## 2 Materials and Methods
### Experimental Verification  
Experiments are employed to validate the model in this research to confirm the dependability and precision of the numerical simulations. The equipment selected for this paper is the DH5927N Dynamic Signal Test and Analysis System (Donghua Testing Technology Co., Ltd., China's Jiangsu), as depicted in Figure 6. The components required include a rubber hammer, acceleration sensor (Sensitivity of 1006 mv/g), and a transmission data line. The center of the pile top is excited by the rubber hammer, and the signal is collected using the accelerometer. The experiment used concrete precast piles to help control the size of the defect. Strength grade of C30 with a ratio of (cement: sand: stone: water) 1:1.57:3.82:0.99 for the whole part of the pile and a strength grade of C15 with a ratio of (cement: sand: stone: water) 1:3.99:6.48:1.71 for the discrete part of the pile. The properties of the materials required in the tests are shown in Table 4 below. The low strain reflected wave method is studied based on the one-dimensional fluctuation theory when L > 5 D piles are slender rods [33]. To avoid interference from reflected waves and shear displacements at the soil boundary, it was supposed that the diameter of the dirt encircling the pile was 10 times larger [25]. As the defective pile could not be used in the actual project even after the test, based on the idea of environmental sustainability and to lower the cost of the test, the characteristics of the model pile were L = 1 m, D = 0.1 m. The soil size around the pile is 1 m in the horizontal direction and 1.5 times the pile length in the vertical direction. A total of 5 model piles were made for this experiment, as depicted in Figure 7a: complete pile, diameter expanding pile, diameter shrinkage pile, segregation pile, and broken pile. PVC pipes were used as molds with diameters of 60 mm, 100 mm, and 140 mm, where 60 mm and 140 mm were used for cross-sectional changes of shrinkage \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Elastic Modulus E (Pa)** & **Density \(\rho\) (kg/m\({}^{3}\))** & **Poisson Ratio v** \\ \hline C30 concrete & 3.0 \(\times\) 10\({}^{10}\) & 2400 & 0.17 \\ C15 concrete & 1.5 \(\times\) 10\({}^{6}\) & 2400 & 0.2 \\ soil layer 1 & 1 \(\times\) 10\({}^{10}\) & 2250 & 0.25 \\ soil layer 2 & 0.015 \(\times\) 10\({}^{9}\) & 1930 & 0.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material parameter table. Figure 6: Dynamic signal analysis system and components. and expanding piles, respectively. After completion of pouring, 28 days of sprinkling and covering maintenance was carried out, and the mold was unmolded after completion of maintenance. The scenario of the model pile was set as in Table 2 and compacted every 20 cm to ensure compaction, as in Figure 7b. The top surface of the pile was cleaned and leveled without debris before the experiment. The coupling agent, such as butter or rubber cement, was used to adhere the sensor close to the top of the pile to ensure that the sensor was perpendicular to the top surface. The excitation point is the pile's center, and the sensor acquisition point is arranged at 2/3 R [34]. Figure 8 depicts the data capture procedure. The validity of the numerical simulation is shown in this work by comparing the simulation's findings with those from experiments, see Figure 9. Due to the different excitation methods of simulation and practical, the phase (i.e., where the wave peak occurs) has changed. However, it was observed that the reflection law of the stress wave is the same  
Figure 8: Pile integrity testing field. Figure 7: Pile models: (**a**) 1# complete pile, 2# diameter expanding pile, 3# diameter shrinkage pile, 4# segregation pile, and 5# broken pile; (**b**) Placement of model pile. when it passes through soil changes and defects during propagation. As shown by the computation outcomes in Table 5, it can be found that the error of the actual results and the numerical simulation results are tiny, which shows the validity of the numerical simulation.