Deep-Water Waves. Water waves are called deep-water waves when the depth of the

Question

Deep-Water Waves. Water waves are called deep-water waves when the depth of the water is much greater than the wavelength of the wave. The speed of deep-water waves depends on the wavelength as follows:

Suppose you are on a ship at rest in the ocean, observing the crests of a passing sinusoidal wave. You estimate that the crests are 75 m apart.


Approximately how much time elapses between one crest reaching your ship and the next? (s)
3.9
4.9
6.9
14

The captain starts the engines and sails directly opposite the motion of the waves at 3.0 m/s. Now how much time elapses between one crest reaching your ship and the next?
2.7
5.4
6.9
25

In the deep ocean, a longer-wavelength wave travels faster than a shorter-wavelength wave. Thus, a higher-frequency wave travels ______ a lower-frequency wave.
faster than
at the same speed as
slower than

Explanation / Answer

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension. As a result, water with a free surface is generally considered to be a dispersive medium. Surface gravity waves, moving under the forcing by gravity, propagate faster for increasing wavelength. For a given wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with this, capillary waves only forced by surface tension, propagate faster for shorter wavelengths. Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves.

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