Define longitudinal wave motion.  What are the essential conditions required for the formation of longitudinal wave motion?

It is that wave motion in which the particles of the medium through which the wave is travelling vibrate in a direction parallel to the direction of the motion of the wave.

In the figure, the wave is travelling from left to right, and the particles of the  medium vibrate in the horizontal direction simple harmonically.  It represents the longitudinal wave motion.

For their propagation, the medium must possess:

(i) elasticity

(ii)  inertia

(iii) absence of frictional resistance.