The Energy Of Ocean Waves

Waves of Many Kinds

At any one moment, the energy in waves in the whole world ocean is only about one-third as great as the energy in currents.1 All the same, waves are more visible than currents, and they display the ocean's enormous energy much more vividly. Sometimes—as when you watch a stormy sea from a protected shore—they are exhilarating. But when you look up from a small boat at waves rearing over you, they are terrifying.

Waves vary in many ways. They vary in what causes them to form and grow and in what makes them die out and disappear. They obviously vary in size and also in period, the time it takes for one wave crest to succeed another at a given point. Taking into account these differences and others, waves can be classified as follows:

Type 1: Wind waves (ripples, "ordinary" waves, and swells). Type 2: Internal waves, below the surface of the water. Type 3: Tsunamis, which are usually, but not always, caused by earthquakes.

Type 4: Solitary waves, whose solitariness puts them in a class by themselves. Type 5: The tides: to many people's surprise, the rise and fall of a tide is a wave. It could be called, but never is, a "tidal wave," a term often used, incorrectly, for a tsunami.

Type 6: Planetary waves: these are not up-and-down waves like all the others. The term refers to the directional changes in a horizontal current that swings alternately to left and right; that is, they resemble Rossby waves in the atmospheric jet streams (see chapter 3).

We consider the first four of these wave types in the following sections, paying particular attention to what causes them (gives them their energy) and how they are dissipated (lose their energy). Type 5, the tides, merit a chapter to themselves (chapter 8). Type 6, planetary waves, are caused by ocean currents responding to the Coriolis effect; though they are technically "waves," they are not waves in the ordinary sense and contain no energy of their own. They are not considered further.

Wind Waves: Ripples and "Seas"

Wind is the commonest cause of waves. Making and sustaining waves on oceans and lakes is one of the ways winds dispose of their energy. It has been truly said that wave energy is "wind energy that has been temporarily trapped in the waves."2

The first question to consider is, How do waves get started? How does the wind exert pressure on calm water? Once the first small ripples have formed, it is easy to see how continued wind pressure will enlarge them. It is not so easy to see how a horizontal water surface is affected by a horizontal wind blowing over it. Nowhere does the wind blow against the water: Why should wind and water do anything more than slide past each other?

This is not a simple problem, and several solutions have been suggested. According to one theory, an apparently smooth sea surface is always dimpled by small variations in air pressure from place to place, and the dimples provide tiny slopes for the wind to act on. A more recent theory proposes that ripples are first caused by downdrafts of wind. The wind is never perfectly horizontal everywhere, and downdrafts produce cat's-paws, the dark patches of ruffled water you see scattered here and there on a calm sea as soon as the air begins to move. The ripples in the cat's-paws provide the slopes that horizontal winds can act on to build up sizable waves.

Ripples and waves don't keep growing: there always comes a time when they disappear and a calm sea is restored. What causes the waves' collapse when the wind dies down and the energy maintaining them stops? Two forces act to flatten an undulating liquid surface: gravity and surface tension. Gravity is by far the more important force, but surface tension alone is enough to even out the smallest undulations. Indeed, this is the technical difference between ripples and waves. Ripples (this means ripples in a water surface, not ripples in sand) are waves so small that surface tension suffices to flatten them.

It is impossible to specify precisely the maximum size of a ripple—equiva-lently, the minimum size of a true wave—since it depends on the surface tension of the water, which depends in turn on the water's salinity and temperature. The surface tension of salty water is greater than that of fresh water, and the surface tension of cold water is greater than that of warm water. Ripples usually have wavelengths (the crest-to-crest distance) of less than 2 cm; waves usually have wavelengths of more than 10 cm. A wavelet of intermediate wavelength may be either a ripple or a wave, depending on the surface tension.

Waves larger than ripples are too large for surface tension to flatten; instead, the force of gravity levels the sea surface when the wind stops blowing; gravity is described as the restoring force, and the waves are formally called gravity waves. Gravity waves (and, a fortiori, ripples) are also leveled by a change of wind: "A sudden reversal of the wind at sea literally knocks the waves flat."3 (Note that this has nothing to do with the "gravity waves" of modern physics, which are postulated periodic variations in the force of gravity.)

