## 1 Introduction  
Among acoustic techniques, ultrasonic echography stands out as a valuable tool in various fields for the non-destructive inspection of materials including neat liquids, plastics, metals, composites, concrete, and cements [1, 2] and references therein. Furthermore, ultrasonic waves can be used to sono-chemically synthesize materials and for welding purposes. In terms of ultrasound intensity, the applications are separated into two main groups, namely the low- and the high-intensity ultrasonic applications. In the low-intensity or equivalent high-frequency ultrasound group, the low-energy ultrasonic waves propagate through the medium providing information dynamic analysis, rheology, and process monitoring [3]. Ultrasonic relaxation spectroscopy also utilizes low-power ultrasound waves to deliver information for the structure of the studied material in the medium- to long-range order. These studies include proton-transfer reaction [4, 5, 6] and association-dissociation mechanisms [7],conformations between isomers [8], complexation processes [9; 10; 11], backbone segmental motion and normal mode relaxation mechanisms in polymeric solutions [12; 13], and other processes related to reaction engineering [14]. In the high-intensity or low-frequency ultrasound group, the sound waves propagate through the medium inducing locally high pressures and temperatures. This methodology is utilized in several laboratory and industrial applications involving processes such as emulsification, welding, cleaning, fatigue testing, chemical reactions, heat generation, and others [15]. In general, ultrasonic measurements are rapid, accurate, non-destructive in the high-frequency region, and can be fully automated for industrial uses but require experienced operators that master the specific methodology and be able to precisely interpret the experimental results. The situation in most of the cases is perplexing since the results are subject to severe disturbances including beam deviation and/or splitting, mode conversion due to the presence of anisotropic and heterogeneous domains in the material, etc. These phenomena are reflected in acoustic spectra as spurious echoes and noise. On the other hand, the numerical simulation of ultrasonic transmission experiments is a powerful tool to evaluate the non-destructive inspections of materials performed by means of acoustic techniques or to progress the optimal design of ultrasonic inspection systems. The simulation involves the transmission of the acoustic waves into materials and the subsequent detection and analysis of the reflected or transmitted signal. The relevance of the simulation data with the experimental results is directly related to the ability of the numerical approach to handle reflection, dispersions, scattering and mode conversion phenomena, and the accuracy of the input parameters of the code [16; 17]. Fluorophosphate (FP) glasses exhibit improved properties compared with the corresponding fluoride and phosphate counterparts. The addition of phosphates into the fluoride systems enhances its glass-forming ability and the final products are promising materials for laser and optics applications [18; 19; 20]. The short-range structure of FP glasses and the medium-range order were studied extensively by means of Raman spectroscopy [21; 22]. The structure is dominated by PO\({}_{4}\) tetrahedra with 2, 3, and 4 non-bridging oxygen atoms in each tetrahedron corresponding to Q\({}^{2}\), Q\({}^{1}\), and Q\({}^{0}\) species, respectively. A recent review on the structure of FP glasses can be found in [23]. Aluminosilicate glasses doped with lanthanum (LAS) exhibit high glass-transition temperature and refractive indices, high hardness, and are chemically durable [24; 25]. Trivalent rare earth cations, such as La\({}^{3+}\), serve as charge compensators for three neighboring aluminum oxide tetrahedra [AlO\({}_{4}\)]\({}^{5-}\) or for one aluminum oxide octahedron [AlO\({}_{6}\)]\({}^{9-}\)[26; 27].