## 5 Results and Discussion  
### Fluorophosphate Glasses  
The experimental and the calculated longitudinal and transversal ultrasonic velocities as well as the elastic properties of the fluorophosphate glasses investigated in the present study are summarized in Table 3. The first row for each mole fraction corresponds with the  
Figure 3: Schematic representation of the model. The two transducers are used to send and receive the ultrasonic signals. Alternatively, only one transducer can be used acting both as emitter and receiver. The absorbing layers can be used only if it is required for elimination of the reflected signals. The implementation of the boundary conditions, where any stress is absent, the vacuum condition is considered, and both elasticity moduli \(B_{x}\) and \(B_{y}\), are set to zero. experimental results, while the second row represents the corresponding calculated value through the proposed numerical simulation model of the wave propagation in these glasses. The structure of the pseudo-binary fluorophosphate glasses is analogous to that reported for multicomponent mixed fluoride-phosphate glassy systems. The study of the structure of these glasses has been performed and presented in detail in [21,22]. In the present work, we only used the experimental longitudinal and transversal values of sound velocity and the experimental elastic properties of these glasses reported in the literature to compare these values with our simulation results. In an effort to facilitate the validation of the simulation procedure, we present in Figure 4a,b the experimental and the calculated values of the longitudinal and transversal sound velocities, respectively, that correspond to the fluorophosphate glasses at room temperature. It is widely accepted that the structure of a glass material is completely different compared with the corresponding crystalline material. Glass only has a short-range order of atoms, while crystals have a long-range order. When studying the structure of the specific fluorophosphate glasses [21,22], our aim was to study in detail the structure of the amorphous phase and this is the reason why we restricted our analysis only in the glass formation range. Crystalline samples can have altered structural properties and they may lead to the abnormal propagation of waves compared to the corresponding glasses of the same composition; thus, causing additional scattering or absorption of the ultrasonic waves. This comparison of the acoustic wave propagation in crystalline and amorphous materials could not provide any new information. In terms of elastic properties, it is proper to examine separately the crystalline and the glassy systems, despite their identical chemical composition. The incorporation of metaphosphate units into the fluoride structure results in the formation of Al(O,F)\({}_{6}\) octahedra that link the two sub-networks by bridging the metaphosphate and fluoride structural units [21,22] and references therein. In the high metaphosphate region, the corresponding phosphate sub-network is dominated by [PO\({}_{2}\)(O\({}_{\rm T}\))\({}_{2}\)]\({}^{-}\) units, the so-called Q\({}^{2}\) groups. In the low metaphosphate region, the corresponding network is less polymerized through the formation of pyrophosphate [PO(O\({}_{\rm T}\))\({}_{3}\)]\({}  
isolated orthophosphate [P(O\({}_{\rm T}\))\({}_{4}\)]\({}^{3-}\) (Q\({}^{0}\)) structural units [21,22] and references therein. This transition from the fluoride, which is more ionic in character, to the phosphate network, which is covalent, causes a constant increase in the glass transition temperature with increasing the phosphate content in the glasses. A maximum T\({}_{\rm g}\) is detected at \(\sim\)80 mol% in Sr(PO\({}_{3}\))\({}_{2}\) due to the distinct crosslinking capability of the Al\({}^{3+}\) and Sr\({}^{2+}\) ions. The Al\({}^{3+}\) cations crosslink the short metaphosphate structural units more effectively than the Sr\({}^{2+}\) cations. Thus, the crosslinking through Al\({}^{3+}\) ions leads to a more rigid glass structure.