## 3 Construction of the Noise Prediction Model
### Finite-Length Incoherent Moving Line Source Model
If a line source element is considered to have dipole directivity, then the square of the sound pressure propagated by the sound wave emitted by the element to reception point \(R\) can be expressed as  
\[\mathrm{d}{p^{\prime}}^{2}=\frac{\rho c_{0}w\big{(}x^{\prime}\big{)}\cos^{2} \theta}{4\pi r\big{(}x^{\prime}\big{)}^{2}}\mathrm{d}x^{\prime}=\frac{\rho c_ {0}d_{0}^{2}w\big{(}x^{\prime}\big{)}}{4\pi r\big{(}x^{\prime}\big{)}^{4}} \mathrm{d}x^{\prime}, \tag{6}\]  
where \(\cos\theta=d_{0}/r\). Without the loss of generality, the directivity factor of \(\mathrm{d}{p^{\prime}}^{2}\) can be expressed as \(\cos^{2m}\theta\), and \(m\) values of 0, 0.5, and 1, respectively, yield the directivity of no direction, intermediate directivity, and dipole directivity. Then, the square of the sound pressure of the sound wave propagated by the element to \(R\) is  
\[\mathrm{d}{p^{\prime}}^{2}=\frac{\rho c_{0}w\big{(}x^{\prime}\big{)}\cos^{2m} \theta}{4\pi r\big{(}x^{\prime}\big{)}^{2}}\mathrm{d}x^{\prime}=\frac{\rho c_{ 0}d_{0}^{2m}w\big{(}x^{\prime}\big{)}}{4\pi r\big{(}x^{\prime}\big{)}^{2+2m}} \mathrm{d}x^{\prime}. \tag{7}\]  
If the sound source is a uniform and incoherent line source, then the square of the sound pressure at \(R\) at time \(t\) is the linear superposition [21] of the squares of the sound pressures when the sound waves emitted by each element reach the receiving point at time \(t\), so the SPL at \(R\) at time \(t\) can be expressed as  
\[ \begin{split}& SPL(t)=10\log_{10}\left(p^{\prime}(t)^{2}/{p^{ \prime}}_{ref}^{2}\right)\\ &=10\log_{10}\left(\int\limits_{-l/2}^{l/2}\frac{\rho c_{0}wd_{0} ^{2m}(1-M_{0}^{2})^{2+2m}}{4\pi{p^{\prime}}_{ref}^{2}\left(\sqrt{\left(x^{ \prime}+v_{0}t\right)^{2}+(1-M_{0}^{2})\cdot r_{0}^{2}}-M_{0}\cdot\left(x^{ \prime}+v_{0}t\right)\right)^{2+2m}}\text{d}x^{\prime}\right)\,\end{split}  \tag{8}\]  
where \({p^{\prime}}_{ref}\) is the reference sound pressure, with a value of 20 \(\mu\)Pa, and \(l\) is the length of the line source. A moving object causes a stress change in the surrounding fluid, which can be regarded as a pair of equal and opposite dipoles acting on the same spatial position, that is, a quadrupole sound source. If the linear sound source element is considered to have equiaxial quadrupole directivity and the plane formed by a pair of dipoles that make up the quadrupole is parallel to the \(y\)-\(z\) plane, then the square of the sound pressure propagated by the sound wave emitted by the element to the reception point can be expressed as  
\[\text{d}{p^{\prime}}^{2}=\frac{\rho c_{0}w(x^{\prime})\cos^{2}\theta\cos^{2} \gamma}{4\pi r{(x^{\prime})}^{2}}\text{d}x^{\prime}=\frac{\rho c_{0}h_{0}^{2} d_{0}^{2}w(x^{\prime})}{4\pi r{(x^{\prime})}^{6}}\text{d}x^{\prime}. \tag{9}\]  
Similarly, the SPL at the reception point at time \(t\) can be expressed as  
\[ \begin{split} SPL(t)&=10\log_{10}\left(p^{\prime}(t)^ {2}/{p^{\prime}}_{ref}^{2}\right)\\ &=10\log_{10}\left(\int\limits_{-l/2}^{l/2}\frac{\rho c_{0}wh_{0} ^{2}d_{0}^{2}(1-M_{0}^{2})^{6}}{4\pi{p^{\prime}}_{ref}^{2}\left(\sqrt{\left(x^ {\prime}+v_{0}t\right)^{2}+(1-M_{0}^{2})\cdot r_{0}^{2}}-M_{0}\cdot\left(x^{ \prime}+v_{0}t\right)\right)^{6}}\text{d}x^{\prime}\right)\end{split}  \tag{10}\]