We discuss the representation of solutions of hyperbolic systems in the time-domain using translating exact or approximate solutions: plane waves, curvelets, and what we call complete plane waves. The latter are half-space representations which both propagate and decay. We show that they lead to highly efficient local radiation boundary condition sequences satisfying optimal complexity estimates. Precisely, an auxiliary variable formulation with degrees-of-freedom per boundary point suffices to achieve an error less than up to time T assuming sources are separated by from the artificial boundary. We also consider the direct use of all three expansions in algorithms to propagate waves to remote locations.