Yes, wave can accelerate . The electron matter wave in the atom is an example of a wave that keeps changing its direction. The steady state condition of quantum mechanics, however, requires that such a wave forms eigenmodes of the Hamilton operator, which would have a fixed and quantized energy.
Accelerating charges create changing electrical and magnetic fields. Changing electric fields create magnetic fields and changing magnetic fields create electric fields. This interaction of induced electrical and magnetic fields leads to the propagation of electromagnetic waves. Electromagnetic waves can propagate in free space.
