  Black holes found in binaries move at very high velocities relative to our
own reference frame and can accelerate due to the emission of gravitational
radiation. Here, we investigate the numerical stability and late-time behavior
of linear scalar perturbations in accelerating black holes described by the
$C-$metric. We identify a family of quasinormal modes associated with the
photon surface and a brand new family of purely imaginary modes associated with
the boost parameter of the accelerating black hole spacetime. When the
accelerating black hole is charged, we find a third family of modes which
dominates the ringdown waveform near extremality. Our frequency and time domain
analysis indicate that such spacetimes are stable under scalar fluctuations,
while the late-time behavior follows an exponential decay law, dominated by
quasinormal modes. This result is in contrast with the common belief that such
perturbations, for black holes without a cosmological constant, always have a
power-law cutoff. In this sense, our results suggest that the asymptotic
structure of black hole backgrounds does not always dictate how radiative
fields behave at late times.
