## 4 Results and Discussions
### Ejector Nozzle
#### 4.1.1 Stationarity and Ergodicity  
Table 1 evaluates whether the process can be regarded as wide-sense stationary for both the baseline (B) and eject nozzle (E) configurations by presenting the mean, variance and standard deviation of the segments. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Segment Length** & **Overlapped** & **Mean for All** & **Standard** & **Mean of the** & **Standard** \\  & **[Samples]** & **Segments** & **Segments** & **Deviation of the** & **Variance of All** & **Deviation of the** \\  & & & & **Mean** & **Segments** & **Variance** \\ \hline  & 10,000 & 335 & \(-\)0.0000022 & 0.00476377 & 0.00789353 & 0.0005773 \\  & 20,000 & 167 & \(-\)0.0000011 & 0.00672744 & 0.00787076 & 0. The absolute mean for all segments is less than \(10^{-5}\), which is close to zero, indicating the data are centered around zero. This could suggest that the signal oscillates around zero, with no significant drift (n.b. which is true as the data are normalized to the maximum value). The variance of all segments is relatively stable and is less than 0.008, indicating that the spread of the data remains consistent across different number of segments. The standard deviation of the mean increases slightly, suggesting moderate variability in the signal over larger segments. A low standard deviation of the variance indicates consistent spread within segments, while an increase reflects differing spread across segments. The variance and autocorrelation values fluctuate within 0.002, indicating minimal variability. Signals with near-zero mean and variance are centered and show little fluctuation, suggesting relative constancy. Overall, if the mean and variance of a signal are both close to zero, it indicates that the signal is centered around zero and exhibits very little variability or fluctuation. This could imply that the signal is relatively flat or constant, lacking significant changes or dynamics. Now, regarding the autocorrelation values, they are the represented along with the overall mean and overall variance for each segmentation (n.b. from Table 1) in Figure 7. A total of \(N_{lag}=100\) lags is considered, represented by \(\left(\tau_{k}\right)_{k=0,N_{lag}}\), and we randomly select 10 lags to represent the mean of the autocorrelation for each segmentation based on segment length. The graphical representation shows that the mean remains nearly constant across all segmentations, with small fluctuations suggesting minimal repeating patterns or periodic behavior. Thus, if the mean, variance and autocorrelation remain stable over time, the signals can be reasonably considered stationary, or at least stationary in the wide sense. The analysis confirms that the signals demonstrate characteristics of wide-sense ergodicity (WSE), which implies they are also wide-sense stationary (WSS). This property allows statistical parameters, such as mean, variance and autocorrelation, to be derived from a single sample function, making it feasible to analyze real-world signals where ensemble averaging is impractical. From Table 2, the comparison between time averages and ensemble averages shows that the RMSE values for mean and variance are sufficiently small. This indicates strong agreement between these averages, validating the ergodic nature of the signals in terms of their central tendency and spread. Figure 7: Cont. Table 3 further examines autocorrelation, revealing slightly higher RMSE values compared to the mean and variance. While this suggests minor variations in the evolution of signal patterns, the values remain small enough to support the conclusion that the signals are ergodic, or at least ergodic in a wide sense. Overall, the results justify treating the signals as wide-sense ergodic, allowing time averages to be reliably used for further acoustic analysis. To conclude, the trimmed selected signals from the original ones are ergodic, and all the results concerning acoustic quantities could use just the trimmed data signal.