The problem of discrete observation of the wave equation on bounded domains in euclidean space is considered. Wallace and Wolf (1991 b) studied this problem for the heat equation using an infinite but discrete set of spatial samples for one time. They showed that the heat equation satisfied a certain type of discrete observability, which can be thought of as approximate discrete observability. In contrast to the heat equation, the wave equation has an additional initial condition that requires the use of an extended sampling scheme, and changes the structure of the problem. The conditions under which approximate discrete observability holds for a given set of samples are proven, as is the existence of samples that satisfy those conditions. Acuity of observation is established, providing general analytic bounds on the error in using only finitely many samples of the solution to the wave equation.