We apply the discontinuous Petrov-Galerkin (DPG) method to linear acoustic waves in space and time using the framework of first-order Friedrichs systems. Based on results for operators and semigroups of hyperbolic systems, we show that the ideal DPG method is well-posed. The main task is to avoid the explicit use of traces, which are difficult to define in Hilbert spaces with respect to the graph norm of the space-time differential operator. Then, the practical DPG method is analyzed by constructing a Fortin operator numerically. For our numerical experiments, we introduce a simplified DPG method with discontinuous ansatz functions on the faces of the space-time skeleton, where the error is bounded by an equivalent conforming DPG method. Examples for a plane-wave configuration confirms the numerical analysis, and the computation of a diffraction pattern illustrates a first step to applications.