## 2 Numerical Methods and Target Model
### Wavenumber-Frequency Analysis  
The wavenumber-frequency diagram of surface pressure fluctuation can be obtained from the three-dimensional Fourier transform of the surface pressure in the space-time domain. It shows the distribution of the magnitude of the specific wave component in terms of wavenumber \(k\) and frequency \(\omega\) in the wavenumber-frequency domain. One of the advantages of using the wavenumber-frequency spectrum instead of the space-time diagram is that the phase speed of the specific wave component constituting surface pressure fluctuations in the space-time domain can be determined. The phase speed \(v_{p}=\omega/k\) can be used as a criterion to decompose the surface pressure fluctuation into its incompressible and compressible components. The former is convected at the velocity proportional to mean flow velocity \(U_{0}\), and the latter does so at the vector sum of mean flow velocity and the sound speed \(c_{0}\). The spatial-temporal pressure field on a train wall can be characterized using the correlation function. \[R(\mathbf{x};\mathbf{\xi},\tau)=\big{\langle}p^{\prime}(\mathbf{x},t)\cdot p^{\prime}(\bm {x}+\mathbf{\xi},t+\tau)\big{\rangle} \tag{5}\]  
where \(\mathbf{\xi}\) is the distance vector, \(\tau\) is the time delay, and \(p^{\prime}\) is the perturbed pressure. The correlation function of the spatial-temporal pressure field can be converted to the wavenumber-frequency domain using a Fourier transform. The Fourier transform from the space-time domain to the wavenumber-frequency domain is written as  
\[S(\mathbf{k},\omega)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty} ^{\infty}R(\mathbf{\xi},\tau)e^{-i(k\mathbf{\xi}+\omega\tau)}d\xi_{1}d\xi_{2}d\tau \tag{6}\]  
The power spectral density (PSD) of the wall pressure fluctuations can be estimated using the periodogram method in the following form:  
\[S(\mathbf{k},\omega)=\lim_{T,L_{1},L_{2}\rightarrow\infty}\frac{1}{L_{1}}\frac{ 1}{L_{2}}\frac{1}{T}\left\langle\dot{\rho}_{L,T}(\mathbf{k},\omega)\cdot\dot{\rho }_{\mathbf{L},T}^{*}(\mathbf{k},\omega)\right\rangle \tag{7}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Equations** & **Discretization Method** \\ \hline Pressure & Second order \\ Momentum & Bounded central differencing \\ Energy & Second order upwind \\ Transient formulation & Second order implicit \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Discretization schemes of governing equations. where  
\[\hat{p}_{L,T}(\mathbf{k},\omega)=\int_{-T/2}^{T/2}\int_{-L_{1}/2}^{L_{1}/2}\int_{-L_{2 }/2}^{L_{2}/2}p^{\prime}(\mathbf{x},t)e^{-i(\mathbf{kx}+\omega t)}dx_{1}dx_{2}dt \tag{8}\]  
and \(\hat{p}_{L,T}^{*}\) is a complex conjugate of \(\hat{p}_{L,T}\). In Equation (8), \(L_{1}\), and \(L_{2}\) are the length of space domain in each direction, respectively, and \(T\) is the time period. Finally, the three-dimensional discrete Fourier transform equation used to obtain the three-dimensional averaged modified periodogram can be written in the following form [7; 8]. \[ \begin{array}{c}S_{m,n,\rho}=S(m\Delta f,n\Delta k_{x_{1}},\omega\Delta k_{x_ {2}})=\\ \frac{\Delta x_{1}\Delta x_{2}}{N_{1}N_{x_{1}}N_{x_{2}}}\sum_{k=0}^{N_{1}-1} \sum_{l=0}^{N_{x_{1}}-1}\psi_{klj}P_{klj}e^{-2\pi n(\frac{km}{N_{l}}\frac{ml}{ N_{x_{1}}}-\frac{\phi j}{N_{x_{2}}})}\Bigg{|}^{2}/\frac{1}{N_{i}N_{x_{1}}N_{x_{2}}} \sum_{k=0}^{N_{x_{1}}-1}\sum_{l=0}^{N_{x_{2}}-1}\Big{|}w_{klj}\Big{|}^{2}\end{array}  \tag{9}\]  
where \(w_{klj}\) is a discretized Hanning window. In the three-dimensional periodogram, the incompressible and compressible parts can be separated by using the slanted Dirac cone defined as  
\[\omega=c_{0}\sqrt{k_{x_{1}}^{2}+k_{x_{2}}^{2}}+k_{x_{1}}U_{0} \tag{10}\]  
where \(U_{0}\) is free stream velocity.