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why do waves slow down in shallow water

The calculation of the speed of surface waves on a fluid is a little lengthy to do in full generality for all fluid heights and all wavelengths. When you say that you "dropped it" do you mean you dropped the whole tray full of water on the floor, or did you drop something into the tray to make some waves? I would set up something with a card or a board on one end, shaken side to side so as to make waves of a desired wavelength -- dropping something in the tray results in a complicated mixture of waves of different wavelengths, which travel at different speeds. I found a full expression for the speed of a wave in an incompressible fluid with no viscosity (there are no such fluids, but water is pretty close to this approximation for small trayfuls at normal temperatures and pressures). The speed of surface waves is:
from Fetter and Walecka, "Theoretical Mechanics of Particles and Continua", McGraw-Hill, 1980. Here, g is the acceleration due to gravity, 9. 81 m/s**2, lambda is the wavelength of the waves under study, and h is the depth of the fluid.

For very shallow fluids (compared to the wavelength), the speed increases proportionally to the square root of the depth, and for very deep fluids, the speed increases with the square root of the wavelength. In words why this is the case -- for a shallow fluid, the motion of the fluid is mostly side-to-side, and in order to accumulate more fluid in one place (to make the crest of the wave), each little bit of fluid must travel a little farther than it would have to in deeper water. When a wave passes, the bits of fluid, if you could watch one at a time, travel in ellipses. For shallow water, the ellipses are stretched out horizontally, and in very deep water, they are very nearly circular. So for a wave of the same height (top to bottom of the ellipse), the bits of water must travel farther in the shallow tray than the deep tray. Because the waves of the same height in shallow and deep water exert the same pressure differences due to gravity to get the water moving (although the motion is different due to the fact that the bottom is there), similar forces push and pull on the water.

To get the water moving farther and faster with the same force takes a longer time for each push, and hence a slower speed for the wave, in the shallow water. Note: this speed formula assumes that the waves are small -- for waves whose heights are comparable to the depth of the tray, you will get even more complicated behavior -- the most spectacular of these is the formation of "breakers" where the waves will curl over and crash as they do on beaches, making for good surfing. (published on 10/22/2007) As mentioned earlier, Tsunami waves become dangerous only when they get close to the coast: the height of a Tsunami wave grows larger as the water becomes more and more shallow in a wave shoaling process. If we look at the natural causes of this phenomenon, weвll realize the shoaling process is strictly related to the wave " ". An increase in wave amplitude results in "shoaling" when waves, including tsunamis, run from deep to shallow water.

This is significant in coastal regions. This phenomena occurs because of the force from the seabed as it becomes shallower. This slows down the wave: the shallower the water, the slower the wave. Even when tsunamis have only a small amplitude (less than a meter) they can shoal up to many meters high as they hit shallow water. When a tsunami hits shallower coastal waters, the trough or base of the wave contacts the beach floor. As a consequence, the leading edge of the tsunami slows dramatically due to the shallower water, but the trailing part of the wave is still moving rapidly in the deeper water. The wave is compressed and its velocity slows below 80 kilometers per hour. Its wavelength diminishes to less than 20 kilometers and its amplitude is magnified many times. This piling up of tsunami energy results in growth of the wave height. As the tsunami still has a long period, this results in the tsunami taking minutes to reach it's maximum height, which could be hundreds of meters high! The form of the adjacent geography to deep water (open bays and coastline), can shape the tsunami into a step-like wave with a steep braking face.

The wave height as it crashes upon a shore depends almost entirely upon the submarine topography offshore. Steeper shorelines produce higher tsunami waves. Large tsunamis have been known to rise to over 100 feet! Note 400 mph in 4000 m depth equals a large wavelength (213 km) and low amplitude. It is less than a meter usually as it passes through deep water. Because of the factors of low amplitude in deep water and large wavelength, tsunamis are often not noticed in mid-ocean. As the tsunami hits shallower water, the velocity slows, wavelength decreases and the waves height (amplitude) increases. Tsunami waves can grow up to 30 meters in height as they hit the shoreline and are followed by more waves that may even be more dangerous. Not just the crests but also the troughs can be treacherous as they form currents which take both people and entire buildings out to sea as seen in the recent tsunami in Japan.

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