remains at near 0 degrees.
The sampling frequency is an important factor, since it determines the degree of resolution of the group velocity. As these examples are performed for short propagation distances, the time that the wave modes take to cover that distance is small. Thus, low sampling frequencies are not able to determine the group velocity accurately. Faster modes require higher sampling frequencies to get the same resolution as slower modes. In this case of 5 cm spacing, the propagation time for the S
0 mode at 200 kHz is 9.3 µs; a sampling frequency of 1 MHz will have a time resolution of 1 µs which yields a velocity resolution for this particular example of 556 m/s; with a sampling frequency of 10 MHz, the velocity resolution will be 58.4 m/s; and with a sampling frequency of 100 MHz, the velocity resolution will be 5.8 m/s. This resolution issue can be observed in
Figure 7; at lower velocities the curves are smoother, and at higher velocities (S
0 mode) the curves have poorer velocity resolution.
5. Finite Element Analysis
A three-dimensional FE analysis was performed in Abaqus to simulate signals. While the previous analysis required the input of the dispersion curves to create the propagated signals, this FE analysis does not require prior knowledge of the dispersion characteristics.
FE models were created simulating guided wave propagation at five different frequencies in an aluminium plate (1000 × 1000 × 3 mm). The transmitter and receivers were modelled as ideal point transducers. The transmitting point was placed at the centre of the plate, where the coordinate system was established. A force was applied in the x-direction to simulate a shear transducer. This generated S0 and A0 modes along the x-axis and the SH0 mode along the y-axis. The input signal was a 5-cycle sine with a Hanning window at central frequencies of 60, 80, 100, 120 and 140 kHz. Multiple receiver points were located on the positive x-axis and positive y-axis from the centre of the plate to the edge every 1 cm, resulting in 50 receivers at each axis. The time and spatial discretization for the finite element model was established at 10-2 µs and 1 mm, respectively, in order to ensure convergence and to obey the Nyquist sample-rate criterion. The mesh was formed by C3D8 elements.
The FE analysis software provides the results decoupled for each axis direction, so it is straightforward to evaluate each wave mode separately. In
Figure 9, the results have been computed in cylindrical coordinates to show clearly the three wave modes of propagation at each axis. In
Figure 9a, showing radial displacement, S
0 and A
0 modes are depicted, S
0 being the faster mode. In
Figure 9b showing tangential displacement, the SH
0 mode propagates along the
y-axis, and in
Figure 9c, showing out-of-plane displacement, the A
0 mode is clearly present.
Since the plate is relatively small, overlaps between wave modes and echoes from edges are produced, as can be seen in
Figure 10. This is a particular problem at lower frequencies (60 and 80 kHz) due to the length of the pulses. Therefore, an additional optimisation step has been added to minimize the error introduced by the overlapping. The FE model has 50 receiver points, every 1 cm along the axis so the five least-overlapped signals are selected for analysis. The selection is performed by analyzing the amplitude at the beginning and end of the envelope of the wavepacket of interest; the signals with the lowest amplitude at those regions are selected. Then, the signal processing described in
Section 3 is applied for all the pair combinations between the five selected receivers. Once the phase and group velocities are calculated for each combination, the velocities are averaged to give the final velocity estimates. Any outliers that are introduced during selection of the five signals are eliminated. As shown in
Figure 11, the phase velocity matches very well with the theoretical values from Disperse
®. In the group velocity graph, the results correlate well with the theoretical velocities, having a noticeable variation in the S
0 value at 60 kHz. This variation occurs at 60 kHz due to the overlap between the S
0 mode and the A
0 mode, which is more pronounced for the longer pulses at this frequency. This error does not appear for the A
0 mode because the signals used for the analysis were out-of-plane displacements, where the S
0 mode is practically imperceptible. Note that the results improve as the frequency increases, since the wavelength decreases and mode separation increases. Overlapping of pulses is an issue for this technique, as it changes the phase and waveform of the analyzed mode, highly distorting the results and impeding its application.
The results for the phase velocity are better than for the group velocity, more notably when overlapping occurs. This is because of the phase-shift term in the denominator in (9). This contribution in the phase velocity equation minimizes the erroneous calculated value of the propagation time and also provides more resolution on the calculation of the phase velocity, removing the stepped shape seen in the group velocity.
7. Conclusions
This paper has presented a new methodology for creating dispersion curves based on measuring the phase difference and time lag between two pulses acquired at different distances. Firstly, the theoretical basis of the method has been presented where the phase velocity has been related to the phase at the pulse centre for systems of limited dispersion; then, a signal processing methodology to extract the phase shift and time lag between two signals and subsequently calculate the phase and group velocities has been outlined. Three different tests were performed to validate the formulation and methodology; using synthesized signals, signals from an FE model and experimental signals. In the three cases the performance achieved agrees well with the theoretical values.
There are some limitations to the proposed method for the extraction of the phase shift to calculate the dispersion curves. Highly dispersive modes, like high-order modes, are more challenging to evaluate; to mitigate this issue, much shorter spacing distances between transducers should be established to minimize the mode-shape distortion. Overlapping between wavepackets limits the method’s applicability, since it distorts the apparent phase of the mode being analyzed. Hence, relatively large samples are required to avoid overlapping between wavepackets and edge echoes. The size of the sample can be smaller at higher frequencies. The velocity–frequency spectrum that can be evaluated using this method will be determined by the excitation mechanism used to generate the guided waves in the solid media. With appropriate experimental design, errors of less than 1% are obtained.
For future work, the proposed technique will be applied to more complex structures, such as composite plates where the elastic constants are not easy to obtain. In this case, by studying the phase and group velocities as a function of propagation direction, the full angular dependency of the dispersion curves will be obtained.