We consider wave propagation problems over two-dimensional domains with piecewise-linear boundaries, possibly including scatterers. We assume that the wave speed is constant and that the initial conditions and forcing terms are radially symmetric and compactly supported. We propose an approximation of the propagating wave as the sum of some special space-time functions. Each term in this sum identifies a particular field component, modeling the result of a single reflection or diffraction effect. We describe an algorithm for identifying such components automatically based on the domain geometry. To showcase our proposed method, we present several numerical examples, such as waves scattering off wedges and waves propagating through a room in the presence of obstacles. Software implementing our numerical algorithm is made available as open-source code.