# A Simple Method to Estimate the In Situ Performance of Noise Barriers
## 3 Fdtd Simulations  
As commented above, the FDTD method has been chosen in this work. A 2D acoustic simulation scheme using FDTD has been developed, schematically illustrated in Figure 2, hereafter referred to as XZ simulation model. Although further details of the numerical technique can be found in the literature, for completeness, the fundamental concepts are given next. The acoustic FDTD model with no sound sources is governed by the equations for conservation of momentum and continuity. In a homogeneous medium with no losses, these can be written as:  
\[\frac{\partial p}{\partial t}+k\overrightarrow{\
abla}\cdot\overrightarrow{u}=0 \tag{9a}\]  
Figure 2: Schematic representation of the FDTD model, used in the XZ simulation, including indication of the considered boundary conditions. \[\overrightarrow{\
abla}p+\rho_{0}\frac{\partial\overrightarrow{u}}{\partial t}=0 \tag{9b}\]  
where \(p\) is the pressure field, \(\overrightarrow{u}=(u_{x},u_{z})\) is the vector particle velocity field, \(\rho_{0}\) is the mass density of the medium and \(k=\rho_{0}\,c^{2}\) is the compressibility of the medium. The partial derivatives (space and time) of pressure and particle velocity can be approximated by well-known central finite difference schemes; for instance, in the case of the sound pressure derivative with respect to \(x\), this approximation can be given as  
\[\left.\frac{\partial p}{\partial x}\right|_{x=x_{0}}\approx\frac{p\Big{(}x_{0 }+\frac{\Delta x}{2}\Big{)}-p\Big{(}x_{0}-\frac{\Delta x}{2}\Big{)}}{\Delta x} \tag{10}\]  
with \(\Delta x\) being the spatial interval between two consecutive points in the \(x\) direction. In the case of 2D problems, three grids must be defined, namely one grid for pressure and one grid for each of the particle velocity components (\(x\) and \(z\)). To minimize the significance of higher order terms, those grids are'staggered', which means that, for example, the meshes for the \(x\) and \(z\) components of the particle velocity are shifted a distance of \(\Delta x/2\) and \(\Delta z/2\) with respect to the pressure mesh; the same principle is applied to the time discretization, in which the particle velocity meshes are shifted \(\Delta t/2\) in time with respect to the pressure mesh. A set of update equations can then be obtained to compute the values of pressure and particle velocity after repetition for a given number of time steps. A detailed description of all relevant equations is given in [17]. In order to avoid unwanted numerical reflections in the considered rectangular numerical calculation domain, a Perfectly Matched Layer (PML) [18] is located at three of the four numerical domain boundaries, depicted in Figure 2. Bear in mind that the sound can be reflected on the ground, corresponding to the lower horizontal boundary. Without loss of generality, in this work, a completely rigid floor will be considered, in other words, total reflection is assumed on the ground surface. With these boundary conditions, the numerical domain is excited by a plane wave travelling from left to right. The acoustic measurements have been numerically recorded at the microphones, located at the shadow area of a classical noise barrier. This FDTD model assumes that all the propagation domain is air, except the ground surface and the acoustic barrier that are completely stiff and reflective. The described FDTD simulation model can be easily adapted to verify the accuracy of the simplified model discussed above. The modification consists of adding the possibility of simulating the transmission of sound through the noise barrier without the simulation of the full vibroacoustic problem. Figure 3 includes a schematic representation of this modification. The signal obtained at the pressure points (labelled as \"recorded\" in the figure), on the left side of the acoustic screen, is taken and re-injected at the x-velocity points (labelled as \"injected\" in the figure) that define the screen contour on the right side of the screen once multiplied by a transmission factor (\(\tau\)).