Ripples are of minor importance in the context of energy: their energy is trivial, so in what follows we concentrate on waves. Once they are large enough, waves interact with the wind; the feedback reinforces the waves and makes them grow higher, at the same time increasing the rate at which energy is transferred from the atmosphere to the ocean. So long as the wind continues to blow without abating, the waves grow higher and higher, though not, obviously, without limit. If a wave becomes too steep for the incoming wind energy to sustain it, its crest topples over and it becomes a whitecap.

The height to which waves can grow depends on three things: the wind speed; the fetch, which is the distance the wind has blown across the ocean without interruption; and the duration, which is the length of time the wind has been blowing with no change of speed or direction. For example if a wind of 5 m s-1 (the speed) blows over open sea for 20 km (the fetch), it will take 2.3 hours (the duration) for the waves to grow into a fully developed sea, after which they grow no higher; their average height will be about 0.25 m, and there will be a scattering of whitecaps.4 A wind of 15 m s-1, known as a mod-

Height meters

Cartooncity Neighborhood Drawing

Figure 7.1. Typical profile of a fully developed sea. Note the exaggerated vertical scale. MSL is mean sea level.

Figure 7.1. Typical profile of a fully developed sea. Note the exaggerated vertical scale. MSL is mean sea level.




Figure 7.2. (a) Typical profile of swell. (b) A sine wave, showing its dimensions. A wave's height is the vertical distance from crest to trough; its amplitude is half the height. MSL is mean sea level.

a b erate gale, with a fetch of 480 km, produces a fully developed sea in twenty-two hours. The average height of the waves is about 1 m, most of them are whitecaps, and streaks of foam, or spindrift, begin to blow from their crests. Note that these examples give average wave heights; about one-tenth of the waves will be twice as high or higher. In a fully developed sea resulting from a strong wind, an appreciable fraction of the energy is dissipated by the breaking of the wave crests into whitecaps; then the viscous shearing ("friction") caused by turbulence in the whitecaps produces heat.

The profile of a fully developed sea, indeed of any "sea" in the sailor's sense of the word, meaning a rough sea, is markedly irregular, as figure 7.1 shows. The waves vary tremendously in height, and their wavelengths also vary to some extent, though not nearly so much as their heights. This irregularity makes the physics of real waves much more difficult to investigate than that of the simple, "pure" waves (sine waves) shown in figure 7.2. The waves that come closest to sine waves in real life—often they are true sine waves—are swells.


Wind waves, which in large numbers make a "sea," are found where the wind is blowing. They advance in the direction of the wind and keep on moving. Once they are out of the area where they were generated, they "settle down" (in a manner to be described below) and become swell. Figure 7.2a shows the profile of a typical low swell; for comparison a pure sine wave is shown in figure 7.2b, which also shows the terminology used to describe it.

Another note on terminology: the word "wave," by itself, sometimes includes both wind waves and swells; and when there is no risk of misunderstanding it is used as an abbreviation for wind wave.

Swells typically have longer wavelengths and longer periods than wind waves. Few wind waves have wavelengths greater than 130 m, whereas swells are often several hundreds of meters long. Most wind waves have periods in the range 0.2 to 10 s (ripples have periods of less than 0.2 s), whereas most swells have periods in the range 10 to 30 s. These measurements don't define wind waves and swells, however; they are merely typical, and it is quite possible for a wind wave to be longer than 130 m or slower than 10 s and for a swell to be shorter or faster. The defining difference between a wind wave and a swell is origin. A "sea," by definition, is the mass of wind waves in the area where the wind generated them. A swell is the waves (or a single wave, the same word for both) that has traveled outside the generating area.

