Earthquakes are commonly associated with ground shaking that is a result of elastic waves traveling through the solid earth. However, near the source of submarine earthquakes, the seafloor is “permanently” uplifted and down-dropped, pushing the entire water column up and down.
The potential energy that results from pushing water above mean sea level is then transferred to horizontal propagation of the tsunami wave (kinetic energy). For the case shown above, the earthquake rupture occurred at the base of the continental slope in relatively deep water. Situations can also arise where the earthquake rupture occurs beneath the continental shelf in much shallower water.
Note: In the figure, the waves are greatly exaggerated compared to water depth! In the open ocean, the waves are at most several meters high spread over many tens to hundreds of kilometers in length.
Within several minutes of the earthquake, the initial tsunami (Panel 1) is split into a tsunami that travels out to the deep ocean (distant tsunami) and another tsunami that travels towards the nearby coast (local tsunami). The height above mean sea level of the two oppositely traveling tsunamis is approximately half that of the original tsunami (Panel 1). (This is somewhat modified in three dimensions, but the same idea holds). The speed at which both tsunamis travel varies as the square root of the water depth. Therefore the deep-ocean tsunami travels faster than the local tsunami near shore.
Several things happen as the local tsunami travels over the continental slope. Most obvious is that the amplitude increases. In addition, the wavelength decreases. This results in steepening of the leading wave–an important control of wave runup at the coast (next panel). Note also that the deep ocean tsunami has traveled much farther than the local tsunami because of the higher propagation speed. As the deep ocean tsunami approaches a distant shore, amplification and shortening of the wave will occur, just as with the local tsunami shown above.
Panel 4—Run up
As the tsunami wave travels from the deep-water, continental slope region to the near-shore region, tsunami run up occurs. Run up is a measurement of the height of the water onshore observed above a reference sea level. Contrary to many artistic images of tsunamis, most tsunamis do not result in giant breaking waves (like normal surf waves at the beach that curl over as they approach shore). Rather, they come in much like very strong and very fast tides (i.e., a rapid, local rise in sea level). Much of the damage inflicted by tsunamis is caused by strong currents and floating debris. The small number of tsunamis that do break often form vertical walls of turbulent water called bores. Tsunamis will often travel much farther inland than normal waves.
source: The Influence of Building’s Height and Lay-out to Tsunami’s Run Up