# Bar-Wave Calibration of Acoustic Emission Sensors
## 3 Results and Discussion
### Displacement
In Figure 5a, calculated group velocities for bar waves are shown together with that of S\({}_{0}\)-mode plate wave for 6.4 mm-thick Al. The bar-wave data are divided into three types: red points represent the modes dominated by z-direction displacement, blue points those by y-direction displacement and black points those by x-direction displacement. The corresponding phase velocity dispersion curves are given in Figure 5b, using the same color codes, while Figure 5c provides the group velocity data only for the z-displacement dominated modes. Figure 3: (a) FFT magnitude spectrum of normal displacement at L = 300 mm at the center; (b) FFT magnitude spectra of normal displacement at L = 300 mm: Center-blue curve, 1 mm off-center-light green, 2 mm off-center-dark red, 3 mm off-center-dark blue, 5 mm off-center-purple, Average of five curves-red. Figure 2: (a) Normal displacement in nm vs. time measured at L = 300 mm at the center of the test bar; (b) Normal displacement in nm vs. time measured at L = 300 mm at 5 mm off-center. Figure 4: Displacement waveform from Figure 3a and its Choi-Williams transform spectrogram for 40–200 \(\upmu\)s segment up to 1 MHz. Some peak positions are marked as cw1 to cw6. Figure 5: _Cont._  
Because of the complexity, no standard naming of wave modes is available. In Figure 5c, six initial peaks are indicated by pn (\(n\) = 1-6) and with a prime for a subsequent peak of the same mode. The two lowest modes, p1 and p2, were absent in Figure 4. The next mode, p3, appears to correspond to the weak cw2 peak in Figure 4, since the velocity at the initial arrival of cw2 (4.6 mm/us) is close to the calculated p3 peak velocity of 4.7 mm/us. The positions of p4' and p5 peaks match well to the arrival of cw3 (3.5 mm/us and 280 kHz). Similarly, the position of p3' peak matches well to the arrival of cw4  
Figure 5: (**a**) Dispersion curves of group velocity vs. frequency for bar waves calculated using [PERSON]’s theory and semi-analytical finite-element (SAFE) algorithm [9]. Red points represent the modes dominated by z-direction displacement, blue points those by y-direction displacement and black points those by x-direction displacement. The group velocity of S\({}_{0}\)-mode plate wave for 6.4 mm-thick Al is drawn in as a black curve; (**b**) The corresponding phase velocity dispersion curves; (**c**) The group velocity vs. frequency for bar-wave modes dominated by z-direction displacement. Some peak positions are noted by p1 through p6. (2.7 mm/\(\upmu\)s and 310 kHz). This p3' mode (and perhaps p6 mode) appears to fit the position of cw5. At the position of cw1 maximum intensity peak, three or four dispersion curves are present and are likely to contribute to the cw1 peak. However, these curves are difficult to trace to the lower frequency modes. At the position of cw6 (2.4 mm/\(\upmu\)s), a concentration of many modes exists from 470 kHz to 1 MHz. Again, these modes are difficult to identify. The nature of cw6 peak is examined further in the following section and in Appendix A. From the CWT spectrogram and the dispersion curves, it is seen that the number of observable modes becomes large above 500 kHz. Many modes at lower velocities extend displacement signals to 200 \(\upmu\)s and beyond and reduce the amplitude. The dispersion curves given in Figure 5 can be scaled in terms of frequency when the ratio of thickness to width remains identical. When the thickness and width are doubled, the frequency is halved. This behavior is common with other dispersion curves for guided waves and the above results can be utilized when other bar dimensions are selected while keeping the thickness-to-width ratio of 0.25.