Figure 7.3. Movement of the water in a deepwater wave. The wave is moving from left to right. The two panels are from the same viewpoint (note the identical background scenery). (a) A floating log is on the crest of a wave traveling to the right. (b) Half a period later, the wave has advanced half a wavelength to the right, and the log is in the trough; its orbit is the large circle at the surface. The water molecules M and N, at depth, circle in their own smaller orbits in synchrony with the log.

Before considering the way a "confused sea" (as sailors call it), with high, irregular, foam-tipped waves, becomes converted into a smooth, gentle, regular swell, a digression is necessary on the physics of waves in general.

The first point to emphasize is that when waves travel across the sea, what is traveling is the shape, or form, of the wave, not the water itself. This is obvious if you watch an object—a log, say—floating on a choppy sea. Although the waves travel steadily forward, the log does not travel with them; it stays in more or less the same place, bobbing up and down as the crests and troughs of the waves pass beneath it.5 Therefore the water is not flowing in the direction of the wind, and it obviously is not stationary: How, then, does it move? The answer is shown in figure 7.3. If you could observe the movement of an individual molecule of water, it would be seen to move in circles, as the figure shows. Note that a molecule at the surface—or a floating log—does not go below the surface as it circles, any more than the foot of a person riding a bicycle goes below the pedal: foot and pedal circle together, with the foot attached to the top. In the same way, a water molecule in the sea surface, or a log floating on the sea surface, acts as part of the surface and circles with it.

Water below the surface circles too, in time with the circling water above, but the circles become smaller and smaller at increasing depths; at a depth equal to one wavelength, the circle has a diameter less than one-five hundredth of the diameter at the surface. The speed at which each molecule of water circles is normally less than the speed at which the waveform travels; if it becomes greater, the wave breaks.6

Now consider the energy of a wave. Notice first that it has potential energy (PE) because the water is not flat; the PE would disappear if the water were to become flat in response to the restoring force—gravity—acting on it, but so long as it is not flat it has PE. It also has kinetic energy (KE) because of the circling motion of the water in the wave. The total energy of a wave is the sum of its potential and kinetic energies. The PE and the KE of a single wave are equal. As the trough of a wave rises, its KE is converted to PE; then, as the crest sinks, its PE is converted into the KE of the water's circling motion. The latter is dissipated by viscous drag. The energy lost is made good by the wind so long as it is still blowing. If the wind dies down, or if the waves travel out of the windy area, they lose both energy and height and change their form, as described in the next section.

The total energy in sea waves is given by the formula energy = 1255.68 H2 joules per square meter (J m-2), in which H represents the height of the waves in meters.7 The factor 1255.68 is 1/8 x 1,024 kg m-3 (the density of seawater) x 9.81 m s-2 (the acceleration due to gravity).

The formula gives the energy in joules per square meter of sea surface. It gives more meaningful results with swells than with waves; the waves in a "sea" are very variable in height (see fig. 7.1) and very irregular in shape; swells, on the other hand, are notably uniform in both height and shape, and the shape is often close to a pure sine curve.

Two examples: the energy in waves 1 m high is 1,255.68 J m-2. In waves half as high (50 cm) the energy is only one-fourth as much, or 313.92 J m-2, because it depends on the square of H.

It has been estimated that, at any instant, the energy in the surface waves of the whole world ocean is 1018 J. When these waves break on shore they release heat—but not much. If all the heat were used to heat the water, with none being lost, it would take 90,000 years to raise the temperature of the world ocean by 1°C. The rate at which wave energy is converted to heat by the waves breaking on all the world's shorelines is believed to be about 2 x 1012 J s-1 (equivalently, 2 billion kilowatts). The rate at which the sun heats the oceans is 1,500 times as great.8

Most of the energy in wind waves and swells is dissipated when they break on the shore. But a fraction is lost while they are still out in the deep ocean— if it were not for this loss, the sea would never be calm.

How Waves Die Down

As noted already, the waves in a big "sea" lose much of their energy because of the viscous drag in the turbulent water. What becomes of the remainder? Unsurprisingly, it is also lost because of viscous drag in all the rest of the constantly moving water. What is surprising is the slowness of the loss. The energy captured from a windstorm and carried away from it in smooth swells lasts a long time.

