# A Simple Method to Estimate the In Situ Performance of Noise Barriers
## 4 Model Validation
### Comparison with \"Overlapping Model\"  
Concerning noise barriers based on sonic crystals, there are several works where methods of combining different numerical simulations have been proposed in order to avoid the need to perform 3D numerical simulations to obtain the noise barriers' insertion loss curve [23; 24]. In these methods, called by the authors as \"overlapping models\", the 3D problem is divided into two 2D schemes, namely in the \(XY\) and \(XZ\) planes, obtaining the \(IL\) values of the 3D case as:  
\[IL(f)=10\log\left(\left|\frac{p_{without}}{p_{withXY}+p_{withXz}}\right|\right) ^{2} \tag{11}\]  
where \(p_{without}\) is the sound pressure without the acoustic screen, \(p_{withXY}\) is the sound pressure with the acoustic screen in the \(XY\) plane simulation and \(p_{withXZ}\) is the sound pressure with the acoustic screen in the \(XZ\) plane simulation. In the aforementioned works, the authors distinguish between the contributions of incident sound and scattered sound. Notice that the summation in the denominator is complex, i.e., it takes into account the phase changes in each of the 2D orthogonal simulations. It should be noted that this approximation is not valid in the limiting cases where the barrier does not provide any shielding, since in that case \(p_{without}=p_{withXY}=p_{withXZ}\), resulting in a value of \(-6\) dB when it should be \(0\) dB. It is also assumed that the acoustic path between sources and receivers is identical in the two 2D orthogonal simulations that are combined to obtain the full model. Figure 9 illustrates the results for the two approximate models together with the exact solution obtained in a 3D simulation. In general, an excellent agreement is observed between the results of the model proposed in this work and the 3D simulations. This is not the case for the model of [PERSON] et al. [23; 24]. As already indicated, this model also combines two numerical simulations, but the underlying philosophy is quite different. If insertion loss concept is translated into transmission coefficients it may be easier to compare the \"overlapping method\" proposed by [PERSON] et al. [23; 24] and the simple method proposed in this paper (Equation (5)). A global transmission coefficient, \(\tau_{G}\), can be defined as follows,  
\[IL=10\log\left(\frac{1}{\tau_{G}}\right) \tag{12}\]  
In other words,  
\[\tau_{G}=10^{-IL/10} \tag{13}\]  
In the case of the \"overlapping method\" proposed by [PERSON] et al. [23; 24], this coefficient is calculated as:  
\[\tau_{G\ OM}=\left(\left|\frac{p_{withXY+p_{withXZ}}}{p_{without}}\right| \right)^{2} \tag{14}\]  
where \(OM\) stands for the \"overlapping method\" proposed by [PERSON] et al. [23; 24]. Assuming incoherence between the two 2D models, it can be written as follows,  
\[\tau_{G\ OM}=\tau_{XY}+\tau_{XZ} \tag{15}\]  
This can lead to values of the transmission coefficient outside the physically meaningful limits, above \(1\). Moreover, taking into account the phase of the coefficients in Equation (12) could lead to a value of \(4\) in the case of a completely transparent screen, when it should be \(1\). For a better comparison between the two methods, we can restate the model proposed in this work in terms of the transmission coefficients obtained in the \(XY\) and \(XZ\) simulations. So, in Equation (5), \(\varepsilon\) will be substituted by \(\tau_{XZ}\) and \(\tau\) will be substituted by \(\tau_{XY}\). Therefore, the global transmission coefficient can be expressed as:  
\[\tau_{G}=(1-\tau_{XZ})\tau_{XY}+\tau_{XZ}=1-(1-\tau_{XY})(1-\tau_{XZ})=\tau_{XY}+ \tau_{XZ}-\tau_{XY}\;\tau_{XZ} \tag{16}\]  
This transmission coefficient is bounded between 0 and 1, as opposed to the \"overlapping model\". This is especially noticeable for high values of the transmission coefficients, which for example appear in the low frequency range for small values of the diameter of the scatterers (top plots in Figure 9).