  By writing the complete set of $3 + 1$ (ADM) equations for linearized waves,
we are able to demonstrate the properties of the initial data and of the
evolution of a wave problem set by Alcubierre and Schutz. We show that the
gauge modes and constraint error modes arise in a straightforward way in the
analysis, and are of a form which will be controlled in any well specified
convergent computational discretization of the differential equations.
