## 4 Results Analysis  
Figure 3 shows the averaged values of ultrasonic wave velocity, amplitude, and frequency. Standard deviations, which reflect test errors, are also presented. By using the outputs, the dynamic modulus of each asphalt concrete was calculated using Equation (1) and is presented in Table 1. [PERSON]'s ratio was assumed to be 0.3 for all four mix types.  
As shown in the table, the dynamic moduli measured in the laboratory were much smaller than those calculated by using the ultrasonic wave velocity. This was within expectations, since the two test methods are different in nature; in the laboratory, the specimens were tested with a relative low frequency (10 Hz) and with viscoelastic properties, whereas the ultrasonic test was conducted at a high frequency (50 kHz) and with very little force acting on the sample by vibration, therefore significant deformation was not caused. However, the laboratory-measured and calculated moduli shared the same ranking; for the laboratory measurement, the AC-20 mix experienced the highest magnitude of modulus, followed by AC-13 and SMA-13 mixes, whereas the AM-20 showed the smallest value and the calculated dynamic moduli showed exact the same trend.  
Figure 4a shows a snapshot of a wave field generated by the ultrasonic probe in model AC-13. The top of the picture indicates the pavement surface, whereas the bottom of the picture means the  
Figure 3: Output of ultrasonic wave test.  
Figure 2: Schematic diagram of wave propagation through pavement (**a**) without cracks, and (**b**) with cracks. Cracks are denoted as black strips.  
bottom of the asphalt concrete. It can be seen that wave propagation is symmetrical and continuously distributed along the pavement depth direction. It is also observed that the ultrasonic wave propagated through model and diffused outward in a ring shape with concentrated and good directionality and followed a straight line alongside the wave incident direction. Figure 4b shows the extracted time-displacement curve that was obtained from the highlighted point at the bottom of the pavement. Such a curve is required for the calculation of simulated wave velocity. The figure shows that the first wave was detected at a time of 5.069 E-5 s and then turned into sinusoidal waveform, with its amplitude near the sensor decreasing due to the attenuation phenomena.  
The ultrasonic wave velocity propagated through the model that originated from the transmitting end and continued to the receiving end was calculated. Taking the AC-13 mix as an example, node #379 at the receiving end on the ultrasonic propagation path was selected and its time-displacement curve was obtained. The time that the first wave of ultrasonic propagation was detected was 5.069 E-5 s. Also, the total height of AC was 0.15 m, so the ultrasonic wave velocity can be calculated by using the relationship of time, distance, and velocity, as given in Equation (5). The final calculated velocity was 2958 m/s.  
\[V=\frac{H}{T}, \tag{5}\]  
Figure 4: (**a**) Wave propagation along pavement depth direction and (**b**) time-displacement curve.  
where \(V\) (m/s) represents ultrasonic wave velocity, \(H\) is the height (m) of the model, and \(T\) is the time (s) taken to detect the first wave. Similarly, the wave velocities of ultrasonic wave propagation through other asphalt concrete models were calculated and are summarized below.