Since a tsunami has a large wave length, tsunamis act as a shallow-water wave even in deep oceanic water. Shallow-water waves move at a speed that is equal to the square root of the product of the acceleration of gravity (9.8 m/s^2) and the water depth. For example, in the Pacific Ocean, where the typical water depth is about 4000 m, a tsunami travels at about 200 m/s (about 712 km/hr or 442 mi/hr) with little energy loss even for far distances, while at a water depth of 40 m, the speed is 20 m/s (about 71 km/hr or 44 mi/hr), much slower, but still very fast to run away from.
Interesting article at Wikipedia: I had learned prior to this that these waves traveled at 400 mph or so, but didn’t realize that they changed so dramatically as they reached shallow water. I have to wonder how useful a warning system would have been: I think we would have seen less loss of life in the tourist areas, but I suspect the native people would have borne the brunt of this.