## 2 Methods
### Analytical and Numerical Design of a Novel Noise Barrier
This can be done by iteration, gradually increasing the effective height in steps, applying the Kurze-Anderson formula in each frequency band for each step to calculate the \(\Delta L\) in dB [35], and finally, combining energetically the sound pressure levels without the barrier, the design insertion loss, and the A-weighting correction to obtain the A-weighted overall sound pressure level with the barrier. Subtracting this value from the overall A-weighted SPL without the barrier gives the overall design insertion loss in dB(A):  
\[IL_{Design}=SPL_{without}-10\cdot log\sum_{i=1}^{n}10^{SPL_{without,i}-\Delta L _{i}+C_{i}}[\text{dB}(\text{A})] \tag{2}\]  
Figure 5: Schematic for the noise barrier’s pre-dimensioning using the Kurze–Anderson formula to determine the effective height, \(\text{h}_{eff}\), given the target insertion loss [35]. where \(SPL_{without,i}\) is the measured sound pressure level in the \(i\)-th frequency band, and \(C_{i}\) is the A-weighting correction in the \(i\)-th frequency band. To consistently apply this pre-dimensioning, Terracini Street was divided into four sections where the noise barrier was expected to have a constant height based on the buildings' height and distance from the road (see Figure 6). In each section, the target insertion loss (\(IL_{Target}\)) is the maximum level difference between the measured values (over the whole time frame and at points 1-3 in Figure 4) and the noise level limit imposed by the Italian Regulation [21]. Therefore,  
\[IL_{Target}=SPL_{without}\ -SPL_{limit}\ \ [\text{dB(A)}] \tag{3}\]  
where \(SPL_{without}\) is the measured sound pressure level (see Figure 3), and \(SPL_{limit}\) is the maximum allowed sound pressure level according to the Italian Regulation. \(IL_{Target}\) assumes the following values as follows:  
In section A (ref. measurement point 1), \(IL_{Target}=10\) dB(A);  
In section B (ref. measurement point 2), \(IL_{Target}=13\) dB(A);  
In section C (ref. measurement point 2), \(IL_{Target}=13\) dB(A);  
In section D (ref. measurement point 3), \(IL_{Target}=11\) dB(A). Then, the above-described iterative procedure was applied for each section, and the effective height \(\text{h}_{\text{eff}}\) was determined in each case. Next, the factor \(K\) to be multiplied by the distance \(B\) between the barrier and the receiver to obtain the barrier half-length \(W=\text{K}\cdot B\) is red, from the work of [PERSON] and [PERSON] [35]. Finally, the length of the barrier was determined as \(L=2\cdot W\). See Table 4. \begin{table}
\begin{tabular}{c|c c c c c c c} \hline \hline \multirow{2}{*}{**Section**} & \multicolumn{2}{c}{\(IL_{Target}\)} & \multirow{2}{*}{**A [m]**} & \multirow{2}{*}{**B [m]**} & \multirow{2}{*}{**K**} & \multirow{2}{*}{**h\({}_{\text{min}}\) [m]**} & \multirow{2}{*}{**h\({}_{\text{eff}}\) [m]**} & \multirow{2}{*}{**H [m]**} & \multirow{2}{*}{**L [m]**} \\  & \multicolumn{1}{c}{[dB(A)]} & & & & & & & \\ \hline
**A** & 8 & 13.31 & 21.93 & 3.5 & 3.96 & 1.1 & 5.06 & 140 \\ \hline
**B** & 13 & 10.00 & 10.20 & 6.5 & 1.99 & 2.2 & 4.19 & 130 \\ \hline
**C** & 13 & 11.49 & 11.66 & 6.5 & 5.45 & 2.2 & 7.65 & 130 \\ \hline
**D** & 11 & 12.04 & 12.21 & 5.0 & 5.45 & 1.6 & 7.05 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Significant parameters and related values used for dimensioning the noise barriers under study in each section. Figure 6: Schematic showing the four sections along Terracini Street used to design the noise barrier. The result of this analytical pre-dimensioning study was implemented by a FEM (finite element method) model in the specific context using the Comsol Multiphysics software (v.6.1) in order to quantitatively appreciate the effect of the combination of noise barriers and low-noise asphalt. The FEM method determines an approximation of the solution of a system of partial differential equations in a region of space with well-defined boundary conditions [36].