[en] The interaction of a charged particle with a semi-infinite medium bounded by a flat surface is analyzed as a problem of metal optics. The reflection at the surface of the electromagnetic fields excited by the passing beam is investigated including retardation and spatial dispersion in the hydrodynamic approximation. The soft decay of the electron density at the metal-vacuum interface is modeled. Using measured values of the dielectric function, the calculations are found to agree with experimental results of electron-energy-loss distributions for Ag targets in interaction at a distance. Further predictions of the theoretical model are discussed