Longitudinal Wave
Definition
A wave where the medium vibrates back and forth along the same direction the wave travels — like a slinky being pushed and pulled.
Technical definition
A wave in which particle displacement is parallel to the direction of propagation, consisting of alternating compressions and rarefactions. Sound waves in air are the classic example.
Example
Clap your hands: the air molecules bunch together and spread apart along the direction the sound travels to your ears — compressions and rarefactions racing outward at 343 m/s, with each molecule only shuffling back and forth.
Longitudinal does not mean 'along a rope'. A wave on a shaken rope is transverse — the rope moves up and down while the wave moves sideways. The test is the angle between the medium's motion and the wave's travel: parallel means longitudinal (sound), perpendicular means transverse (light, rope waves). Get the geometry right and the label follows.