A new kind of magnetohydrodynamic instability and waves are analyzed for a current sheet in the presence of a small normal magnetic field component varying along the sheet. These waves and instability are related to the existence of two gradients of the tangential (Bτ) and normal (Bn) magnetic field components along the normal (nBτ) and tangential (τBn) directions with respect to the current sheet. The current sheet can be stable or unstable if the multiplication of two magnetic gradients is positive or negative. In the stable region, the kinklike wave mode is interpreted as so-called flapping waves observed in Earth's magnetotail current sheet. The kink wave group velocity estimated for the Earth's current sheet is of the order of a few tens of kilometers per second. This is in good agreement with the observations of the flapping motions of the magnetotail current sheet.