ON LAMB MODES AS A FUNCTION OF ACOUSTIC EMISSION SOURCE RISE TIME #

Transcription

1 ON LAMB MODES AS A FUNCTION OF ACOUSTIC EMISSION SOURCE RISE TIME # M. A. HAMSTAD National Institute of Standards and Technology, Materials Reliability Division (853), 325 Broadway, Boulder, CO and University of Denver, Department of Mechanical and Materials Engineering, Denver, CO Abstract A study was carried out to examine Lamb-wave modal content as a function of the acoustic emission (AE) source rise-time. The study used a validated finite-element code to model the source operation and subsequent wave propagation up to a distance of 480 mm in a 4.7-mm thick aluminum plate with large transverse dimensions. To obtain the large propagation distances with sufficient transverse dimensions so that plate edge reflections did not arrive during the significant part of the direct AE signal, an axisymmetric code was used. The buried dipole AE sources were located at three different depths below the top surface of the plate, where the pseudo-ae sensors were located. These sensors provided the out-of-plane displacement as a function of time. The rise-times for the different finite-element runs varied from 0.5 µs to 15 µs. The resulting data were high-pass filtered at 40 khz and re-sampled with a time step of 0.1 µs. The intense portions of the Lamb modes were determined by use of a Choi-Williams transform (CWT) for the range of source rise-times, three different source depths and the signal propagation distances. Higher Lamb modes were observed in the CWT results for the shortest rise-times, but fundamental mode frequencies still dominated for all the rise-time cases and different source depths. Thus, only the fundamental modes need to be considered in the determination of accurate signal arrival times. Keywords: AE source rise-times, Choi-Williams transform, Lamb modes, thin plate Introduction In the analysis of burst-type acoustic emission (AE) signals obtained with wideband sensors (nearly flat frequency response), the use of intense Lamb mode/frequency combinations has produced an improvement in the accuracy of source location calculations [1-3]. This result comes from the determination of signal arrival times that are independent of a voltage threshold and that correspond to the same group velocity at each sensor in an array. A previous paper [4] used finite-element modeling to study the effects of different source rise-times on the peak amplitudes and frequency content of the far-field AE signals in a plate. The results of that study demonstrated a large drop in the peak amplitude as the source rise-time increased. The purpose of this paper is to extend the previous examination to the modal content of the out-of-plane AE signals that result from sources with different rise-times. Finite-element modeling (FEM) offers several key advantages: i) generation of AE signals from buried selfequilibrated sources at different depths (below the top surface) in a plate, ii) generation of AE signals with specific source rise-times; iii) obtaining the exact out-of-plane displacement versus # Contribution of the U.S. National Institute of Standards and Technology; not subject to copyright in the United States. Trade and company names are included only for complete scientific/technical description; endorsement is neither intended nor implied. J. Acoustic Emission, 28 (2010) Acoustic Emission Group

