What is the difference between transverse and longitudinal waves and what is the relation between motion of wave and compression, rarefraction.????

Transverse Waves: Displacement of the medium is perpendicular to the direction of propagation of the wave.

To understand this it is good to think of a rope being held still by person B and being moved up and down by person A. The direction of propagation is from person A to B, so you will see the waves move along this way. But the displacement will be up and down.

Can travel in solids, but not in liquids and gas.

eg. Electromagnetic radiation

Longitudinal Waves: Displacement of the medium is parallel to the direction of propagation of the wave.

A good example for this is a slinky being pushed along the table, the propagation will be along the table and so will the displacement of all the 'rings'.

Can travel through all states of matter.

eg. Sound waves

longitudinal waves have compressions and rarefactions. Compression. A compression is a region in a longitudinal wave where the particles are closest together. Rarefaction. A rarefaction is a region in a longitudinal wave where the particles are furthest apart.