  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
$\omega$, but is especially efficient for high-frequency problems. It is based
on a time-domain approach and consists of three steps: \emph{i)} computation of
a suitable incoming plane wavelet with compact support in the propagation
direction; \emph{ii)} solving a scattering problem in the time domain for the
incoming plane wavelet; \emph{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 \emph{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 $\omega$. We also present a new algorithm for computing the
Fourier transform in \emph{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.