2 time from a pseudo sensor (a perfect, wideband, point-contact sensor with no resonances), iv) obtaining the sensor results at different known propagation distances from the source and v) use of a specimen that did not produce reflections from the specimen edges arriving at the sensors during the major portion of the direct arrival of the AE displacement waves from the source. Relevant Details on the Specimen Domain and Finite-element Modeling A circular aluminum plate, 4.7-mm thick with a radius of nearly 1000 mm, served as the sample domain. The large in-plane dimension allowed propagation distances of up to 480 mm such that the direct arrival of the significant portion of the AE waves was not altered by reflections from the plate edge. A total of nine selected source rise-times varied from 0.5 µs to 15 µs. The basis for the selection of this range in rise-times was previously presented [4]. Because the 0.5-µs rise-time generated frequencies well above the frequency sensitivity range of even special AE sensors, shorter rise-times were not examined. The sources in the continuous mesh domain were dipoles (self-equilibrating forces of two oppositely directed single-cell monopoles with one cell between them) by use of the equivalent body force concept for displacement discontinuities [5]. The forces (each monopole of the dipole had a force of 1 N) were applied with a cosine bell temporal time dependence T(t) given by 0 for t < 0, T(t) = ( cos [ t / ] ) for 0 t, and (1) 1 for t >, where was the source rise-time. The same very small cell size (54 µm) and time step (7.7 ns) was used for all the FEM runs. These values were chosen to meet the requirements of the validated finite-element code [6] for the shortest rise-time source. An axisymmetric version of the code was used to enable carrying out the modeling in such a large domain with the very small cells and time steps with reasonable computing resources. The finite-element code was run for a small period of time beyond 300 µs from the start of the operation of the AE source. The dipole sources were oriented out of plane, with the center of the dipole located at one of three different depths below the top surface of the plate where the sensors were located. The depths were 2.35 mm (mid-plane), 1.32 mm (mid-depth) and 0.24 mm (near the plate top surface). The entire AE signals were numerically processed with a 40 khz (eight-pole) high-pass Butterworth filter, followed by resampling from the original time step to 0.1 µs per point (comparable to the digitization rates often used in AE measurement systems). The aluminum material properties used for the FEM calculations were bulk longitudinal and shear velocities of 6320 m/s and 3100 m/s, respectively, and a density of 2.7 kg/m 3. Determination of Frequency/Time Content Choi-Williams transform (CWT) results were used to enhance the identification of the AE signal Lamb modes, to indicate highly excited frequency regions within the modes, and to obtain the CWT magnitudes (coefficients) in these regions. The CWT results were obtained by use of the AGU-Vallen Wavelet freeware [7] with the key parameter settings being maximum frequency up to 2000 khz, frequency resolution = 2.4 khz, 112 terms in the damping summation, and an exponential damping parameter of 20. The time-domain setting of the number of samples was up to about 3000 for the furthest propagation distance. In the color CWT figures, red indicates the highest-intensity region of the CWT coefficients. By use of the known propagation distances, the appropriate group velocity curves [8] were superimposed on the CWT results to identify the Lamb modes. In some cases, the relevant group-velocity curves for the figures in this 42

3 Fig. 1. Comparison of CWT results for in-plane versus out-of-plane source orientation. Scaling of 0 to 1000 khz for vertical axis and 0 to 150 µs for horizontal axis. paper are shown at the bottom of the CWT results, so they do not hide the intense regions in these results. Out-of-plane versus In-plane Source Orientation Based on the CWT results from the three-dimensional FEM calculations shown in Fig. 1, the use of an out-of-plane dipole source orientation provides frequency/time results similar to those from in-plane dipoles for sources at different depths. The results in Fig. 1, for the same material and plate thickness, were for a source rise-time of 1.5 µs and a propagation distance of 180 mm. Again, these CWT results were calculated from 40-kHz high-passed, out-of-plane displacement data. Even with the use of a smaller domain and a longer rise-time (allowing a larger cell size and longer time steps), considerable computer resources were required to make the threedimensional runs. The results described above indicated that the out-of-plane sources could be used in this study, resulting in a significant reduction in required computer resources. Modal Content versus Source Rise Time Mid-plane Source Location Figures 2 and 3 show the CWT results (first columns) as well as the out-of-plane displacement signals (second column of Fig. 2; zero time corresponds to the start of the dipole source function) and their frequency domains (second column of Fig. 3) for a selected set of rise- 43

4 Fig. 2. CWT (first column) and AE out-of-plane displacement signal (second column) for a selection of rise-times for the source at the mid-plane and at a propagation distance of 480 mm. Note: only the group velocity curves are present at the bottom plot of the first column. times for the signals at a propagation distance of 480 mm. Because the FEM code did not apply material attenuation, the displacement results at a propagation distance of 480 mm were used exclusively in these figures for all the rise-time cases. This distance provided the best separation of the dispersive modes. Note that the CWT results are duplicated in both figures so that these 44

5 results can be directly compared by the reader to the AE signals and the frequency-domain results without making the figures so small that they lose their illustrative role. Also, prior to the calculation of the fast Fourier transform (FFT) results, the waveforms were shortened and terminated at a zero to remove any effects from the first edge reflection that occurs at about 280 µs for this propagation distance. (Note that the FFTs were calculated with a rectangular window after each terminated signal was extended with zeros to obtain a total of 4096 points). This reflection Fig. 3. CWT (first column) and corresponding spectra (second column) for a selection of risetimes for the source at the midplane and at a propagation distance of 480 mm. See note in Fig. 2 caption. 45

