  One may believe that front velocities of waves in a given theory coincide
with the UV limit of phase velocities for any dispersion relations. This
implies that IR physics is irrelevant to the discussion of propagation speed of
waves. We first consider a theory that contains higher spatial derivatives in
the wave equation and prove that front velocities coincide with the UV limit of
phase velocities, at least, if parity is conserved. However, we also show that
front velocities do not coincide with the UV limit of phase velocities in
general dispersion relations. We explicitly give several examples in which
front velocities are superluminal owing to an IR or intermediate energy scale
property of dispersion relations even if the UV limit of phase velocities is
luminal. Our finding conveys the important caution that not only UV physics but
also IR physics can be significant to superluminality.
