Matter effects in oscillations of neutrinos traveling short distances in matter

It is well known that when the distance t traveled by neutrinos in matter is short, matter effects in oscillations of neutrino flavour are small and decrease with decreasing t more rapidly than the oscillations effects themselves. We discuss the reason for this and demonstrate that under certain circumstances this statement is no longer correct. In particular, we show that if neutrinos propagate significant distances in vacuum before entering matter (or after exiting it), matter effects in short-t neutrino oscillations can be significantly enhanced. Implications for oscillations of solar and atmospheric neutrinos with nearly horizontal trajectories inside the earth and for accelerator experiments are considered. We also comment on neutrino oscillations in matter due to flavour changing neutral currents.