Wave motion is a propagatlon of energy through a material medium due to repeated periodic motion.Transverse waves cannot be propagated through gases. Why?

The motion that repeat themselves are called periodic motions but for a simple harmonic motion, the force must be proportional to the displacement and it is directed towards the centre of motion. A particle executes a SHM of amplitude 'a'. At what distance from the mean position is its kinetic energy equal to its potential energy?