## 2 Materials and Methods  
### Glass Formation  
The preparation of the FP glasses was described in detail elsewhere [21; 22]. In brief, the fluoride strontium phosphate-based glasses correspond to xSr(PO\({}_{3}\))\({}_{2}\)-(1-x)(0.62 MgF\({}_{2}\)-0.38 AlF\({}_{3}\)) with x = 0, 0.04, 0.06, 0.1, 0.15, 0.2, 0.3, 0.4, 0.8, 0.9, and 1. The prepared glasses have a constant ratio of AlF\({}_{3}\)/MgF\({}_{2}\). As starting materials we used high purity Sr(PO\({}_{3}\))\({}_{2}\), MgF\({}_{2}\), and AlF\({}_{3}\) that were used as received. The mole fractions of the fluorophosphate glasses are presented in Table 1.  
The glass formation was achieved by means of the standard melt-quenching technique. The materials were melted in crucibles at temperatures above their corresponding melting points depending on the composition and kept at these temperatures until achieving full homogenization and then fast quenched on preheated plates near the glass transition temperature to avoid internal stresses in the resulting glassy materials. Subsequently, the samples were cooled down to room temperature with a cooling rate of \(\sim\)5 K/min. Annealing was also performed for 1 h near their respective glass transition temperature. Stable glasses were not obtained in the mole fraction range 0.45 \(<\) x \(<\) 0.65. The obtained glasses were fully amorphous and with a low OH content. Before ultrasonic pulse-echo measurements, the samples were cut into circular disks with \(\sim\)10 mm diameter and 3-5 mm thickness and highly polished to facilitate the ultrasonic wave propagation in the glassy samples. The non-parallelism between the opposite faces of the studied sample is a key factor that may affect the accuracy of the sound velocity measurements. Thus, special attention was paid to the sample during cutting. The parallelism of the opposite faces was verified by means of a dial gauge.  
In this study, we did not prepare the lanthanum aluminosilicate glasses. We only used the experimental longitudinal and transversal values of sound velocity reported in [30] to estimate the experimental elastic properties of these glasses and compare these values with our simulation results. In that work, the authors prepared lanthanum aluminosilicate glasses corresponding to 15 La\({}_{2}\)O\({}_{3}\)-xAl\({}_{2}\)O\({}_{3}\)-(85-x)SiO\({}_{2}\) and 25 La\({}_{2}\)O\({}_{3}\)-yAl\({}_{2}\)O\({}_{3}\)-(75-y)SiO\({}_{2}\), with x and y = 15, 20, 25, 30, 35 mol\(\%\). All the experimental details for the glass formation can be found in [30]. In the present work, our aim was not to prepare and study the structure of the FP and LAS glasses, but to evaluate our proposed simulation model by comparing the experimental and simulated elastic properties for two distinct glass families. Therefore, the main contribution of this paper is the treatment of the numerical time-domain modeling of 2D acoustic wave propagation in a viscoelastic medium by implementing the elastodynamic finite integration technique (EFIT). The mole fractions of the lanthanum aluminosilicate glasses are presented in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline xSr(PO\({}_{3}\))\({}_{2}\) & (1–x)(0.62 MgF\({}_{2}\)–0.38 AlF\({}_{3}\)) \\ \hline
0.04 & 0.96 \\
0.06 & 0.94 \\
0.1 & 0.90 \\
0.15 & 0.85 \\
0.2 & 0.80 \\
0.3 & 0.70 \\
0.4 & 0.60 \\
0.8 & 0.20 \\
0.9 & 0.10 \\
1 & 0 \\ \hline \end{tabular}
\end{table}
Table 1: Mole fraction of the fluorophosphate glasses used in this study.