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The highest part of the wave is called the crest. The lowest part is called the trough. Wave height is the overall vertical change in height between crest and trough. The distance between two successive crests (or troughs) is the length of the wave, or wavelength.
While the passage of each wave is associated with an up and down motion, the water is actually moving in a circular motion. It is this orbital motion of the water that causes an object to bob up and down, forward and backward, as waves pass under it.
However, this motion is not exactly circular but trochoidal (imagine a curved line formed by tracing a point on a rolling wheel. In other words, while the motion is circular, there is still overall forward motion. Even over deep water, where a wave moves in an almost-closed circular path, there is still a tiny forward motion with the passage of each wave, particularly large waves.
In deep water, as the depth increases, the wave motion changes fairly rapidly. The curved trochoidal shape at the surface flattens further down, and the total motion decreases. This flattening and decreasing continues with increasing water depth until all that remains is a small back and forth movement. Eventually, when that motion decreases to one-half of the wave's total length, it will cease as well. In shallow water, the same flattening of the circular wave motion occurs, but there is no decrease in the forward/backward motion.
As a deep-water wave reaches shore, it begins to interact with the bottom. When the depth of the water is one-half of the wave's length, drag (friction) from the bottom will cause the waves to slow down, grow taller, and become shaped like peaks. These wave peaks will continue to grow until they become unstable and, moving faster than the water below, break forward.
The speed at which a wave moves through the water is dependent on the wave's length and the depth of the water. Generally, the longer the wavelength, the faster it moves. Tsunamis can have extremely long wavelengths (60 miles/100 km or more) and thus move around 550 mph (900 km/h). Learn more about Tsunamis.