We introduce a new numerical method for solving time-harmonic acoustic scattering problems. The main focus is on plane waves scattered by smoothly varying material inhomogeneities. The proposed method works for any frequency ω, but is especially efficient for high-frequency problems. It is based on a time-domain approach and consists of three steps: i) computation of a suitable incoming plane wavelet with compact support in the propagation direction; ii) solving a scattering problem in the time domain for the incoming plane wavelet; iii) reconstruction of the time-harmonic solution from the time-domain solution via a Fourier transform in time. An essential ingredient of the new method is a front-tracking mesh adaptation algorithm for solving the problem in ii). By exploiting the limited support of the wave front, this allows us to make the number of the required degrees of freedom to reach a given accuracy significantly less dependent on the frequency ω. We also present a new algorithm for computing the Fourier transform in iii) that exploits the reduced number of degrees of freedom corresponding to the adapted meshes. Numerical examples demonstrate the advantages of the proposed method and the fact that the method can also be applied with external source terms such as point sources and sound-soft scatterers. The gained efficiency, however, is limited in the presence of trapping modes.