## 3 Aerodynamic Noise Analysis of the Panel
### Compressible Numerical Simulation  
#### 3.2.1 WFS Analysis  
(a) Layout of Receiver Points and Cross-PSD  
A total of 1384 receiver points are arranged on the panel, as shown in Figure 22. The distance between each receiver point along the direction of airflow, i.e., x direction, is 0.001 m, and the distance between each receiver point in the longitudinal direction is 0.002 m. In Figure 22, P1 represents 571 receiver points along the airflow direction at the symmetrical line of the side mirror. P2 represents 571 receiver points at 1/2 of the side mirror diameter along the X axis. A total of 121 receiver points are arranged at T1 and T2 locations, which are 0.4 m and 0.6 m away from the coordinate origin. The cut-off frequency is related to the distance between monitoring points, which can be expressed by Equation (17). Since the minimum distance between the two receiver points is 0.001 m, by assuming the convective velocity \(U_{c}=0.55U\), the cut-off frequency is 8800 Hz. \[f_{cut}=\frac{U_{c}}{2.5\Delta r} \tag{17}\]  
where, \(\Delta r\) is the distance between the two receiver points along the direction of flow. Figure 21: Radiated SPLs of the panel under the deterministic modal force with and without acoustic component. Figure 22: The arrangement of the receiver points on the panel. After obtaining the FSP of 1384 receiver points, the pressure of time series receiver points need to be transformed into the pressure in frequency-domain by the Fourier transform. \(p\left(\overset{\rightarrow}{r},t\right)\) is a random static ergodic signal in time and space domain. Its Fourier transform is as follows:  
\[p\left(\overset{\rightarrow}{r},\omega\right)=\frac{1}{2\pi}\int_{0}^{T}p \left(\overset{\rightarrow}{r},t\right)\mathrm{e}^{i\omega t}dt \tag{18}\]  
where, \(T\) is the time interval. The cross-PSD is defined as:  
\[S_{p;p}(\overset{\rightarrow}{x_{i}},\overset{\rightarrow}{r},\omega)= \underset{T\rightarrow\infty}{lim}\frac{2\pi}{T}E[p(\overset{\rightarrow}{x_ {i}},\omega)p^{\ast}(\overset{\rightarrow}{x_{j}}=\overset{\rightarrow}{x_{i }}+\overset{\rightarrow}{r},\omega)] \tag{19}\]  
where, \(E\) is a mathematical expectation. In uniform field, the cross-PSD is only a function of distance \(\overset{\rightarrow}{r}\), i.e., \(S_{p;p_{j}}(\overset{\rightarrow}{x_{i}},\overset{\rightarrow}{r},\omega)=S_ {p;p_{j}}(\overset{\rightarrow}{r},\omega)\). Its spatial Fourier transform is the expression of the cross-PSD in frequency-domain and wavenumber domain, which is usually called the WFS. \[\varPhi(\overset{\rightarrow}{k},\omega)=\frac{1}{\left(2\pi\right)^{2}}\int \int_{-\infty}^{+\infty}S_{p;p_{j}}(\overset{\rightarrow}{r},\omega)\mathrm{ e}^{-i\overset{\rightarrow}{k}\overset{\rightarrow}{r}}d\overset{ \rightarrow}{r} \tag{20}\]  
It can be seen that the key to solving the WFS is to solve the cross-PSD. In this paper, the Welch method is used to estimate the average power spectrum:  
\[S_{xx}(k)=\frac{1}{L}\sum\
olimits_{i=1}^{L}S_{xx}^{i}(k)=\frac{1}{MUL}\sum \
olimits_{i=1}^{L}\sum\
olimits_{n=0}^{M-1}X_{N}^{i}(n)w(n)\underset{N}{W_{N }^{-km}}^{2} \tag{21}\]  
where \(X\) (\(n\)) is a finite sequence of \(N\) points, \(M\) is a sample point, \(L\) is a segment number, \(0\leq n\leq M-1\), \(1\leq i\leq L\), \(U=\frac{1}{M}\sum\
olimits_{n=0}^{M-1}w^{2}(n)\) is a normalization factor, \(w(n)\) is a data window. (b) WFS Analysis  
The WFS analysis is used to calculate the FSP distribution of the panel in time and space domain. According to the SPL and space-time Fourier transform of the receiver points calculated by the compressible numerical method, the WFS of the panel can be calculated by Equation (21), as shown in Figure 23. The wavenumber in the positive direction of \(k_{x}\) is along the airflow direction, and the velocity in the negative direction represents the opposite direction of the airflow.