6 can be observed most clearly for the longer rise-times. Except for the 15-µs rise-time CWT result, the group-velocity curves were not superimposed on the results. Instead, these curves are shown at the bottom of the CWT results. For the 15-µs rise-time CWT, the S 0 curve was superimposed to show that the intense arrival appears to be both late and extended in time. To examine this situation, the CWT was calculated for this rise-time and propagation distance without applying any frequency filter. In this case, the S 0 mode peak intensity was at 41 khz (9% late in time, based on the thin-plate extensional velocity). For the 40-kHz high-pass data, the S 0 mode peak intensity was at 61 khz (25 % late in time). Thus, the 40-kHz filter contributes to the late arrival. In addition, since the period of the signal in this low frequency range is relatively long, as shown in Fig. 2, it is possible that this influences the CWT calculation and also contributes to the late and extended arrival in the CWT result. A number of observations for the mid-plane source depth can be made. First, for the 0.5 µs rise-time, a portion of the higher-frequency part of the S 1 mode can be seen the CWT result (see arrow in Figs. 2 and 3). However, relative to the CWT magnitude of the most intense portion of the S 0 mode at a frequency of 539 khz, its peak magnitude, which occurs at a frequency of 1430 khz, is only 18 % of that S 0 peak magnitude. At rise-times of 0.75 µs and longer, the S 1 mode is no longer clearly visible. Second, at a rise-time of 3 µs and greater, the most intense portion of the CWT moves to the initial arrival portion of the S 0 mode as compared to a CWT peak arrival later in the signal for shorter rise-times. Third, as can be observed from the second column of Fig. 2, for all the rise-time cases, the time of arrival of the peak CWT intensity corresponds to the highest or nearly highest-amplitude portion of the signal. Further, in Fig. 2 (second column), this high-signal-amplitude arrival time shifts from late in the signal to early as the rise-time increases. Fourth, the most intense region of the CWT corresponds to the highest or near-highest amplitude frequency region of the spectra results, as shown in the second column of Fig. 3. Finally, it is worth noting that for the 3-µs rise-time, the second most intense portion of the S 0 mode, with a peak at a frequency of 630 khz, is directly in a region where the S 0 and S 1 group velocity curves first intersect. The above discussion demonstrates that determining signal arrival times at the most intense portions of the CWT corresponds to both the signal amplitudes and signal frequencies that are among the largest in the signals. These are the regions of the AE signals where the best signal-tonoise ratio (based on the electronic preamplifier noise) would be present. Modal Content versus Source Rise Time Near Top Surface Source Location Figures 4 and 5 show the CWT results (first columns) as well as the out-of-plane displacement signals versus time (second column of Fig. 4) and the frequency domains (second column of Fig. 5) for a selected set of rise-times for the signals at a propagation distance of 480 mm. The same comments relative to the duplication of the CWT in the figures and the calculation of the FFTs apply to these figures as were made relative to Figs. 2 and 3. Clearly, for this source depth and all the rise-times, there is a strong dominance of the lower-frequency regions of the A 0 mode. As the rise-time decreases, higher-frequency portions of the A 0 mode are apparent in the CWT, and as the rise-time increases, the most intense portion of this mode gradually decreases in frequency with a significant change in the arrival time of the peak CWT intensity due to the shape of the A 0 group velocity curve. This trend of arrival times is observed in the time domains of Fig. 4. Specifically, the change in the most intense frequency of the A 0 mode in the CWT result goes from 66 khz for the 0.5-µs rise-time to 51 khz at the 15-µs rise-time, while the respective change in the arrival time of the peak intensity changes from about 200 µs to about 230 µs. 46

7 Over the same range of rise-times, the peak magnitude of the spectra hardly changes from about 50 khz. At these lower frequencies there is likely some effect of the 40-kHz high-pass filter. As expected, the frequency content (see Fig. 5) in the spectra above 400 khz gradually fades as the rise-time increases. Fig. 4. CWT (first column) and corresponding out-of-plane displacement AE signal (second column) for a selection of rise-times for the source near the top surface and at a propagation distance of 480 mm. See note in Fig. 2 caption. 47

