Читать книгу Waves and Beaches - Kim McCoy - Страница 25

Wave tanks allow us to change the wave period or wave height independently. One of the first things to be noticed is that the wave length depends upon wave period, whereas wave height does not. A wave’s height is related to the amount of energy that created the wave. Several equations are introduced in the next paragraphs—hang on, like waves these too will soon pass.

Under ideal laboratory conditions, a pure sine wave can be produced. We discover that the wave length L (in feet) is equal to 5.12 times the square of the period.

Which can be simplified to 5.12T2 (or in meters L = 1.56T2) and where L is the wave length in feet, g is the acceleration of gravity (32.2 feet per second, per second), and T is the period in seconds. Thus a one-second wave would be 5.12 feet (1.56 m) long; and a ten-second wave would be 512 feet (156 m) long. Wave velocity, the distance a wave travels per unit time, is usually designated by C (also known as “celerity”), and expressed as . A one-second wave verifies the relationship by moving at 5.1 feet per second (1.5 m/sec).

But, as the period is increased, one soon finds that these relationships for length and velocity do not hold exactly in this wave tank for periods of over one second. Why? Because in the shorter-period range we have been dealing with deep-water waves. A shallow-water wave is one that is traveling in water whose depth is less than half its wave length; that is, if the depth of water is small compared to the wave length, the effect of the bottom is sufficient to alter substantially the character of the waves. With the still-water depth in our tank at 2.5 feet (76 cm), increasing the period has produced waves which “feel the bottom” and are affected by it. The mathematical term “hyperbolic tangent” (tanh) might be difficult for some readers; however, the full expression for wave velocity, water depth (d) is taken into account:

The final term contains the ratio or water depth/wave length.

Most wave researchers find it convenient to describe waves in terms of their and use a simplified version of that equation. For a of more than 0.5 (a deep-water wave), the hyperbolic tangent of it is so close to 1 that it can be neglected, as we did earlier. On the other hand, if the depth is quite small compared to the wave length () of the hyperbolic tangent; tanh can be replaced by simply . Then, after cancellation, the wave-velocity expression becomes much more simple: . This relationship is used for shallow-water waves.

This, too, is very convenient, especially when one is working with seismic sea waves, which are so long that, for them, even the deepest ocean is shallow water. But for values of between 0.05 and 0.5 we must use the longer form of the equation for deep-water waves.

We first thought that the wave height seemed to be independent of both the period and wave length, but further experiments show this is not quite so. If we hold the period constant at one second (with the wave length remaining 5.12 feet or 1.56 m) and gradually increase the wave height, we discover that waves higher than 0.75 feet (20 cm) have unstable crests. That is, they tend to break as they travel down the tank. On repeating the experiment with other heights and lengths we discover that the angle at a wave crest may not be smaller than 120 degrees, or the wave will break. Stated in another way, the wave height may not be greater than one-seventh of the wave length.

This relationship (ratio) of height to length () is called wave steepness (see figure 9).

FIGURE 9: Maximum wave steepness is 1/7. Waves become unstable as the wave-crest angle decreases to less than 120 degrees.

In reviewing the notes on these past experiments, we find that as the waves became steeper they also increased slightly in speed until at 1:7, the maximum, they moved perhaps 10 percent faster than the theoretical speed. However, because ocean waves rarely achieve such steepness—only in violent storms—it is customary to neglect this increase.

When the waves move onto the abruptly shoaling beach at the end of the tank, they change character in another way. As the depth decreases, the waves are said to peak up; that is, their height increases rapidly. At the same time, the shallow water causes the wave length to decrease, and the result is a suddenly steepened wave. In a very short distance, the crest angle decreases below the critical 120 degrees and the wave becomes unstable. The crest, moving more rapidly than the water below, falls forward, and the waveform collapses into turbulent confusion, which uses up most of the wave’s energy. Small amounts of this energy is converted into heat by frictional forces, just like rubbing your hands together. Trapped air produces sounds heard as the rumbling surf above the grinding noises of cobbles and the swishing of moving sand.

Stellar Nightwatch

Navigation has been revolutionized by the Global Positioning System (GPS), yet adequate older methods remain embedded in nature. In December of 2007, we sailed across the Atlantic guided by waves and stars. Our course was set southwest from the island of La Palma off the coast of Africa, across the Atlantic to the Caribbean some 2,500 nautical miles away. When beyond sight of land after night had fallen, the stars, planets, and waves guided us. The rising, culmination, and setting locations of stars and their visibility times provided us with heading and latitude. The long-period swell gave us direction and speed. Each surging wave dipped our bow, lifted our stern, and altered the vessel’s course a few degrees back and forth while slowly passing under us. Above, the nighttime sky was our celestial compass. We kept the upright mast swaying between Mars and Jupiter, the luff of the mainsail followed the constellation Pleiades as the forestay pointed at a star low on the horizon. We forgot about the boat’s compass; we didn’t need it. Far from shipping lanes, our course was in the waves—each a cipher to our destination. All were there for us through the darkness until dawn. – KM