In finite element elastic impact analysis, the propagation of impact-induced waves through finite element meshes normally causes spurious components in the numerical solutions. In this paper the possible spurious wave reflection at the impact interface is investigated, where the impact interface is modelled by using the penalty function method. The amplitudes of the reflected and transmitted waves are derived as functions of frequencies and wavenumbers, which are related by the dispersion relations of the finite element meshes. It is observed that significant spurious wave reflections may appear at the impact interface when a small penalty parameter is used. The current investigation is restricted to one-dimensional cases with constant strain or linear strain elements.