8 Fig. 5. CWT (first column) and corresponding spectra (second column) for a selection of risetimes for the source near the top surface and at a propagation distance of 480 mm. See note in Fig. 2 caption. Modal Content versus Source Rise Time Mid-depth Source Location Figures 6 and 7 show the CWT results (first columns) as well as the out-of-plane displacement versus time (second column of Fig. 6) and frequency domains (second column of Fig. 7) 48

9 Fig. 6. CWT (first column) and corresponding out-of-plane displacement AE signal (second column) for a selection of rise-times for the source at the mid-depth and at a propagation distance of 480 mm. See note in Fig. 2 caption. for a selected set of rise-times for the signals at a propagation distance of 480 mm. The same comments relative to the duplication of the CWT in both figures and the calculation of the FFT apply to these figures as were made relative to Figs. 2 and 3. Again at the two shortest rise-times, there is evidence in the CWT results of a higher mode (A 1, higher-frequency regions of the fundamental modes and an intense region due to mode intersections (second intersection of the S 0 49

10 and A 1 modes). As was the case with the mid-plane source at 0.5-µs rise-time, a higherfrequency region at 542 khz of the S 0 mode is still the most intense. Compared to the magnitude of the S 0 mode at that frequency, the magnitude of the CWT coefficients at the intersection of the S 0 and A 1 modes at 837 khz is 93 %, and the magnitude of the A 1 mode at 993 khz is 61 %. The Fig. 7. CWT (first column) and corresponding spectra (second column) for a selection of risetimes for the source at the mid-depth and at a propagation distance of 480 mm. See note in Fig. 2 caption. 50

11 A 0 mode also is visible (for this rise-time) in the low frequency region with a comparative magnitude of 35 % at its most intense point at 63 khz. Clearly, for this source depth, both the axisymmetric and anti-symmetric modes are apparent. For all the rise-times, the spectra show a low-frequency peak that is the largest in magnitude at the longer rise-times, and it nearly has the largest magnitude at the shorter rise-times. In contrast to the mid-plane source, at the longer rise-times the largest-magnitude region of the CWT of the AE signal is in the low-frequency region of the A 0 mode rather than the low-frequency region of the S 0 mode at an earlier arrival time for the mid-plane source. At the rise-times of 2.3 µs or longer, the AE signal peak amplitude corresponds to the most intense portion of the A 0 mode in the CWT. It is interesting that the spectra for the two shortest rise-times show a high magnitude in a low-frequency region with a peak at approximately 50 khz, but this frequency is less apparent in the CWT and the AE signal. This situation is likely to be the result of the continued presence in time of the A 0 mode in the CWT for these two rise-times (see arrows to A 0 in Figs. 6 and 7). Modal Content Versus Propagation Distance Figure 8 illustrates how the CWT results change as a function of selected propagation distances from 480 mm to 60 mm for two source depths for the sources with a rise-time of 0.5 µs. The first column is for a mid-depth source, and the second column is for a mid-plane source. The cases for the shortest rise-time were selected, since they excite the largest number of modes. In addition, the near top source was not selected for illustration, because only the A 0 mode strongly dominated all the rise-time cases at all the propagation distances. Examination of the first column of Fig. 8 shows that as the propagation distance decreases for the mid-depth source, similar dominant excited modal features can be identified down to a propagation distance of 180 mm. At 60 mm, the modal results are not clear in this figure. To clarify the CWT results in the first column of Fig. 8 for the shortest propagation distances, Fig. 9 shows in the first column the CWT results for the mid-depth source at distances of 120 mm and 60 mm. In addition, the same results are shown in the second column for the mid-plane source where the time axis is only from zero to 80 µs in both columns. The first column of Fig. 9 shows that the closeness in time of the multiple modes that are excited results in distortion of the CWT results above about 600 khz for both the 120 mm and 60 mm distances. To determine whether intense arrivals below this frequency corresponded to the correct Lamb wave group velocity, the arrival times of the CWT peak magnitudes were determined at 564 khz for all the propagation distances. This frequency was that of the peak CWT magnitude for the S 0 mode at a propagation distance of 180 mm. A plot (not shown) of propagation distance versus these arrival times had a slope of 1.85 mm/µs, with all the data points on the straight line including those at 120 mm and 60 mm. This velocity is within 1.2 % of the associated Lamb-wave S 0 mode group velocity at that frequency. Thus, in spite of the distortion in the CWT results, the correct mode arrival times can be obtained at the most intense mode/frequency combination below 600 khz. If the second columns of Figs. 8 and 9 are examined, the same dominant mode and frequency combinations for the S 0 and S 1 modes are apparent at each propagation distance from 480 mm down to 60 mm. Thus, when multiple frequency modes are not sufficiently excited over the same frequency range (as was the case in the first column of Fig. 9), the CWT results are not as distorted at the shorter propagation distances. Again, as a check on the application of the use of the CWT magnitude peaks at certain frequencies to obtain arrival times, the arrival times at the peak 51

