  It is well known that waves propagating in a nontrivial medium develop
``tails''. However, the exact form of the late-time tail has so far been
determined only for a narrow class of models. We present a systematic analysis
of the tail phenomenon for waves propagating under the influence of a {\it
general} scattering potential $V(x)$. It is shown that, generically, the
late-time tail is determined by spatial {\it derivatives} of the potential. The
central role played by derivatives of the scattering potential appears not to
be widely recognized. The analytical results are confirmed by numerical
calculations.
