# A Simple Method to Estimate the In Situ Performance of Noise Barriers
## 4 Model Validation
### Application to Noise Barriers Based on Sonic Crystals  
In the last few years, several authors have demonstrated the applicability of sonic crystals as noise barriers with the additional advantage of being almost transparent to wind and water [21]. Figure 6 illustrates a prototype of this kind of noise reduction devices. Sonic crystals are defined as heterogeneous materials formed by periodic arrangements of acoustic scatterers, commonly cylindrical elements, embedded in air and organized in a regular lattice distribution with a minimum distance between the centres of the scatterers, called lattice constant. These open acoustic screens present a new wave control mechanism due to the arrangement of the scatterers, which provides the appearance of bandgaps, defined as frequency ranges where the propagation of waves is forbidden or blocked. As a result, these structures can be effectively used as noise barriers providing significant attenuation in the bandgaps' frequency range.  
In order to validate the proposed model for the case of noise barriers based on sonic crystals, the transmission coefficients, obtained from a classical 2D numerical simulation of the sound propagation through the sonic crystal using the FDTD numerical model (in the plane XY, see Figure 7), have been used (in the plane XY, see Figure 7). These results have been combined with Equation (5), using the non-interfered values of \(\varepsilon_{i}\) obtained from the XZ numerical simulation commented in the previous section, for the case of infinite acoustic insulation. To validate the proposed model results, they have been compared with the ones obtained from a full 3D numerical simulation model [22] (see Figure 7 with a schematic representation of a sector of the 3D noise barrier). These results are summarized and can be compared in Figure 8. Again, for the sake of clarity, only three representative values of the diameter of the cylinders forming the sonic crystal barrier are used. The presented cases are the following: scatterers diameter (d) equal to 50%, 75% and 90% of the lattice constant (\(lc\)). The lattice constant (\(lc\) = 0.17 m) has been chosen to make the first band gap appear around 1 kHz, which is the more relevant band for traffic noise according to [4].  
From the results illustrated in Figure 8, it can be seen that the agreement is very good for medium and high frequencies. Nevertheless, in the range of low frequencies, as commented before, the effects of coherence between the interfered and non-interfered contributions worsen the results being compared. However, the small deviations at low frequencies are negligible when the overall insulation indices are being computed for the noise barriers.  
Figure 5: Relationship between global IL (\(DL_{IL}\)) and \(DL_{R}\) indexes. Black dashed line indicates the ideal behaviour of a noise barrier if diffraction at the upper edge is not considered (infinite height). Continuous line represents the variation of \(DL_{IL}\) index as a function of \(DL_{R}\) index obtained by the numerical FDTD simulation. Dots represent the global IL index obtained by the simple method proposed in this work (Equation (5)).  
Figure 8: Values of IL index of sonic crystals noise barriers with different scatterer diameters. The lines correspond to 3D numerical simulations [22]. Lines with marks correspond to predictions using the model proposed in this paper (Equation (5)). Three values of the scatterers’ diameter (d) are plotted relative to the lattice constant (lc). (d = 50%, 70% and 90% of the lattice constant).  
Figure 6: Prototype of a noise barrier based on sonic crystals.  
Figure 7: Schematic view of the simulation planes.