12 magnitude of the CWT for the S 1 mode at 1430 khz were determined for all the distances. The same type of plot for this data had a slope of 2.20 mm/µs, which differed by less than 1 % from the value obtained from the Lamb-mode group-velocity curve for the S 1 mode at that frequency. Thus, even this relatively small intensity mode arrival (for example about 17 % of the CWT peak S 0 mode arrival at the propagation distance of 360 mm) provided accurate arrival times. Fig. 8. CWT results versus propagation distance for 0.5-µs rise-time. First column for mid-depth source and second column for mid-plane source depth. 52

13 Fig. 9. CWT results versus distance at the two shortest propagation distances. First column for mid-depth source and second column for mid-plane source depth. Fig. 10. CWT results at 480 mm for the indicated two rise-times and three source depths. 53

14 Modal Content to Possibly Indentify Sources with Different Rise Times The changes in the intense modal regions as the source depth changes imply that, unless the sources with distinctly different rise-times all originate at the same depth, the modal pattern of intense mode/frequency combinations cannot easily be used to distinguish the different risetimes. To demonstrate this directly, Fig. 10 shows the CWT results at 480 mm for sources at the three different depths with two different rise-times of 1.5 µs and 4 µs (rise-times differ by a factor of 2.7). In this figure, it is clear that if the sources were at the same depth, it would be straightforward to distinguish the different rise-time sources by the differences in modal intensity at different frequency points in the modes excited. For example, in the mid-plane source case, the 4-µs rise-time excites a lower frequency portion of the S 0 mode, which is not sufficiently excited in the 1.5-µs rise-time case. In the least straightforward case for the near top source, the calculation of the ratio of the CWT peak magnitudes (for the A 0 mode) at the frequency of 60 khz divided by that at 300 khz would serve to distinguish the two different rise-time sources. In the CWT example in the figure, this ratio for the 1.5-µs source is 4.7, and for the 4-µs source it is 150. However, if the source depths varied, a much more complicated approach would be needed, and to verify such a proposed approach, a more extensive database of possible source depths would be required. To examine another aspect of the dependence of the modal content on the source rise-time, the CWT magnitudes of the intense mode/frequency combinations were examined as a function of the source rise-time and the source depth. The normalized results are shown in Fig. 11 for a propagation distance of 480 mm. The selected intense combinations for each depth were as follows: i) mid-depth S 0 at 537 khz, A 0 at 63 khz and S 0 intersection with A 1 at 837 khz, ii) near top surface A 0 at 66 khz and iii) mid-plane S 0 at 539 khz. These combinations were selected from the CWT results for the 0.5 µs rise-time cases shown in Figs. 2 through 7. Figure 11 clearly shows that the three high-frequency mode/frequency combinations experience a rapid fall-off of the CWT magnitude as the rise-time increases. The most rapid falloff is the mid-depth source CWT magnitude at 837 khz followed closely by the mid-plane source result at 539 khz and the mid-depth result at 537 khz. It should be noted that these two similar frequency values are those provided by the software, but they clearly are from the same mode/frequency region. These results with rise-time increases are expected, because longer rise-times do not sufficiently excite these higher frequencies. It should be noted that the modal arrivals of the three higher mode/frequency combinations could not be identified in the CWT results for rise-times greater than about two microseconds. The falloff in the CWT magnitude with increasing rise-time of the two lower-frequency combinations was nearly identical, because they involved the same mode, A 0, for the two different source depths. The slower fall-off with increasing rise-time for these cases is expected due to the relatively low frequency of this A 0 modal region. The potential ability to use these results to identify sources with different rise-times must be combined with the results of CWT-determined mode attenuation with propagation distance. Figure 12 demonstrates how typical intense mode/frequency combinations attenuate with propagation distance for the 0.5-µs rise-time sources at two different depths. In this figure, the attenuation of the A 0 mode at 66 khz for a near top source and the S 0 mode at 539 khz for a mid-plane source are compared to the amplitude-based geometric attenuation of a wave spreading in-plane. The higher-frequency combination has slightly more attenuation than the lowerfrequency combination, but both decay significantly more than the geometric attenuation alone. Thus, the CWT magnitudes of the intense mode/frequency arrivals experience an additional source of attenuation beyond the expected geometric attenuation. Kerber et al. [9] recently exam 54

