  We perform a complete Fourier analysis of the semi-discrete 1-d wave equation
obtained through a P1 discontinuous Galerkin (DG) approximation of the
continuous wave equation on an uniform grid. The resulting system exhibits the
interaction of two types of components: a physical one and a spurious one,
related to the possible discontinuities that the numerical solution allows.
Each dispersion relation contains critical points where the corresponding group
velocity vanishes. Following previous constructions, we rigorously build wave
packets with arbitrarily small velocity of propagation concentrated either on
the physical or on the spurious component. We also develop filtering mechanisms
aimed at recovering the uniform velocity of propagation of the continuous
solutions. Finally, some applications to numerical approximation issues of
control problems are also presented.
