## 3 Construction of the Noise Prediction Model  
### Finite-Length Incoherent Moving Line Source Model  
Figure 10 provides a schematic of the acoustic radiation from a finite-length moving line source within the global coordinate system \(O\)-\(xyz\). The line source moves along the positive \(x\)-axis at velocity \(v_{0}\), with the reception point \(R\) located at global coordinates (0, \(d_{0}\), \(h_{0}\)) and maintaining a perpendicular distance \(r_{0}\) from the source axis. The blue cylinder represents the infinitesimal element of a line source; its coordinates are (\(x^{\prime}\), 0, 0), and the radiated acoustic power is \(w(x^{\prime})\)dx\({}^{\prime}\). \(\tau\) is the moment at which the line source element emits sound waves, \(t\) is the moment at which \(R\) receives the sound waves emitted by the line source element at time \(\tau\), and the center of the line source moves exactly to point \(O\) at time \(t=0\). The spatial global \(x\)-axis coordinates of the element at times \(\tau\) and \(t\) are \(x^{\prime}+v_{0}\tau\) and \(x^{\prime}+v_{0}t\), respectively. The distance between the element and \(R\) is \(r\), and the angles between the lines from the element to \(R\) and the \(x\)-, \(y\)-, and \(z\)-axes are \(\varphi\), \(\theta\), and \(\gamma\), respectively. \(r\) is the distance that the sound wave travels at the sound speed \(c_{0}\), which can be expressed as  
\[r=c_{0}(t-\tau)=\sqrt{\left(x^{\prime}+v_{0}\tau\right)^{2}+r_{0}^{2}} \tag{1}\]  
Figure 10: Schematic of acoustic radiation from finite-length moving line sources. Figure 9: Three-dimensional vortex structures visualized by velocity-colored Q-criterion around the Maglev train at 430 km/h. After squaring Equation (1), it can be arranged into a quadratic equation about \(\tau\):  
\[(1-M_{0}^{2})\tau^{2}-2\bigg{(}t+\frac{M_{0}x^{\prime}}{c_{0}}\bigg{)}\tau+t^{2}- \frac{{x^{\prime}}^{2}+r_{0}^{2}}{c_{0}^{2}}=0, \tag{2}\]  
where the Mach number \(M_{0}=v_{0}/c_{0}<1\), and the following solution for \(\tau\) can be obtained:  
\[\tau=\frac{M_{0}x^{\prime}+c_{0}t\pm\sqrt{(x^{\prime}+v_{0}t)^{2}+(1-M_{0}^{2} )\cdot r_{0}^{2}}}{c_{0}(1-M_{0}^{2})}. \tag{3}\]  
Substituting the above formula into Equation (1) obtains  
\[r=c_{0}(t-\tau)=\frac{-M_{0}(x^{\prime}+v_{0}t)\mp\sqrt{\big{(}x^{\prime}+v_{0 }t\big{)}^{2}+(1-M_{0}^{2})\cdot r_{0}^{2}}}{1-M_{0}^{2}}. \tag{4}\]  
As \(\sqrt{\big{(}x^{\prime}+v_{0}t\big{)}^{2}+(1-M_{0}^{2})\cdot r_{0}^{2}}>|x^{ \prime}+v_{0}t|>|-M_{0}(x^{\prime}+v_{0}t)|\), \(r\) is always greater than 0 when a positive sign is taken before the root of the upper formula, and \(r\) is always less than 0 when a negative sign is taken. As \(r\) is a distance, it only has meaning when \(r\) is a positive real number and a positive sign is taken before the root of the upper formula. If the directivity of the sound source is not considered, the square of the sound pressure generated by the sound wave emitted by the microelement propagating to \(R\) in the free field is [19; 20]  
\[\mathrm{d}{p^{\prime}}^{2}=\frac{\rho c_{0}w\big{(}x^{\prime}\big{)}}{4\pi r \big{(}x^{\prime}\big{)}^{2}}\mathrm{d}x^{\prime}=\frac{\rho c_{0}w\big{(}x^{ \prime}\big{)}\big{(}1-M_{0}^{2}\big{)}^{2}}{4\pi\bigg{(}\sqrt{\big{(}x^{ \prime}+v_{0}t\big{)}^{2}+(1-M_{0}^{2})\cdot r_{0}^{2}}-M_{0}\cdot\big{(}x^{ \prime}+v_{0}t\big{)}\bigg{)}^{2}}\mathrm{d}x^{\prime}. \tag{5}\]  
When a Maglev train runs at high speed, the fluid on the boundary is subjected to pulsating pressure on the surface of the car body, and the pulse power source can be regarded as a dipole sound source whose dipole moment direction is consistent with the side normal direction of the car body.