## 3 Results and Discussion  
A finite element analysis-based numerical simulation was performed to simulate the feasibility of using effective density detection in determining soil voids and the overall density of the porous soil. The simulation was used to visualize the transient sound wave propagation in the four soil with different conditions. As Figure 2 shows, the models of the initial study were categorized into healthy soil (A), soil with a region of soft soil Figure 2B, soil with a region of softer soil Figure 2C, and healthy soil with an internal small air gap (of about 0.5 \(\lambda\) in size) Figure 2D. In the axial propagation of the wave in the detection setup, the low-frequency sound wave pulse has a small amplitude reflection at the interface between Air and soil layers. In the soil layers, the sound wave propagation was delayed in the case of Figure 2B,C comparing with healthy soil (A). Moreover, in the case of Figure 2D, the small air void caused a scattering effect on the propagating wave without any clear temporal delay. The sound wave was reflected back into the transducer with a larger amplitude at the interface between the soil layer and the rock layer under it. The reflected wave propagated in the opposite direction of the wave emission direction. The backward trip of the wave undergoes another temporal delay in the case of Figure 2B,C, and once scattering effect in case (D). By measuring the reflected signal pressure over a roundtrip propagation of the wave, a significant difference was found between the four cases, as shown in Figure 3.  
Figure 3: **(A)** Temporal reflected signal collected at the surface of the probe in all four cases mentioned in Figure 2. “No cavity” was the healthy soil condition in case (**A**). “One cavity” indicated the soil with a small air void in case (**D**). “Soft soil” was the case (**B**), which has a soft soil zone in the soil layer, which has similar properties. “Softer soil” was the case (**C**), which has a soft soil zone in the soil layer which has dissimilar properties. The window width was 6 ms. The wave completed a roundtrip in the simulated model in the calculated time length. (**B**) was the zoomed-in view between 0.4 ms and 1.8 ms. (**C**) was the zoomed-in view between 2 ms and 4 ms. (**D**) was the zoomed-in view between 4.125 ms and 5 ms. In Figure 3, Figure 3A shows the entire range of the temporal response of the propagating wave. Figure 3B-D shows the zoomed-in response from the various temporal regions plotted in Figure 3A. Figure 3B shows that the echoes from the two reflections can be expected from the interface between the air and soil and its second roundtrip envelope. Figure 3C depicts the zoomed-in time window, and the propagation of the wave occurs within the soil layer's internal region. No reflection pulse can be observed from the healthy soil line. The first and third echo on the green line occurred from the front and back interface between softer soil and healthy soil. The second and fourth pulses were the second roundtrip of the first and third reflection that occurred inside the softer soil cavity. The only two visible reflections on the soft soil were the echo from the two mismatch interface around the anomalous zone and occurred along the direction of the propagating wave. The red line has two lower amplitude echoes occurring from the air cavity's front and back interface on the wave propagation direction. In Figure 3D, there were three major reflection envelopes on each line. The first echo was reflected from the rocky layer's front interface, and the second echo was from the back boundary of the rock layer. The third reflection envelope occurred from the back surface and underwent a roundtrip internally within the rocky layer. Figure 3A showed the temporal measurement of the reflected sound wave at the probe surface regarding the normalized sound pressure for all the four proposed cases.