The problem of discrete observation of the wave equation on compact homogeneous spaces is considered. In contrast to previous work on the heat equation, an additional initial condition requires the use of an extended sampling scheme and changes the structure of the problem. The conditions under which approximate discrete observability holds are proven, as is the existence of samples which satisfy those conditions. Acuity of observation is established, providing general analytic bounds on the error in using only finitely many samples. Also, under certain strict conditions which depend on the sample points, the error is shown to go to zero through a set of Sobolev conditions. The results are somewhat similar in statement to those obtained for bounded domains in Euclidean space in an earlier paper, but the methods used to obtain them are different and rely heavily on the representation theory of the underlying space.