  We introduce a new method for the numerical approximation of time-harmonic
acoustic scattering problems stemming from material inhomogeneities. The 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 $\omega$, as shown in the
numerical experiments. 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.
