A tsunami is not just one giant wave, but a group of waves with an extremely long wavelength and period. These waves begin by dispersing energy from displaced water outwards.
Near the source of energy, a tsunami can have wavelengths greater than 300 miles long with periods as long as an hour. Typically, the wavelength and period increase proportionally to the depth of the water. One period is the time between two waves. Due to the great length of the waves, tsunamis behave as shallow water waves, meaning the ratio between the water depth and wavelength is very small. In deep water, tsunamis are capable of traveling at very high speed. The acceleration of a tsunami is the product of the acceleration of gravity multiplied and the water depth. A tsunami in the Pacific Ocean can travel at speeds up to 450 mph, over half the speed of sound.
As the waves approach the shore, the depth gradually decreases, decelerating the tsunami. However, due to the conservation of energy, the waves are forced to grow taller until reaching heights between 30 and 100 feet. Some of the energy is lost to friction and turbulence, but the waves still have a significant amount of energy and can reach many feet inland.