The way waves turn to swells and then fade away is not as simple as it seems. Any train of waves, however jagged its shape, can be analyzed into the sum of a number (usually an infinitely large number) of different component waves, each of them a sine wave like that in figure 7.2b. The component sine waves differ from one another in period, in amplitude, or in position relative to the others—usually in two or all three of these attributes. Figure 7.4 shows a train of waves made up of only three sine waves, as an example (an unnaturally simplified example, to make the figure clear). The heavy line in the upper panel shows the form of the waves to be analyzed.9 The three sine waves in the lower panel are its component waves.

If the original waves leaving a windy area were to match those shown in the upper panel, their component waves would slowly become separated, as shown in figure 7.5a. The example, as explained, is artificial; in real life the pattern is like that in figure 7.5b: waves of many wavelengths are present, sorted by wavelength with the longest, highest waves in the lead. The sorting hap-

Wave height

Wave height

What Causes Erratic Waves Ocean
Figure 7.4. Analyzing a train of irregular waves (a) into its component sine waves (b). MSL is mean sea level. The component waves are added as shown. H is the height of the water surface above MSL; H = x + y - z, in which x, y, and z are the heights of the three component waves.

pens because long waves—those with long period and long wavelength— travel faster than short waves.10

The sorting process is known as wave dispersion. The waves lose energy, and consequently height, because of internal viscous shearing, and the loss happens much faster in short waves than in long; as a result, the waves in the rear shrink faster than those in the vanguard and fade away sooner, leaving the long-wavelength waves as temporary survivors. In theory, a wave with a period of four seconds would have to travel the enormous distance of 23,000

Slower Faster

Slower Faster

Figure 7.5. The dispersion (sorting out) of waves of different wavelengths; the longer its wavelength, the faster a wave travels. (a) Dispersion of the irregular waves of figure 7.4a. The component waves have separated: the dotted line shows the shortest, slowest waves; the dash-dot line shows intermediate waves; and the dashed line shows the longest, fastest waves. (b) A more lifelike dispersed wave train: the waves vary continuously from long, high, and fast-moving in the lead (right) to shorter, lower, and slower in the rear (left).

Figure 7.5. The dispersion (sorting out) of waves of different wavelengths; the longer its wavelength, the faster a wave travels. (a) Dispersion of the irregular waves of figure 7.4a. The component waves have separated: the dotted line shows the shortest, slowest waves; the dash-dot line shows intermediate waves; and the dashed line shows the longest, fastest waves. (b) A more lifelike dispersed wave train: the waves vary continuously from long, high, and fast-moving in the lead (right) to shorter, lower, and slower in the rear (left).

km before losing half its height, and the journey would take 1,000 hours (nearly forty-two days).11 Contrast this with the fate of a wave with a period of one second, which would lose half its height after traveling a mere 12 km, a trip taking 4.3 hours. These numbers illustrate the durability of long waves. Storm waves generated in Antarctica have been observed to travel all the way to the shores of the Alaska panhandle 10,000 km away, arriving as low swells.

Because loss of energy is so exceedingly slow in long swells, the longest ones keep going almost indefinitely; they continue to travel while their height dwindles to a centimeter or two; at this stage their slopes are too gentle for the wind to affect them, and contrary winds cannot stop them. It seems likely that big swells never have enough time or distance to die away entirely. Their end comes when they break on a distant shore.

At the Beach

The energy in wind waves and swells is ultimately dissipated when they reach a shore. If they encounter a gently sloping beach, rising at an angle of 10° or less, practically all their energy is spent immediately: the waves break. A wave slows down on moving into shallow water, and its energy is temporarily conserved because it gains height—it rears up. This dooms the wave, however: in rearing up, it becomes so steep that its crest topples forward—it becomes a breaker. This is because the waveform's forward speed has become less than that of the water circling within it. This happens when the wave has grown to a height equal to about one-seventh of its wavelength.12

If the slope of the beach is gentle enough, the waves breaking on it lose all their energy. The loss takes place in several ways: some of the energy is converted to turbulence, which abruptly speeds up viscous drag and hence the conversion of energy to entropy; some energy drags beach material—sand and shingle—up the beach and down again; some energizes longshore currents that drag beach material along the shore (see chapter 9); the last of the energy is dissipated as noise—the crash of breakers and the roar of rolling, sliding shingle. The final dissipation of the energy in a big wave is noisy and spectacular.

