## 2 Numerical Methods
### Noise Propagation Method  
Sound speed in an ambient flow is written as \(c_{eff}=c+v_{x}\) where \(c\) is the constant sound speed at a given air temperature, \(v_{x}\) is the wind speed component through the propagation path. The wind speed components may come from field measurements, empirical expressions or numerical simulations. For an incompressible wind flow, the air density is assumed as a constant, the wave equation in a moving atmosphere reads [12]  
\[\left[\
abla^{2}+k^{2}(1+\epsilon)-\frac{2i}{\omega}\frac{\partial v_{i}}{ \partial x_{j}}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}+\frac{2 ik}{c_ {0}}v\cdot\
abla\right]P^{\prime}(r)=0 \tag{9}\]  
where \(k=\omega/c_{0}\), \(\omega\) is the radian frequency of sound wave, \(c_{0}\) is the reference sound speed, \(P^{\prime}(r)\) denotes the monochromatic field of sound pressure and \(\epsilon=\left(c_{0}/c\right)^{2}-1\). If the effect from ambient flow is neglected, this equation can be further reduced to Helmholtz equation. The ANSource of a wind turbine is placed on the left boundary of the computational domain depicted in Figure 2. A classical approach to represent a monopole ANSource is to distribute the source along the vertical direction which is also called starting field. In this study, the rotor noise source is located at the hub height as sketched at the left side of the boundary. Based on the initial source strength, the equation or the sound pressure amplitude is computed along the \(x\)-direction, such that solution is renewed from \(P(x)\) to \(P(x+\Delta x)\). The boundary condition applied on the ground surface is calculated from the impedance of the ground. The empirical impedance model for absorbing materials developed by [PERSON] and [PERSON] [21] is used in this study,  
\[Z=1+0.0511\left(\frac{\sigma}{f}\right)^{0.75}+i\ 0.0768\left(\frac{\sigma}{f} \right)^{0.73} \tag{10}\]  
where \(Z\) is the normalized acoustic impedance, \(\sigma\) represents the flow resistivity and \(f\) is the frequency. The similar approach can be applied on the top surface. A constant of the normalized acoustic impedance \(Z=1\) is implemented on the top grid, such that the vertical waves through the top surface is vanished without any reflection. However, to sufficiently eliminate wave reflections, an absorbing layer is defined on top of the grid. As shown in the figure, the grey region sketched on top of the domain is the absorbing layer. In the study, the width of such an absorbing layer is defined as \(50\lambda\). This implies that for lower frequency sound propagation, a larger absorbing layer is needed. The sound pressure along the vertical line is updated from step \(n\) to \(n+1\) until the receiver position is reached. The solution procedure starts with the separation of the forward and backward propagation waves. Here, only the forward wave propagates from source towards receiver is considered (in the positive \(x\)-axis), which is in the downwind direction. Detailed mathematical manipulations of the equation are found in [12; 13]. According to the methodologies described in Sections 2.1 and 2.2, the noise spectrum of a given wind turbine is the logarithmic sum of sound pressure level (SPL) from all the blade elements such that  
\[L_{p}(f)=10 log_{10}\left(\sum_{i}^{n}10^{0.1 SPL^{i}_{total}}\right) \tag{11}\]  
where \(n\) is the number of the blade elements and \(SPL^{i}_{total}\) is the SPL of the \(i\)th blade element, which accounts for all the noise mechanisms. The SPL of wind turbine ANSource \(L_{p}(f)\) is the initial condition for calculating long-path ANPropagation. The sound power level \(L_{w}(f)\), which is a representation of wind turbine total noise strength, it does not change with propagation range. The SPL at the receiver is  
\[L_{p}(f)=L_{w}(f)-10 log_{10}4\pi D^{2}-\alpha D+\Delta L \tag{12}\]  
where the first term on the right side is the sound power level, the second term \(-10\log_{10}4\pi D^{2}\) is the geometric attenuation term that corresponds to the sound intensity loss when the sound wave sphere is expanding.