# Experimental Analysis and Simulation of a Porous Absorbing Layer for Noise Barriers
## 2 Materials and Methods
### Noise Barier Modelling  
In this work, an attempt was made to understand whether the calculation and simulation models allow taking into account, adequately, the presence of different types of noise barriers, including reflective and absorbing barriers, as well as traditional (tall) and low-height barriers (with about 1.5 m). For barriers with an absorbing surface, the properties of the porous concrete material with expanded clay analyzed before, and with a thickness of 0.08 m [41; 42], were considered. Different scenarios were simulated, considering a simple reference configuration without any acoustic barriers, and then the same scenario with the presence of acoustic barriers with heights of 1.5 m and 4 m. The distance between the barriers and the noise source varies according to their height, with the low-height barriers being positioned 2.83 m from the source, and the traditional tall barriers 7.5 m from it. These distances were defined assuming a possible railway track as the emitting source, and so these distances, for the first case, were based on a minimum allowable distance between the track and the barrier (maintaining the safety conditions), and for the second case, they were defined based on common distances adopted in the railway infrastructures in Portugal [43]. Two different barrier shapes were considered, one of them vertical and the other a L-shaped barriers, and cases with and without an absorbing layer were tested. Figure 9 presents a schematic representation of these configurations. Figure 8: Flowchart of the inversion methodology used to compute the macroscopic parameters. Numerical simulations were performed using two methodologies, the first one based on a commonly used environmental noise simulation software (CadnaA), implementing the CNOSOS-EU methodology [5], and the second adopted a Boundary Element Method (BEM) in the frequency domain [32; 33; 34], developed and implemented in MATLAB. For the first methodology, the emitting sound source was assumed to be a train, and the sound powers were corresponding to a BR 1016 train, with a constant speed of about 80.47 km/h (50 mph) being used for the simulation. Horizontal meshes were made of receivers placed at heights of 1.5 m and 4 m, spaced 1 m apart, and a vertical mesh, transverse to the track, with receivers equally spaced 1 m apart was considered. The source is located at 0.5 m height and the absorption coefficient of the soil was assumed to be 0.3 [5]. For the BEM, the calculation is based on solving the boundary integral equation:  
\[C(\xi)p(\xi)=-i\rho\omega\int\limits_{\Gamma}G(\xi,X)v_{n}(X)d\Gamma-\int \limits_{\Gamma}\frac{G(\xi,X)}{\partial n}p(X)d\Gamma+\sum\limits_{k=1}^{NS} Q_{k}G\left(\xi_{k}^{f},\xi\right) \tag{14}\]  
where \(\Gamma\) represents the boundary surface, \(\rho\) the propagation medium density (1.22 kg/m\({}^{3}\) in the case of air), \(G\left(\xi_{k}^{f},\xi\right)\) the incident field generated by a source located at \(\xi_{k}^{f}\), and \(p(X)\) and \(v_{n}(X)\) represent the unknown pressure and velocity of the particle, respectively. \(C(\xi)\) depends on the boundary geometry and takes the value \(\
icefrac{{1}}{{2}}\) for a smooth boundary. The sound absorption of the different surfaces is imposed through a Robin-type boundary condition, in which the surface acoustic impedance is prescribed as \(Z=\rho c\left(1+\sqrt{1-\alpha}\right)/\left(1-\sqrt{1-\alpha}\right)\), \(\alpha\) being the sound absorption coefficient derived from diffuse field conditions. To allow assigning the necessary impedance conditions to the soil, it is also discretized using boundary elements along a 25 m extension (from x = -5 m to x = 20 m). In the simulated cases, it was decided to locate the sound source at the height of 0.5 m, following the same strategy indicated in the previously described CNOSOSOS-EU model.