It is depends on what kind of wave you are talking about.
Source wave: Due to pressure change so the particle move and produce another pressure change near by, so cause near by particle move,.... it is a cycle but the influence propagate to naer by position as time moves on.
Light (electromagnetic field):
changing electric field produce magnetic field change at near by space ,
changing magnetic field procude electric field change at near by space,
the above process continue , and the EM wave propagate in space.