# The Refined Simulation Methods of Fluctuating Wind and Aerodynamic Forces Relative to Moving Trains
## 2 The Wind Spectrum of Moving Trains Under Different Wind Environments
### Inhomogeneous and Non-Stationary Wind Environments
The separation distances \(\Delta x\) and \(\Delta y\) in Figure 1b are expressed as follows:  
\[\Delta x =(V_{tr}\tau-\Delta\eta)cos\varphi_{0} \tag{4}\] \[\Delta y =(V_{tr}\tau-\Delta\eta)sin\varphi_{0} \tag{5}\]  
Based on the above relations, Equation (1) is further expressed as follows:  
\[R_{\widetilde{u}\widetilde{u}}^{\widetilde{\cdot}\widetilde{\cdot}\widetilde{ \cdot}}(\Delta y,\tau,t)=E\Big{[}\widetilde{u}\big{(}x_{j},y_{j},t\big{)} \widetilde{u}\big{(}x_{m},y_{m},t+\tau+\tau^{\prime}\big{)}\Big{]}=R_{ \widetilde{u}\widetilde{u}}\big{(}\Delta y,\tau+\tau^{\prime},t\big{)} \tag{6}\]  
where \(R_{\widetilde{u}\widetilde{u}}(\Delta y,\tau+\tau^{\prime},t)\) is the cross-correlation function of the along-wind fluctuating wind at spatial positions \(j\) and \(m\) at time \(t\) in the non-stationary wind environment. The correlation function of the fluctuating wind between points \(P\) and \(P^{\prime}\) on the moving train and spatial positions \(j\) and \(m\) is established through the above derivation. The correlation function is determined by the inverse Fourier transform of the wind spectrum, thus the equation is as follows:  
\[R_{\widetilde{u}\widetilde{u}}(\Delta y,\tau,t)=R_{\widetilde{u}\widetilde{u }}\big{(}\Delta y,\tau+\tau^{\prime},t\big{)}=\int_{0}^{+\infty}S_{ \widetilde{u}\widetilde{u}}(\Delta y,f,t)\text{cos}\big{[}2\pi f\big{(}\tau+ \tau^{\prime}\big{)}\big{]}df \tag{7}\]  
where \(S_{\widetilde{u}\widetilde{u}}(\Delta y,f,t)\) is the cross-spectrum of the along-wind fluctuating wind at spatial positions \(j\) and \(m\) at time \(t\) in the non-stationary wind environment; \(f\) is the frequency in Hz. For inhomogeneous and non-stationary wind environments, the cross-spectrum of the fluctuating wind in Equation (7) can be rewritten as the following expressions according to the evolutionary spectrum theory [20]:  
\[S_{\widetilde{u}\widetilde{u}}(\Delta y,f,t)=Coh_{\widetilde{u}}(\Delta y,f,t )\sqrt{S_{\widetilde{u}}(y_{j},f,t)S_{\widetilde{u}}(y_{m},f,t)} \tag{8}\]  
\[Coh_{\widetilde{u}}(\Delta y,f,t)=\text{exp}\left(-\frac{fC_{u}|\Delta y|}{0. 5\big{[}U(y_{j},t)+U(y_{m},t)\big{]}}\right) \tag{9}\]  
\[\sqrt{S_{\widetilde{u}}(y_{j},f,t)S_{\widetilde{u}}(y_{m},f,t)}=A\big{(}y_{j},f,t\big{)}A(y_{m},f,t)\sqrt{S_{u}(y_{j},f)S_{u}(y_{m},f)} \tag{10}\]  
where \(S_{\widetilde{u}}(y_{j},f,t)\), \(S_{\widetilde{u}}(y_{m},f,t)\) are the auto-spectrum functions of the along-wind fluctuating wind at spatial stationary positions \(j\) and \(m\) at time \(t\) in the inhomogeneous and non-stationary wind environment, which are evolutionary spectra; \(S_{u}\big{(}y_{j},f\big{)}\), \(S_{u}(y_{m},f)\) are the one-side auto-spectrum functions of the along-wind fluctuating wind at spatial stationary positions \(j\) and \(m\) in the inhomogeneous and stationary wind environment; \(A\big{(}y_{j},f,t\big{)}\), \(A(y_{m},f,t)\) are the modulation functions of fluctuating wind at spatial positions \(j\) and \(m\) in the inhomogeneous wind environment; \(Coh_{\widetilde{u}}(\Delta y,f,t)\) is the coherence function of the fluctuating wind at spatial positions \(j\) and \(m\) at time \(t\) in the non-stationary wind environment; \(C_{u}\) is the decay coefficient of the along-wind fluctuating wind; and \(U(y_{j},t)\), \(U(y_{m},t)\) are the mean wind speeds at spatial positions \(j\) and \(m\) at time \(t\), respectively. The cross-spectrum of the fluctuating wind relative to moving trains in the inhomogeneous and non-stationary wind environment is obtained by the Fourier transform of Equation (7) as follows:  
Figure 1:0  
where \(S_{u\;u}(\Delta\eta,f,t)\) is the cross-spectrum of the along-wind fluctuating wind relative to the moving train at time \(t\) in the inhomogeneous and non-stationary wind environment. Equation (11) is the auto-spectrum of the fluctuating wind at point \(P\) on the moving train at time \(t\) as \(\Delta\eta=0\). It should be noted that when \(\Delta\eta\
eq 0\), the \(|\Delta y|\) in the coherence function is not a symmetrical function at \(\tau=0\).