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Wave
A travelling disturbance/oscillation that transfers energy (and information) through space or a medium without a net transfer of matter.
Medium
The medium is the material through which a wave travels.
Waves such as sound cannot travel without a medium because there are no particles to vibrate.
Transverse wave
A wave in which the oscillations of the medium are perpendicular (at right angles) to the direction the wave travels.
Crests and troughs show positions of vibration, not pieces of matter.
Longitudinal wave
A wave in which the oscillations of the medium are parallel to the direction the wave travels.
Particles move back and forth but do not stay in compressions or rarefactions.
The key difference is direction of particle vibration, not speed or energy.