## 4 Acoustic Field Reconstruction
### Reconstruction of Sausal Noise and Comparison with Experiments  
The acoustic field reconstruction based on the method proposed in Section 4.1 is applied to validate its effectiveness for reproducing the characterization of squeal noise at the location of the antenna microphones. These numerical calculations for the reconstruction of the radiated field are compared with the direct pressure measurements provided by the antenna microphones (by considering the main frequencies of interest, as previously stated in Section 3.2) in order to validate the assumptions of the method adopted, in particular the plane wave and free field assumptions. We notice that the Tikhonov regularization proposed in Equation (4) prevents the reconstruction from exactly matching the experiments, since we use the \(\varepsilon\) value recommended by [PERSON] [31], which is assumed to be well-suited to the method proposed. Figures 11-15 show the reconstruction of the acoustic patterns for frequencies \(f_{1}=610\) Hz, \(f_{2}=1220\) Hz, \(f_{3}=1830\) Hz, \(f_{4}=2440\) Hz and \(f_{5}=3050\) Hz, respectively. The reconstruction of the radiated acoustic field is proposed by examining nine frames displayed evenly over time by considering one period of disc revolution (to be compared to the previous radiated acoustic fields obtained by the direct pressure measurements of the microphones antenna shown in the first nine subfigures of Figures 6-10, respectively). These show that the evolution of the radiated field is repeatable over several rounds and that a nine-frame discretization is sufficient to provide an appropriate representation of the temporal evolution of the acoustic pattern for both level and directivity. Whatever the frequency considered, it can be seen very clearly that the different patterns and their evolution in terms of both directivity and level are very well reproduced. Figure 11: Reconstruction of the radiated field for \(f_{1}=610\) Hz and its temporal evolution for one disc revolution (to be compared with Figure 6)—to be read from left to right, top to bottom. Figure 12: Reconstruction of the radiated field for \(f_{2}=1220\) Hz and its temporal evolution for one disc revolution (to be compared with Figure 7)—to be read from left to right, top to bottom. Figure 13: Reconstruction of the radiated field for \(f_{3}=1830\) Hz and its temporal evolution for one disc revolution (to be compared with Figure 8)—to be read from left to right, top to bottom. Figure 14: Reconstruction of the radiated field for \(f_{4}=2440\) Hz and its temporal evolution for one disc revolution (to be compared with Figure 9)—to be read from left to right, top to bottom. Figure 15: Reconstruction of the radiated field for \(f_{5}=3050\) Hz and its temporal evolution for one disc revolution (to be compared with Figure 10)—to be read from left to right, top to bottom. In order to quantify the error between the direct pressure measurements and the reconstructed acoustic field for each frequency of interest in the squeal noise, the well-known correlation coefficient is used. Considering two families of sequences defined by \(\mathbf{A}=[\mathbf{A}_{1},\mathbf{A}_{2},\ldots,\mathbf{A}_{N}]\) and \(\mathbf{B}=[\mathbf{B}_{1},\mathbf{B}_{2},\ldots,\mathbf{B}_{N}]\), the associated correlation coefficients can be represented in matrix \(\mathbf{R}\) according to the following formula:  
\[\mathbf{R}_{mn}=\frac{\left|(\mathbf{A}_{n}-\bar{\mathbf{A}}_{n})^{\mathrm{T}}.(\mathbf{B}_{m}-\bar{\mathbf{B}}_{m})\right|^{2}}{\|\mathbf{A}_{n}^{\mathrm{ T}}-\bar{\mathbf{A}}_{n}\|.\|\mathbf{B}_{m}^{\mathrm{T}}-\bar{\mathbf{B}}_{m}\|}, \tag{14}\]  
where \(\bar{\mathbf{A}}_{\mathbf{n}}\) and \(\bar{\mathbf{B}}_{m}\) are the mean values of vectors \(\mathbf{A}_{n}\) and \(\mathbf{B}_{m}\). \(\mathbf{R}_{mn}\) is called the scalar correlation coefficient corresponding to the \(n\)th sequence of \(\mathbf{A}\) and the \(m\)th sequence of \(\mathbf{B}\). Each correlation coefficient returns a scalar value between \(0\) and \(1\).