## 4 Acoustic Field Reconstruction
### 3D Reconstruction of Squeal Noise  
As previously discussed in Section 4.2, the acoustic field reconstruction method is proposed to reproduce the squeal noise of the brake system subjected to friction-induced vibration. Its validation is obtained by comparing the estimated acoustic field and the data measurements at the location of the 117 microphones. In the following section, we focus on the ability to reconstruct the radiated sound field anywhere in the space surrounding the braking system (downstream of the antenna). This should provide a more accurate pattern of the acoustic radiated field and thus provide further insights into the 3D radiated acoustic field during squeal events. The main assumption of the 3D sound field reconstruction calculations is that the previous results obtained in the pressure field reconstruction at the measurement points make it possible to extend the reconstruction method to the whole space. More specifically, regarding the 3D sound field reconstruction strategy, it is carried out by defining different 2D planes of interest, on which the propagation of the acoustic radiated field is estimated. In order to achieve this objective, the reconstructed acoustic field is calculated at \(N_{P}\) specific points on the selected 2D plane. Then, the calculation of the pressure field at the \(N_{P}\) assessment points according to the measurements of the 117 microphones is performed as explained in Section 4.1. To obtain a precise representation of the reconstructed radiated field, \(N_{P}\) is chosen to be equal to 150. More specifically, to illustrate the results based on this 3D reconstruction approach, three analyses are carried out on selected specific 2D planes of interest:  
* one section defined along the plane \((0,x,y)\) and located at \(z=2\) m. These results correspond to the reconstruction of the acoustic noise field that is relatively distant from the experimental measurements of the antenna microphones;
* one section defined along the vertical plane \((0,z,y)\) and passing through the center of the disc. This provides a visualization of the acoustic propagation in the vertical plane over a distance of 0 to 4 m with respect to the direction normal to the plane of the brake disc (along z-axis);
* one section defined along the horizontal plane \((0,z,x)\) and passing through the center of the disc (i.e., orthogonal to the chosen vertical plane \((0,z,y)\)). This offers a complementary representation to the previous visualization of the propagation of the acoustic field over a distance of 0 to 4 m.  
Figures 17-21 show the propagation of the acoustic radiated field during squeal events (at the fixed time \(t=24\) s) for the five main frequencies of interest, respectively. This offers an overview of the acoustic levels and directivity involved for each frequency of interest. However, given the potential variability of the acoustic pattern as a function of time (as discussed previously in Section 2), it is advisable to treat these results with caution, as they are a reflection of a single reconstruction of the acoustic field for one specific time \begin{table}
\begin{tabular}{l c c c c c} \hline  & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) \\ \hline \(\underset{m}{\min}[\mathbf{R}_{mm}]\) & 0.85 & 0.8 & 0.99 & 0.98 & 0.99 \\ \(\underset{(m,n)}{\min}[\mathbf{R}_{mm}]\) & 0.07 & 0.37 & 0.81 & 0.95 & 0.86 \\ Evolution vs. time & very strong & strong & low & none & low \\ \hline \end{tabular}
\end{table} 
Table 2: Uses of the indicator \(\mathbf{R}_{mn}\) to characterize the evolution of the acoustic field versus time for one disc revolution. The representations of the squeal noise in the \((0,x,y)\) plane show that the acoustic pattern radiated at \(z=2\) m is different depending on the frequency considered, both in level and directivity.