Waves are broadly classified as disturbances that travel through time and space, typically resulting in a transfer of energy. The two main mathematical descriptions of a wave are its amplitude (the amount of change in oscillations of a wave) and frequency (the amount of oscillations the wave produces over a period of time). The medium is the space through which the wave travels, such as an empty vacuum and the ocean's waters.
The type of wave that can travel through empty space (also known as a vacuum) is an electromagnetic wave. The most common example of electromagnetic waves are light waves. Electromagnetic waves are distinguished by their ability to vibrate without any interaction with the medium (because, by default, a vacuum has no exterior) and instead relies on its own charged particle (a particle with a negative or positive charge, according to its number of electrons) to oscillate (the action of a wave moving back and forth). It should be noted that the concept of a complete vacuum, even in space, is somewhat rare because even "empty" space has small (often subatomic) particles; however, these particles do not effect the transmission of electromagnetic waves.
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Most types of waves do require some sort of interaction with their medium in order to transport. One of the most common examples on Earth are sound waves, which travel through the air. A sound wave is a longitudinal wave, which is a type of wave that moves in a direction parallel to the particles of the medium through which it travels. Unlike electromagnetic waves, longitudinal waves can not travel in a vacuum.
When people visualize waves, they often think of water traveling through the ocean. The ocean is a type of medium that is highly conducive to longitudinal waves. Waves travel without much resistance from the medium because the atoms of water are not tightly packed together. This allows for a wave to easily vibrate in water. Waves in the ocean are frequently instigated by strong winds stirring up the water and transferring energy.