# Physics of Within-Tissue Wave Propagation Generated by Pulse Propagation in the Carotid Artery
## 4 Results
### Load Pressure Conditions: Shock Impulse and Traveling Impulse
The main wave front clearly has an oblique trajectory throughout the thickness of the model, causing a spatial lag along the z-location between the z-location of the main wave front at the lumen and skin level (about 10.48 mm). Maximal displacement magnitude of the main wave front has increased to 0.13 mm at the level of the lumen and 0.10 mm at the level of the skin. At 6 ms, we again observe a secondary wave front at the lumen surface (indicated with f\({}_{\text{r}}\)) due to reflection at the skin surface. Out-of-plane displacement (y-direction) as a function of time and propagation path (z-direction) is shown in Figure 5 for the top of the lumen and skin surface under shock and traveling impulse conditions. The propagation of a single shock impulse is shown in Figure 5a, demonstrating one main wave front and one secondary wave front propagating in the lumen displacement plot with speeds V\({}_{1}\) and V\({}_{2}\) of 4.66 m/s and 11.6 m/s, respectively. At the skin level, the main wave front propagated with a speed V\({}_{1}\) of 9.74 m/s and two secondary waves travelled with speeds V\({}_{2}\) and V\({}_{2}\)\({}^{*}\) of 4.79 and 15.7 m/s, respectively. The speed of secondary wave V\({}_{2}\) deviated only 2.8% from the main wave speed V\({}_{1}\) at the lumen surface. For the traveling impulse condition, Figure 5b demonstrates one single broad wave front propagating at the lumen surface traveling with a velocity V\({}_{1}\) of 6.99 m/s. This is in good  
Figure 4: Radial displacement in the common carotid artery (CCA) model depicting wave propagation at four time instances for (**a**) shock impulse and (**b**) traveling impulse with a velocity of 7 m/s. Note that a cylindrical coordinate system is used in this figure to obtain a more intuitive visual representation of wave propagation. See text for a description of the observed and indicated waves. agreement with the imposed loading conditions (V\({}_{\rm z}\) of 7 m/s). However, at the skin level, two wave fronts are visible. The main wave front travels with a velocity V\({}_{1}\) of 6.99 m/s, in very close agreement with the imposed V\({}_{\rm z}\). The secondary wave traveled with a velocity V\({}_{2}\) of 11.70 m/s. Note that in both panels of Figure 5 wave reflections are not completely absent due to the non-ideal behavior of the infinite elements. For example, wave reflection is visible at a time (7 ms (lumen) and 9 ms (skin) in the TOF plots of Figure 5b. The right panels of Figure 5 show the temporal profile of the y-displacement at two locations along the length of the model (z = 20 mm and z = 30 mm) for the top of the lumen and skin surface, to replicate the signal measured by LDV. Under shock impulse loading conditions, the lumen displacement profile is characterized by multiple peaks and it has a different shape for z = 20 mm compared to z = 30 mm. For the traveling load condition, the displacement patterns have clearly one traveling peak, allowing an easy identification for LDV analysis. For both loading conditions represented in Figure 5, the tracked trajectory using the Radon transform corresponded well with the main displacement peak. Figure 6 shows the spatio-temporal displacement (y-direction) plots at the skin surface when considering traveling pressure pulse velocities V\({}_{\rm z}\) of 4 and 10 m/s. The main wave front propagated with a speed V\({}_{1}\) of 5.05 m/s (26.3%) and 9.75 m/s (-2.5%) for an imposed V\({}_{\rm z}\) of 4 and 10 m/s, respectively. For all loading conditions, the secondary wave fronts propagated with a higher speed V\({}_{2}\) than the main wave front, as indicated on the top of each plot in Figure 6.  
Comparing the results of all imposed traveling pulses in Figures 5 and 6, the maximum amplitudes of the main wave front (indicated with speed V\({}_{1}\)) were 0.06, 0.10 and 0.08 mm for V\({}_{\rm z}\) of 4, 7 and 10 m/s respectively, whereas the maximal amplitude of the secondary wave fronts (indicated with speed V\({}_{2}\)) remained constant (0.030 mm for V\({}_{\rm z}\) of 4, 7 and 10 m/s respectively).