Many realistic problems in computational acoustics involve complex geometries and sound propagation over large domains, which requires accurate and efficient numerical schemes. It is difficult to meet these requirements with a single numerical method. Pseudo-spectral (PS) methods are very efficient, but are limited to rectangular shaped domains. In contrast, the nodal discontinuous Galerkin (DG) method can be easily applied to complex geometries, but can become expensive for large problems. In this paper, we study a coupling strategy between the PS and DG methods to efficiently solve time-domain acoustic wave problems. The idea is to combine the strengths of these two methods: the PS method is used on the part of the domain without geometric constraints, while the DG method is used around the PS region to accurately represent the geometry. This combination allows for the rapid and accurate simulations of large-scale acoustic problems with complex geometries, but the coupling and the parameter selection require great care. The coupling is achieved by introducing an overlap between the PS and DG regions. The solutions are interpolated on the overlaps, which allows the use of unstructured finite element meshes. A standard explicit Runge-Kutta time-stepping scheme is used with the DG scheme, while implicit schemes can be used with the PS scheme due to the peculiar structure of this scheme. We present one-and two-dimensional results to validate the coupling technique. To guide future implementations of this method, we extensively study the influence of different numerical parameters on the accuracy of the schemes and the coupling strategy.