## 3 Results  
### Application of Complex Continuous Wavelet Transform for the Numerical Simulation Results  
Here, we apply the CCWT analysis to the results of numerical simulations presented in Section 2.3 (Figure 3). As a mother wavelet, we use a complex Morlet wavelet with the bandwidth parameter \(B\) of 0.2 and the center frequency \(C\) of 1.2 (Equation (4)).  
Figure 4 shows the application of CCWT for a single-section pile (Model 1). The CCWT energy concentration points and the phase shift locations for each duration of the input force pulse are clearly visible at 0 (Line 1) and 7.5 ms (Line 2), which correspond to the responses of the initial hammer impact and the pile toe, respectively.  
Next, we consider the application of the CCWT on numerical models of driven piles of two sections with different splice types. While analyzing the data in the time domain, the response of a 5 mm sand gap splice (Model 2) was barely visible when using long-duration signal excitations. The CCWT results (Figure 5) obtained for these signals with the input force pulses of 0.5 and 0.8 ms show bright energy concentration points at 0.0, 4.2, and 7.5 ms. For longer input pulses of 1.2 and 2.0 ms, there are no energy concentration points in the splice area (in the vicinity of 4 ms). However, when analyzing the phase spectrum, we observe clear phase changes at 0.0 (Line 1), 4.2 (Line 2), and 7.5 ms (Line 3) for each duration of input force pulse. These phase changes outline the response from the initial hammer impact, 5 mm sand gap splice, and the pile toe, respectively.  
Figure 4: CCWT analysis of the low strain test numerical simulation results for a single-section pile (Model 1) with input force pulses of 0.5, 0.8, 1.2, and 2 ms duration. 1 st row—the simulated signals, 2 nd row—the distribution of energy of the wavelet transform coefficients, 3 rd row—the distribution of phase angles of the wavelet coefficients. Lines 1 and 2 delineate the phase shift moments.  
Figure 5: CCWT analysis of the low strain test numerical simulation results for a pile with a splice of 5 mm sand gap (Model 2) with input force pulses of 0.5, 0.8, 1.2, and 2 ms duration. 1 st row—the simulated signals, 2 nd row—the distributions of energy of the wavelet transform coefficients, 3 rd row—the distributions of phase angles of the wavelet coefficients. Lines 1, 2, and 3 delineate the phase shift moments.  
The CCWT analysis of the numerical simulation results for a pile with a 1 mm sand gap splice (Model 3) is shown in Figure 6. While the splice responses are poorly distinguishable when analyzing the time domain representations of signals and the distributions of CCWT energy, they clearly stand out as phase changes at 4 ms (Line 2) for the durations of input force pulses of 0.5, 0.8, and 1.2 ms.  
The time domain analysis of the simulation results for a pile with a 1 mm air gap splice (Model 4) delineates pronounced responses at 4 and 8 ms, related to first and second reflections from the splice, respectively (Figure 7). No pulse that can be interpreted as a reflection from the pile toe is distinguished neither for the time domain representations of the signals nor for the distributions of the wavelet energy. While there are several distinctive changes in the phase spectrum for different signals with various input force pulses, the only characteristic phase shift that manifests itself upon all input pulses is the one at 7.5 ms (Line 3). This phase change corresponds to a total pile length of 14 m.