Question

This question will deal with tsunamis, long-wavelength waves caused by seismic activity, bolide impacts, landslides, and in rare cases, atmospheric disturbances. Tsunamis can propagate across ocean basins and can be incredibly destructive.

Tsunamis are interesting from a perspective of wave phenomena because when they are in deep water they are almost undetectable. Their wave height is on order of at most a few meters. However, they have very long wavelengths, typically 200 km. This makes their phase speed very fast. And because of the large phase speed, they contain a lot of energy.

For water waves, instead of a spring constant, k, providing the restoring force, the tension is provided by gravity. Assuming the tension due to gravity per unit volume of water is given by:

Where is the density of water (= 1000 kg/m³), V is a control volume defined here as a unit area multiplied by water depth, h, and g is the acceleration of gravity.

Part 1 (1 pts): Rewrite to express the phase velocity of water waves as a function of g and h. (hint: express m in terms of  and a unit area)

Part 2 (1 pts): Your answer from part 1 should be the phase speed of shallow-water waves. Use that to calculate the phase speed (i.e., propagation velocity of a tsunami) across the Pacific Ocean. Assume the mean depth of the Pacific is 5 km.

Part 3 (1 pts): Suppose the tsunami runs onto a coast where the water depth decreases to 50 m from 5000 m. How does the phase speed of the wave change?

Part 4 (1 pts): Assume the period of the wave remains constant as it reaches the shallow water, what is the new wavelength?

Part 5 (1 pts): Green’s Law says that for shallow water waves, wave height between the two depths is given by:  . Use Green’s Law to estimate the new wave height of the tsunami.

Answer 1