# Shear-Actuation and Vibrometer Reception of Penetrating Ultrasonic Guided Wave Modes in Human Tibia
## 3 Results
### Numerical Results  
The transient dynamic finite element simulation confirms that the displacements in the wave field are measurable with an LDV. The displacement components related to those measured in the experiment are plotted in Figure 5. The related contours of \(\sqrt{v^{2}+w^{2}}\) are plotted on the cross-section in Figure 6, where \(v\) and \(w\) are displacement components in the \(y\) and \(z\) directions shown in Figure 3. The longitudinal displacement component \(u\) is excluded from this analysis because the LDV used in the experiment did not sense it. Figure 5: Finite element simulation of wave propagation in a human tibia provided displacement waveforms at the analogous point where LDV measurements were made on the synthetic bone. (**a**) Relative position of normal at the reception region, (**b**) the displacement components \(v\) and \(w\), and (**c**) the displacement component normal to the surface and the Hilbert envelope used to identify peaks. Figure 6: Contour maps of the in-plane displacement magnitude \(\sqrt{v^{2}+w^{2}}\), at different times obtained from finite element simulation of human tibia. The black line at the medial facet is tangent to the surface at the position on the synthetic tibia where LDV measurements were made. To calculate the displacement component normal to the surface as received by the laser head of the LDV, the normal direction to the laser reception points on the reflective tape was calculated. It was found that the normal to the reception point makes an angle of approximately \(\theta=28^{\circ}\) with respect to the y-axis (shown in Figure 5a). The total displacement normal to the surface then becomes equal to \(v/cos(\theta)+w/sin(\theta)\). The individual displacement components are shown in Figure 5a, and the composite signal is shown in Figure 5b; note the similarity between the simulated and experimental time histories shown earlier in Figure 4b. Next, displacement magnitude profiles corresponding to the expression \(\sqrt{v^{2}+w^{2}}\) were extracted for the central cross-section of the diaphysis at multiple time instances corresponding to the peaks in the A-scan for the two wave packets in the composite signal. The Hilbert transform was used to create an envelope and identify a number of displacement amplitude peaks (Figure 5b). Our analysis identifies five peaks in total: three of them correspond to the first wave packet, and the remaining two belong to the second wave packet. The color bar values in Figures 6 and 7 correspond to the expression \(\sqrt{v^{2}+w^{2}}\). To represent the relative location of the reflective tape on the medial surface, a black line segment is shown in Figures 6 and 7. In the two cases corresponding to time values of 84.0 \(\mu\)s and 89.9 \(\mu\)s in Figure 6, the displacement profiles have a localized region of significant intensity at the medial surface partially overlapping with the region where the reflective tape is present. Although the reflective tape is not exactly located at the center of the highest displacement intensity regions, this explains why the LDV is able to detect these waves. The last two displacement profiles for time values of 129.2 \(\mu\)s and 135.5 \(\mu\)s have small but non-zero displacement magnitude distribution on the medial surface. We do not attempt to assess whether these would be detectable. Additionally, wave structures were extracted from SAFE analysis corresponding to the eigenvalues within the propagating guided waves region highlighted in Figure 4c. These wave structures are plotted in Figure 7 for a number of eigenvalues and for the same combination of displacement components \(\sqrt{v^{2}+w^{2}}\).