Finite element methods for the reduced wave equation in unbounded domains are presented. A computational problem over a finite domain is formulated by imposing an exact impedance relation at an artificial exterior boimdary. Method design is based on a detailed examination of discrete errors in simplified settings, leading to a thorough analytical understanding of method performance. For this purpose, model problems of radiation with inhomogeneous Neumann boundary conditions, including the effects of a moving acoustic medium, are considered for the entire range of propagation and decay. A Galerkin/least-squares method is shown to exhibit superior behavior for this class of problems.