## 5 Numerical Results
### Flow-Vibroacoustic Simulation Result of HSM  
Finally, the numerical result obtained by applying the flow-vibroacoustic methods to the HSM was compared with the measured data. Figure 22 compares the calculated interior noise spectrum with the measured one for the HSM using the injected power formula of Equation (60). Over-prediction can be seen in the frequency range between 300 Ha and \(\sim\)2500 Hz, whereas under-prediction can be seen in the frequency range higher than 2500 Hz. This result follows the trend of the wall pressure power spectrum in Figure 9a. The discrepancies between the experimental and numerical results were considered to be due to the error caused by Equation (60), which could not account for the vibration characteristics of the window. To improve the accuracy of numerical result, the power supplied to a structure needs to be determined considering the vibration characteristics of the side window and front windshield. Vibration displacement of a plate can be written as a general superposition of normal mode as follows [29]:  
\[w(\mathbf{x},\omega)=\sum_{M=1}^{\infty}a_{M}(\omega)w_{M}(\mathbf{x}). \tag{69}\]  
Modal pressure expresses the degree to which the excitation field \(p\) spatially coupled to the \(M\)-th mode \(w_{M}\) of the plate and can be defined as  
Figure 21: Comparison of numerical results with analytic results: (**a**) FEM model; (**b**) SEA model. Figure 20: Simplified rectangular cavity model enclosed by a flexible panel. \[P_{M}(\omega)=\frac{1}{A}\int_{A}p(\mathbf{x},\omega)w_{M}(\mathbf{x},\omega)d\mathbf{x}. \tag{70}\]  
Under the assumption that \(p(\mathbf{x},t)\) is a random pressure field, the total pressure supplied to the plate is defined as follows:  
\[P(\omega)=\frac{1}{2}Re\bigg{(}-\int_{A}p(\mathbf{x},\omega)v^{*}(\mathbf{x},\omega)d \mathbf{x}\bigg{)}. \tag{71}\]  
Substituting Equations (69) and (70) into Equation (71), followed by rearranging, we obtain the following equation:  
\[P(\omega)=\frac{\Lambda m_{s}}{2}\sum_{M=1}^{\infty}\eta_{M}\omega_{M}|p_{M}|^ {2}|Y_{M}(\omega)|^{2}. \tag{72}\]  
Lastly, the power input \(\Pi_{k,in}\) in the frequency band is defined as follows:  
\[P(\omega)=\frac{Am_{s}}{2}\sum_{M=1}^{\infty}\eta_{M}\omega_{M}\int_{\omega_{ 1}}^{\omega_{2}}|p_{M}|^{2}|Y_{M}(\omega)|^{2}d\omega. \tag{73}\]  
Assuming the admittance has a pronounced peak when \(\omega=\omega_{M}\), and that the mode bandwidth \(\eta_{M}\omega_{M}\) is small compared to \(\omega_{2}-\omega_{1}\), the integral of the power admittance can be read as follows:  
\[\int_{\omega_{1}}^{\omega_{2}}|Y_{M}(\omega)|^{2}d\omega\approx\frac{\pi}{2m_ {s}^{2}\eta_{M}\omega_{M}}. \tag{74}\]  
The injected power by the surface pressure fluctuations to the window glass can be written in the form  
\[P(\omega)=\frac{\pi A}{4m_{s}}\sum_{M_{1}}^{M_{2}}|P_{M}(\omega)|^{2}, \tag{75}\]  
where \(P_{M}(\omega)\) is the modal pressure. The input power of Equation (75) includes the modal pressure related to the mode shape of the plate. In order to reflect the shape of the actual glass window, some data processing was required. The surface pressure was zero-padded in the regions other than the window. The modal shape obtained from the FEM analysis was regenerated in the rectangular domain. The modal pressure was calculated using the surface pressure and the mode  
Figure 22: Comparison of numerical and measured interior noise levels using the injected power formula of Equation (60). shape generated in the rectangular domain. Figure 23 shows the regenerated pressure and mode shape. Figure 24 compares the calculated interior noise spectrum with the measured one for the HSM using the injected power formula of Equation (56). First, it can be found that the numerical result obtained by combining the total surface pressure field from the high-resolution LES simulation result with the FE-SEA methods showed excellent agreement with the measured data. Figure 19 also shows the numerical results obtained using the individual contribution of hydrodynamic and acoustic surface pressure fields.