15 ined Lamb wave modal attenuation with propagation distance by comparing the results from a chirplet transform (CT) to those from a short-time Fourier transform (STFT). The CT allowed them to use an additional degree of freedom to adjust the window function to the group delay of the signal. The CT-based algorithm thus allowed them to use the known group-velocity curves. With experimental data for an aluminum plate one millimeter thick, they found significantly smaller mode attenuation with propagation distance with the CT-based algorithm as compared to the STFT results. In their work, the source of the out-of-plane displacement signals was a laser ablation on the aluminum plate surface (out-of-plane surface source). A point-based laser measurement system was used to obtain the displacement signals. They also showed further reductions in the attenuation with both the CT and STFT results when synthetic signals were used. Due to the differences in plate thickness and a lack of specific data in this reference, a direct comparison could not be made with the current results for the CWT. However, the results in reference 9 indicate that the added attenuation beyond geometric attenuation is at least in part artificial, as it depends on the signal-processing approach. Fig. 11. Normalized CWT magnitudes of the intense mode/frequency combinations for the three source depths (at a propagation distance of 480 mm) as a function of the source rise-time. It is interesting to note that the added attenuation relative to geometric attenuation for these mode/frequency combinations is about 12 to 15 db for these two cases. This result contrasts with the peak amplitude attenuation of about 3 to 5 db more than geometric attenuation [4]. To further illustrate this aspect, Fig. 12 also includes the attenuation of the peak AE signal amplitude with propagation distance for a mid-plane source with a rise-time of 0.5 µs. It is clear that the peak amplitude does not attenuate as rapidly with propagation distance as the intense mode/frequency combinations. It is worth noting, that the peak signal amplitude attenuation with distance is approximately the same for the different source depths and source rise-times [4]. Fur 55

16 ther, the wavy character of the peak amplitude attenuation with distance is due to differences (at different propagation distances) in the superposition of the multiple modes excited by the short rise-time. It was demonstrated previously [4] that the wavy character decreases when the source rise-time is longer when fewer modes are generated. Also, when the CWT magnitude of a single mode/frequency combination is considered, the attenuation with propagation distance is smooth, as shown in Fig. 12. Thus, unless the propagation distance to the sensors is approximately the same, the attenuation of the CWT magnitude of the intense mode/frequency combinations creates an additional complication to the use of modal information for the identification of sources with different rise-times. Fig. 12. Normalized CWT attenuation of intense mode/frequency combinations and normalized peak amplitude for mid-plane source, 0.5 µs rise-time sources versus geometric attenuation. Comments on Comparisons of Modal Content versus Source Depth In the previous publication [4] which specifically analyzed the frequency spectra dependence on rise-time and source depth, it was pointed out that for rise-times less than 2.3 µs, the middepth spectra could be roughly viewed as a superposition of the peak-frequency regions from the mid-plane and near top spectra results at the same rise-times. Examination of Figs. 2, 4 and 6 demonstrates that such a statement cannot be made relative to the CWT results. For example, in the case of a 0.5-µs rise-time, the primary modal content for the source near the top surface is the A 0 mode, and for the mid-plane source it is the S 0 mode with a small magnitude of the S 1 mode. The mid-depth primary modal content is the S 0 mode, the A 1 mode, the intersection point of these two modes and a small intensity of the A 0 mode. Clearly, this does not correspond to the superposition of the primary modal content of the mid-plane and near top depths at this rise-time. At other rise-times, similar inconsistencies were present. 56

