## 5 Decomposition of Surface Pressure  
The illustrative wavenumber-frequency analysis was conducted by applying Equation (6) to the predicted surface pressure fields on the roof and the left-side wall surfaces of the TC car shown in Figure 15. The sampling rate \(f_{s}\) and the frequency interval \(\Delta f\) were 5000 Hz and 2.03 Hz, respectively. The wavenumber intervals \(\Delta k_{x}\) and \(\Delta k_{z}\) were 0.01954 (1/m) and 0.4379 (1/m), respectively. The power spectral density obtained from the wavenumber-frequency transform is plotted in Figure 16. The slanted Dirac cone (\(-c_{0}+U_{0}\leq\omega/k\leq c_{0}+U_{0}\)) was distinctly identified. To show the characteristics of the wavenumber-frequency diagram more clearly, the two-dimensional spectrum at the cross-sectional plane of \(k_{y}=0\) and \(k_{z}=0\) for the roof and left-side walls is shown in the right side of Figure 16. Strong compressible components were identified, between the characteristic lines of which the slopes corresponded to the phase speeds of \(|-c_{0}+U_{0}|=259.67\) m/s and \(|c_{0}+U_{0}|=426.33\) m/s, while most of the incompressible components were located on the line of slope \(U_{c}\simeq 0.9U_{0}=75.00\) m/s. These distinctly separated regions manifested that the LES simulation effectively captured the acoustic wave propagation as well as the aerodynamic noise generation in the external flow of the high-speed train. Figure 17a,b shows the power spectral density diagrams for the decomposed incompressible and compressible parts, respectively, and the corresponding power spectral density levels which were obtained by integrating the power spectral density over the entire wavenumber ranges in the form  
\[\mathrm{S}(\omega)=\int\int S(\mathbf{k},\ \omega)dk_{x_{1}}k_{x_{2}} \tag{13}\]  
Figure 14: Streamlines near intercoach space: red: from bogie and bogie fairing; blue: from sidewall; green: from roof. Figure 15: Surface regions of TC car for wavenumber–frequency analysis: (**a**) roof wall region; (**b**) left-side wall region. Equation (13) represents the averaged power spectral density levels of pressure over the surface regions shown in Figure 18. It can be seen that the incompressible pressure parts dominated in most of the frequency range, except for 120 Hz. This strong tonal component was caused by the coherent vortex shedding from the wiper, the headlamp, and first bogie fairing, as shown in Figure 17. The pressure spectrum obtained at the indicated locations showed the tonal peak at the frequency of 120 Hz. Figure 16: Three-dimensional power spectral density diagram (left) and diagrams at a given \(k_{y}=0\) and \(k_{z}=0\) (right)—reference pressure 20 \(\upmu\)Pa; (**a**) roof wall region and (**b**) left-side wall region. Figure 17: Pressure spectrum at Point 1 and Point 2: (**a**) locations of Point 1 and Point 2; (**b**) pressure spectrum at Point 1; (**c**) pressure spectrum at Point 2. In order to obtain the compressible and incompressible surface pressure fields in the time-space domain, the inverse Fourier transform was taken for each separated region shown in Figure 16.0 Figure 19 shows the snapshot of incompressible and compressible surface pressure fields obtained in this way, together with the total pressure field for all coaches. It can be seen that the magnitudes of the incompressible components were generally much higher than those of the compressible ones, so that the total pressure field was almost similar to the incompressible pressure field. Although the magnitudes of compressible components were much lower, the present numerical method was able to extract the hidden acoustic pressure field successfully. It is well known that subsonic surface waves support an evanescent wave in a fluid which is always confined to the vicinity of the interface [11]. This fact implies that the acoustic pressure field with higher phase speed than the incompressible pressure field may make more contribution to the interior sound field of a high-speed train, which needs to be investigated in a future study.