## 2 The Bridge-Borne Noise Prediction Model
### The Bridge BEM Model  
In the acoustic boundary element theory, the sound pressure of the analysis area \(\Omega\) in the ideal medium obeys the wave equation [27; 28]. \[\
abla^{2}P(s,t)=\frac{1}{c^{2}}\frac{\partial^{2}p(s,t)}{\partial t^{2}} \tag{3}\]  
where \(\
abla^{2}\) is the Laplace operator; \(P(s,t)\) is the sound pressure (Pa) of point \(s\) at time \(t\) (s); \(c\) is the sound speed (m/s). Considering the independence of time, the sound pressure of a harmonic sound field can be expressed as  
\[P(s,t)=p(s)\mathrm{e}^{-i\omega t} \tag{4}\]  
where \(i\) is the imaginary unit; \(\omega\) is the angular frequency (rad/s). Substituting Equation (4) into Equation (3) yields the sound pressure-based Helmholtz control differential equation:  
\[\
abla^{2}p(s)+k^{2}p(s)=0 \tag{5}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline Mass of the car body & 33,660 & kg \\ Mass of the container (dead load/carrying & 210/850 & kg \\ capacity) & 2235 & kg \\ Mass of the buge frame & 1451 & kg \\ Primary suspension & Longitudinal & \(9.2\times 10^{5}\) & N\(\cdot\)m\({}^{-1}\) \\ stiffness & Lateral & \(9.2\times 10^{5}\) & N\(\cdot\)m\({}^{-1}\) \\ (each side) & Vertical & \(7.5\times 10^{5}\) & N\(\cdot\)m\({}^{-1}\) \\ Stiffness of the axle & Longitudinal & \(4\times 10^{7}\) & N\(\cdot\)m\({}^{-1}\) \\ box rubber joint & Lateral & \(1.2\times 10^{7}\) & N\(\cdot\)m\({}^{-1}\) \\ Secondary & Longitudinal & \(1.13\times 10^{5}/1.23\times 10^{5}\) & N\(\cdot\)m\({}^{-1}\) \\ suspension stiffness & Lateral & \(1.13\times 10^{5}/1.23\times 10^{5}\) & N\(\cdot\)m\({}^{-1}\) \\ (unloaded/loaded) & Vertical & \(1.85\times 10^{5}/2.15\times 10^{5}\) & N\(\cdot\)m\({}^{-1}\) \\ Equivalent three-dimensional stiffness of the & & \(1\times 10^{7}\) & N\(\cdot\)m\({}^{-1}\) \\ restraining wedge & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key parameters of the freight EMU. where \(k\) represents the wave number of the sound wave. For acoustic problems in general bounded spaces without distributed source terms, Equation (5) has the following Helmholtz solution, which represents the sound pressure of the point \(s\) in the sound field, with the integral form of [8], as follows  
\[p(s)=\left\{ \begin{array}{l}\int_{\Gamma}\left[\begin{array}{c}G(s,y)q(y)-\\ \frac{\partial G(s,y)}{\partial n(y)}p(y)\end{array}\right]d\Gamma(y),s\in \Omega,s\
ot\in\Gamma\\ \\ \frac{1}{c(s)}\int_{\Gamma}\left[\begin{array}{c}G(s,y)d(y)\\ -\frac{\partial G(s,y)}{\partial n(y)}p(y)\end{array}\right]d\Gamma(y),s\in\Gamma \end{array} \right. \tag{6}\]  
where [PERSON]'s function \(G(s,y)=\frac{e^{\mathrm{i}R(s,y)}}{4\pi R(s,y)}\) represents the sound pressure at point \(s\) produced by a point sound source with unit intensity located at point \(y\) in the space; \(\Gamma\) is the surface boundary of the sound source, \(n\) represents the outward normal direction on the surface boundary of the sound source; \(c(s)\) is the coefficient related to the geometric characteristics of point \(s\). Discretizing Equation (6) using the boundary element method, the structure radiated acoustic power (W) can be obtained as [27], as follows:  
\[W=-\frac{1}{2}\mathrm{Re}\!\left\{\int_{\Gamma}P_{\Gamma}(s)v_{\Gamma}(s)^{*}d \Gamma(s)\right\} \tag{7}\]  
where \(\mathrm{Re}\) denotes the real part of a complex number; superscript \({}^{*}\) represents complex conjugation; \(v_{\Gamma}(s)\) denotes the normal velocity at point \(s\). Taking the above vibration responses of the FEM mode as the boundary condition, the BEM model of a box girder bridge is established in the software Virtual. Lab 13.6. The mesh size is required to be less than 1/6 of the shortest wavelength of the analysis frequency. As the maximum analysis frequency analyzed in this study is 200 Hz, the maximum mesh size is determined as 0.28 m, and the mesh size of 0.2 m is adopted in the modeling.