Distinguish between longitudinal and transverse waves. Do all waves fall into one or the other of these categories? If not, give an example of one that does not?

Answer 1

In a longitudinal wave the particle displacement is parallel to the direction of wave propagation.
In a transverse wave the particle displacement is perpendicular to the direction of wave propagation.
Longitudinal waves can propagate through either solid , fluid and gas media, but transverse waves can propagate only in solids.

Transverse waves: vibrating guitar string, water waves, doing "the wave" at a stadium
Longitudinal waves: sound waves