The dynamics of a low-intensity plane shock wave during its propagation into a foam layer is numerically investigated for experimental data conditions performed in a horizontal shock tube. To describe the behavior of aqueous foam under weak impact, the model of aqueous foam developed by the authors was used, describing it as an elastic-viscoplastic system taking into account elastic properties in accordance with Hooke’s law, and describes viscoplastic behavior by the Herschel–Bulkley conditions. The model was numerically implemented by creating the new solver in the OpenFOAM software. The analysis of the wave flow dynamics during the propagation of a weak air shock wave into the foam layer is carried out. The features of the elastic precursor formation are shown. Based on the results of calculations, the influence of the initial liquid volume fraction on the elastic-viscoplastic properties of the aqueous foam, on which the structure of the shock wave profile, its intensity and propagation velocity depend, is estimated. The reliability of the obtained calculations is confirmed by satisfactory agreement of numerical solutions with experimental data.