Periodically arranged bands of clouds as seen in the figure often identify gravity waves in the atmosphere. Not all bands of clouds are related to gravity waves but such, that have a perpendicular alignment to the main wind flow.
All gravity waves have in common, hence the name, that the restoring force is gravitation. Deflecting an air parcel vertically in a stable atmosphere leads to an oscillation around its initial state. This behaviour is comparable to a spring pendulum. The only difference is the restoring force (buoyancy vs. spring). For a deeper understanding someone needs to know the Brunt-Väisälä frequency. A special gravity wave is the mountain wave. Usually a mountain wave is released due to the flow across mountains. Gravity waves are also important for gliders. They use the ascending air at the front of the wave.
Mathematically you can consider gravity waves just as other wave phenomenons. To characterize waves you need terms like wavelength and phase speed.