Which hazard generated by earthquakes can be destructive many hundreds of kilometers away from the epicenter?

This hazard is known as a tsunami or tsunami wave. Tsunamis are generated by submarine earthquakes where the movement of the earth's crust displaces a large volume of ocean water. The tsunami waves then race away from this point of disturbance in all directions much like ripples in a pond. These waves can travel for thousands of miles at sea before reaching land. As the waves approach land and start to interact with the ocean floor they slow and the water piles up and looks much like a wall of water approaching the shore. 

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