## 4 Acoustic Field Reconstruction
### Reconstruction of Sausal Noise and Comparison with Experiments
If \(\mathbf{R}_{mn}\) is close to one, this means that there is a very good correlation between the two sequences, while a value close to zero corresponds to a bad correlation. It should be noted that in the present case, vectors \(\mathbf{A}_{i}\) and \(\mathbf{B}_{i}\) (with \(i=1,\ldots,N\)) correspond to the data for one specific frequency of interest, obtained from the direct pressure measurements and the reconstructed acoustic field for the \(117\) microphones (i.e., the size of vectors \(\mathbf{A}_{i}\) and \(\mathbf{B}_{i}\) is \(117\times 1\)). Moreover, as explained previously, the evolution of the acoustic radiated field patterns for one disc revolution is discretized by using nine frames, thus the scalar \(N\) is equal to nine (i.e., the size of matrices \(\mathbf{A}\) and \(\mathbf{B}\) is \(117\times 9\)). The results provided in Figure 16 offer the scalar correlation coefficient \(R_{mn}\) (for \(m=1,\ldots,9\) and \(n=1,\ldots,9\)) between the direct pressure measurements and the reconstructed acoustic field for each frequency of interest and by considering the nine sequences per disc revolution. Thus, the diagonal terms provide information on the quality of the method for reconstructing the acoustic field and its evolution over one revolution of the disc, while the extra-diagonal terms provide only information on the similarity for the two selected samples (i.e., one experimental item of data from the \(m\)th sequence and one reconstructed field from the \(n\)th sequence with \(n\
eq m\)) considered at two different times. By analyzing the results proposed in Figure 16, it can first be mentioned that the effectiveness of the reconstruction method is validated since all the diagonal terms, whatever the frequency considered, are composed of values very close to one. This shows, unambiguously, that for all \(117\) microphones, the evolution of the reconstructed radiated field is in very good agreement with the evolution of the measured acoustic field and thus that the reconstruction of the squeal noise by the method proposed provides an appropriate representation of the temporal evolution of the acoustic pattern for both level and directivity. In addition, due to the fact that the reconstructed radiated field and the direct acoustic data measurements are similar over the nine discretized times, the extra-diagonal terms of \(\mathbf{R}_{mn}\) can be used to quantify the temporal evolution, per one revolution of the disc, for the radiated acoustic pattern for both level and directivity for the five frequencies of interest. Thus, if we consider the results associated with the first two frequencies more specifically (see Figure 16a for \(f_{1}=610\) Hz and Figure 16b for \(f_{2}=1220\) Hz), the values of the extra-diagonal terms indicate a temporal evolution of the sound field pattern over one revolution of the disc. This is in very good agreement with the experimental observations proposed in Section 3.2 which indicate that the variation of the radiated acoustic fields in terms of level and directivity can be linked to the rotation speed of the disc. Moreover, the fact that this phenomenon is considered to be greater for \(f_{1}\) is also reflected in the values of the extra-diagonal coefficients \(\mathbf{R}_{mn}\) indicating values closer to zero for frequency \(f_{1}\). This points to non-similarity between two quantities taken at different temporal instants, characterizing a stronger temporal evolution of the acoustic radiated field. Furthermore, the extra-diagonal terms for the last three frequencies of interest all remain above \(0.8\) (see Figure 16c for \(f_{3}=1830\) Hz, Figure 16d for \(f_{4}=2440\) Hz and Figure 16e for \(f_{5}=3050\) Hz). This indicates that the acoustic pattern changes very little over time. These results are again in very good agreement with the observations proposed previously in Section 3.2.