## 4 Results and Discussion
### Mathematical Modeling  
The objective of this mathematical model is to obtain mathematical equations for calculating the _root mean square_ of the AE signal at the cutting head (\(AE_{RMS\;CH}\)) and at the workpiece (\(AE_{RMS\;WP}\)). These mathematical models are employed to detect anomalous events in the system, providing the normal level of \(AE_{RMS}\) signal for different values of process parameters.  
The chosen method for this investigation is the RSM, a set of mathematical and statistical techniques designed to articulate the connection between the response surface and independent variables (water pressure (\(P\)), abrasive mass flow rate (\(Ma\)), feed rate (\(V\)) and material thickness (\(T\))). The analyzed models included cubic, two-factor interaction, quadratic, and linear models.  
The initial dataset analyzed pertains to \(AE_{RMS\;CH}\). A fitting test was conducted to assess the suitability of various models in representing the experimental data. Among these models, the quadratic model displayed a statistically significant correlation with the data (\(p=0.005\)). This indicates a robust and meaningful representation of the relationship between the process parameters and \(AE_{RMS\;CH}\). The significance level of \(p=0.005\) underscores the improbability of a chance occurrence, affirming the validity of the quadratic model in explaining the observed data. The statistical analysis further revealed a strong fit of the quadratic model to the data, with an _adjusted R-squared_ value of 0.9168. This high _R-squared_ value suggests that approximately 91% of the variability in the dependent variable can be accounted for by the model, rendering it a dependable predictor for the given dataset. The \(AE_{RMS\;CH}\) model equation is provided below:  
\[\begin{array}{rl}AE_{RMS\;CH}=2.69&+0.56\times P-7.0519E-004\times V+8.308 \times M_{a}+0.2037\times T\\ &+3.1011E-006\times P\times V+8E-004\times P\times M_{a}-7.8E\\ &-004\times P\times T-8.08909E-005\times V\times M_{a}-2.58427E\\ &-005\times V\times T-0.17\times M_{a}\times T-5.41867E-005\times P^{2}-1.8516E \\ &-008\times V^{2}-8.10667\times M_{a}^{2}+0.01333\times T^{2}\end{array} \tag{1}\]  
Likewise, for \(AE_{RMS\;WP}\), a significant (\(p=0.0001\)) quadratic model was fitted with an _R-squared value_ of 0.883. About 88% of the variability in the dependent variable can be explained by the quadratic model. The equation of the model is  
\[\begin{array}{rl}AE_{RMS\;WP}=0.37&+1.7647E-003\times P-4.94683E-004\times V -5.19255\times M_{a}\\ &+0.6777\times T+1.5973E-006\times P\times V+5.064E-004\times P\times M_{a}\\ &+1.87E-004\times P\times T+5.4382E-005\times V\times M_{a}-1.90449E\\ &-005\times V\times T-0.232\times M_{a}\times T-4.136E-006\times P^{2}+4.4E\\ &-008\times V^{2}+8.662\times M_{a}^{2}-0.0859\times T^{2}\end{array} \tag{2}\]  
where \(AE_{RMS\;CH}\) is the \(RMS\) of the AE signal at the nozzle [V]; \(AE_{RMSWP}\) is the \(RMS\) of the AE signal at the workpiece [V]; \(P\) is the water pressure [MPa]; \(Ma\) is the abrasive mass flow rate [kg/min]; \(V\) is the feed rate [mm/min]; and \(T\) is material thickness [mm].  
To validate the regression models, 30 experimental trials were conducted. Each model was tested with 10 runs for the CFRP samples of thickness 1, 3, and 5 mm. The maximum deviation between the calculated values using the new mathematical models and the experimental values was observed to be 8.3% for \(AE_{RMS\;CH}\) and 10.5% for \(AE_{RMS\;WP}\).