17 Conclusions Applicable for Choi-Williams transform (CWT) results for a 4.7-mm thick aluminum plate based on rise-times from 0.5 µs to 15 µs and propagation distances from 60 mm to 480 mm: For sources at the mid-plane and mid-depth, as the rise-time decreases, higher Lamb modes become visible in the CWT results. In contrast, for sources near the top surface, as the rise-time decreases, the CWT results are dominated by the A 0 Lamb mode to such an extent that higher Lamb modes are not visible. For all the source depths, rise-times and propagation distances, the intense portions of the CWT results are dominated by different frequency portions of only the A 0 and S 0 fundamental Lamb modes. Thus, only the fundamental modes need to be considered in the determination of accurate signal arrival times which would lead to accurate AE source locations. In an experimental situation, to monitor the intense portions of the fundamental modes for the rise-times and source depths considered here would require a sensor response that is nearly flat with frequency (ASTM E1106) from about 40 khz to about 600 khz. As a function of propagation distance, similar frequency regions of the fundamental modes dominate for a given rise-time and the source depths of mid-plane and near the top. For the mid-depth source at the shortest propagation distances (60 mm and 120 mm) and shortest rise-time (0.5 µs), the excitation of both symmetric and anti-symmetric modes results in some distortion of the CWT results at higher frequencies due to the close arrivals in time of the modes. But, by focusing on the lower-frequency intense portions of the fundamental modes, accurate arrival times can be obtained that correspond to the theoretical Lamb-wave group velocities. The CWT magnitudes of higher-frequency intense mode/frequency combinations falloff rapidly as the source rise-time increases, in contrast to a slower falloff for the lower-frequency intense mode/frequency combinations for this change of rise-times. The rate of attenuation with increasing propagation distance of the CWT magnitudes of the intense mode/frequency combinations is greater than that of the peak signal amplitude. In AE applications, the above conclusions are applicable relative to the potential to detect AE signals, to obtain accurate source locations and to approaches that use the modal content of AE signals to identify AE source types. Acknowledgement The finite-element calculations by Dr. John Gary (retired NIST, Boulder, CO, USA) are gratefully acknowledged. References 1. Hamstad, M.A., K.S. Downs and A. O Gallagher, Practical Aspects of Acoustic Emission Source Location by a Wavelet Transform, 21, 2003, Hamstad, M.A., A. O Gallagher and J. Gary, Examination of the Application of a Wavelet Transform to Acoustic Emission Signals: Part 2. Source Location, Journal of Acoustic Emission, 20, 2002, Hamstad, M.A., Comparison of Wavelet Transform and Choi-Williams Distribution to Determine Group Velocities For Different Acoustic Emission Sensors, J. Acoustic Emission, 26, 2008,

18 4. Hamstad, M.A., Frequencies and Amplitudes of AE Signals in a Plate as a Function of Source Rise-time, Proceeding of the 29 th European Conference on Acoustic Emission Testing, Vienna, Austria, Burridge, R. and L. Knopoff, Body force equivalents for seismic dislocations, Bulletin Seismic Society of America, 54, 1964, Hamstad, M.A., A. O'Gallagher and J. Gary, "Modeling of buried acoustic emission monopole and dipole sources with a finite-element technique," Journal of Acoustic Emission, 17(3-4), 1999, Vallen, J., AGU-Vallen Wavelet transform software version R , Vallen-Systeme GmbH, Münich, Germany, 2009, Available at 8. Vallen, J., Dispersion software version R , Vallen-Systeme GmbH, Münich, Germany, 2009, Available at 9. Kerber, F., H. Sprenger, M. Niethammer, K. Luangvilai and L.J. Jacobs, Attenuation analysis of Lamb waves using the chirplet transform, EURASIP Journal on Advances in Signal Processing, Vol. 2010, Article ID , 6 pages. 58