We discuss the geometric content of the wave equation in two dimensions and their implications for numerical methods. Wavefronts and wakes are natural parts of the evolution of point disturbances and yield a rich geometric structure. We discuss how this structure is preserved or destroyed. In particular wavefronts form sharp edged solutions called cusps under repeated reflection. We illustrate the formation and evolution of these cusps using a basic ray-based method and discuss the implication of these cusps for meshed methods. Finally we give an outlook for future algorithms with respect to preservation of geometric structure of the wave equation in two dimensions on arbitrary closed domains as well as some sound examples rendered with the basic ray method.