## 5 Decomposition of Surface Pressure Fields  
The WFT is applied to the unsteady wall pressure fluctuations on the external walls of the train to decompose those into the compressible and incompressible pressure fields. The illustrative WFT is conducted by applying Equation (12) to wall pressure fluctuations on the sidewall of the TC-car depicted in Figure 18. The coordinate system for the wavenumber-frequency analysis is shown in Figures 18 and A1 (in Appendix A). The train running in the open field is considered. The sampling rate \(f_{s}\) and the frequency interval \(\Delta f\) were 5000 Hz and 4.88 Hz, respectively. The power spectral density obtained from the wavenumber-frequency transform of the pressure field on Plane-A in Figure 18 is plotted in Figure 19. The slanted Dirac cone (\(-c_{0}+U_{0}\leq\omega/k\leq c_{0}+U_{0}\)) and the convected incompressible pressure can be distinctly identified. To highlight the characteristics of the 3-D PSD diagram more clearly, the 2-D PSD diagram at the plane of \(k_{x_{2}}=0\) is shown on the right side of Figure 19. Significant compressible components are identified between the characteristic lines of which the slopes correspond to the phase speeds of \(|-c_{0}+U_{0}|=259.67\) m/s and \(|c_{0}+U_{0}|=426.33\) m/s, while the most incompressible components are located around the characteristic line of slope \(U_{c}\cong 0.7U_{0}=58.33\) m/s. Figure 17: Characterization of aerodynamic noise generation due to flow around HVAC box: (**a**) PSD levels of pressure monitored at probe location (red-dot) on HVAC box of head roof, and (**b**) instantaneous filtered pressure fluctuation of 128.6 Hz at t = 1.12 s.  
Figure 18: Surface region (Plane-A) of TC-car selected for wavenumber-frequency analysis. These distinctly separated regions manifest that the LES effectively captures the aerodynamic noise generation and the acoustic wave propagation in the external flow field of the high-speed train. Figure 20a shows the PSD of the total, compressible, and incompressible pressure fields. The spectrum of the compressible and incompressible pressure is obtained by applying the following formula,  
\[\text{S}(\omega)=\int\int S(k,\ \omega)dk_{x_{1}}k_{x_{2}} \tag{12}\]  
The results demonstrate that the incompressible pressure components dominate in most of the frequency range, except for 122.1 Hz, below 1 kHz. Generally, the incompressible components dominate in the low-frequency range while the compressible ones in the high-frequency range. However, the total component around 122.1 Hz is found to be due to compressible pressure from the decomposed pressure spectrum. Figure 20b shows the snapshot of the compressible and incompressible wall pressure fluctuations reconstructed by applying the inverse WFT to the incompressible and compressible regions separately. It can be seen that the compressible pressure fields form the wave pattern with a larger wavelength, which implies a higher wave speed than the incompressible pressure one that convects approximately at the cruising speed of the train. The wavenumber-frequency analysis is also conducted on the surface pressure fields on the wall planes denoted by the symbols B and C in Figure 21. Figure 22 shows the wavenumber-frequency diagrams of surface pressure fields and the corresponding PSD levels of total, compressible, and incompressible pressure for the [lanes of B and C, which are selected due to their proximity to the first bogie, the HVAC box, and the second pantograph, respectively, which are identified as the primary aerodynamic noise sources. It can be seen in Figure 22 that the tonal components can be observed in the decomposed compressible pressure spectra, whereas any of them cannot be found in the total pressure spectra. The tonal component at 127.0 Hz, which is associated with the bogie and the HVAC box, can be found in the PSD of the compressible pressure field on Plane-B in Figure 22a.