## 2 Materials and Methods
### Low Strain Integrity Testing
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Density, kg/m3** & **Pressure Wave Velocity, m/s** & **Shear Wave Velocity, m/s** & **Rayleigh Coefficients:** \\ \hline Concrete & 2400 & 4000 & 2450 & 20; \(10^{-8}\) \\ \hline Sand & 1500 & 600 & 200 & 200; \(10^{-7}\) \\ \hline Air & 1275 & 300 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The properties of simulated materials. Figure 2: The layouts of synthetic driven pile models with different splice types: (**a**) single-section pile (without a splice)—Model 1; (**b**) 5 mm sand gap—Model 2; (**c**) 1 mm sand gap—Model 3; (**d**) 1 mm air gap—Model 4.  
by Courant-Friedrichs-Lewy. The size of the finite elements was chosen considering the typical dimensions of the model geometry and the wavelength in the material so as to ensure that the calculation error does not exceed 0.5%. To generate the elastic waves, an external vertical force \(F\) in the form of a Gaussian pulse modulated by a [PERSON] window was applied at the center of the pile top to an impact area 30 mm in diameter:  
\[F=-A\mathrm{e}^{-(2\pi\frac{(t-0.5T)}{T})^{2}}\cdot\left\{ \begin{matrix}(0.5-0. 5\cos\left(2\pi\frac{t}{T}\right)),\text{ if }t<T\\ 0,\text{ if }t\geq T\end{matrix} \right\}, \tag{2}\]  
where \(A\) is the maximum force value, \(t\) is the time, and \(T\) is the pulse duration. The acoustic waves were registered with a 25 mm synthetic acceleration sensor offset from the pile axis by 75 mm. The duration of the input force pulses was set to be 0.5, 0.8, 1.2, and 2.0 ms to simulate the typical impacts of hammers of different weights and head materials. Shorter pulses correspond to hitting by a lightweight hammer with a metal tip, while long-duration input pulses correspond to heavy hammers with a rubber tip. Figure 3 shows the results of numerical simulations obtained for each synthetic model layout and each input force pulse. The simulations demonstrate that the length of the single-section pile (Model 1) can be determined with any duration of input pulse force applied (Figure 3a). The simulations for a 5 mm sand gap splice model (Model 2) show a clear response of a pile toe at 7 ms (Figure 3b). Using different input pulse forces, we can observe reflections from a pile splice at 4 ms on almost every signal. While the splice response is clearly detected for short input pulses, it becomes barely visible for the duration of the input pulse of 2 ms. The simulations for a 1 mm sand gap splice model (Model 3) show that both low-frequency (long-duration) and high-frequency (short-duration) signal excitations allow us to determine the total length of the pile, as evidenced by the reflection pulse registered at 7 ms (Figure 3c). Regarding the detection of the splice, with short-duration input pulses, a weak reflection can be observed at 4 ms. With long-duration input pulses of 1.2 ms and 2 ms, no reflection from the pile splice can be distinguished. The time domain analysis of these signals can lead to false conclusions that the pile consists of a single section. The simulation results for a 1 mm air gap splice (Model 4, Figure 3d) show a clear splice reflection at 4 ms. Moreover, a second reflection from the splice can be observed at  
Figure 3: Results of numerical simulations of low strain integrity testing with input pulse forces of 0.5, 0.8, 1.2, and 2.0 ms calculated for the driven pile models: (**a**) single-section pile—Model 1; (**b**) 5 mm sand gap—Model 2; (**c**) 1 mm sand gap—Model 3; (**d**) 1 mm air gap—Model 4. 8 ms. Consequently, we can conclude that the air gap splice represents a significant obstacle to the acoustic waves, and the total length of a pile could not be assessed.