The course of events just described is not the only possibility. If waves reach a steep shoreline, they are partly or wholly reflected back to sea, where they interact with incoming waves to produce confused choppiness. Confused chop-piness—a "sloppy sea"—is a sign that viscous shearing is proceeding vigorously and wave energy is rapidly being dissipated. Thus, one way or another, wave energy inevitably stops at the beach.

Internal Waves

The waves we have considered so far have all been surface waves; the movement of the water dies away a short distance below the surface, becoming negligible about one wavelength down. But that is not to say there is no wave action at greater depths: on the contrary, there are often big internal waves under the surface; they are invisible, as waves, from above. They can be detected, however, if they are not too deep and you know what to look for, because they produce surface slicks.

A slick is a band of smooth water forming a lane across a gently ruffled sea. Several widely spaced slicks can often be seen, all somewhat curved and more or less parallel. They appear because currents flow from the surface downward into the troughs of the internal waves from either side (fig 7.6). If there is an oily surface film produced by ships and boats, or naturally by living organisms in the water, the film thickens where the currents converge. The thickened oil film makes the water appear glassy—hence the slicks, which show best when the water between them is slightly ruffled. Slicks are a common part of the scenery for ocean watchers, scenery made more interesting if you visualize the unseen internal waves below.

Slick Surface Internal Wave
Figure 7.6. Internal waves, with slicks at the surface above the wave troughs. From E. C. Pielou, Fresh Water (Chicago: University of Chicago Press, 1998).

Internal waves form most readily in the shallow waters over coastal shelves.13 They develop where rising and falling tides are channeled by seafloor valleys, becoming concentrated into currents. Where these currents—they could be described as seafloor rivers and streams—flow over topographic irregularities, they develop waves, just as a river on land does when it flows over shallow rocks. Among the causes of internal waves far out to sea are moving low-pressure systems that produce the inverted barometer effect (see chapter 6); quickly repeated changes in wind stress also cause them.

Internal waves are undulations in an internal surface in the sea, just as surface waves are undulations in the "ordinary" surface. The internal surface is the layer known as the pycnocline (see chapter 6), in which the density of the water increases relatively suddenly as you descend from the surface, either because of cooling or because of increased salinity. The more sudden the density change, the more sharply defined the internal surface is, though it is never in-

finitesimally thin, as the air-sea surface is. When the density change is comparatively gradual, the layer in which it takes place hardly merits the name surface; in any case, internal waves can develop whether the density change is abrupt or gradual.

In what follows, we assume the internal surface is well defined. The difference in density between the waters above and below it are orders of magnitude less than the difference in density across an air-sea interface: this is what accounts for the striking dissimilarity between internal and surface waves. Compared with surface waves, internal waves are much higher, have much longer periods, and move much more slowly. These characteristics are most pronounced in the deep ocean, where the density contrasts are even less than they are close to the shore. Internal waves 200 m and more high, with periods of several hours, have been recorded in the open ocean.

An internal wave has much less energy than a surface wave of equal height because the difference in density is so slight. The energy in a surface wave 1 m high is, as we saw earlier, 1,256 J m-2; the energy in a 1 m high internal wave in the open ocean is less than 4 J m-2. The internal wave would have to be 18 m high to have the same energy as the 1 m surface wave.

Internal waves dissipate their energy more quickly than surface waves do and cannot travel nearly as far before dying away. They break when they run up a sloped seafloor in the same way that surface waves break when they run up a sloping beach; internal waves even produce "internal surf."14 And although the energy in underwater breakers is slight compared with the energy in subaerial breakers, it does biologically useful work nevertheless; it is the energy that, by causing turbulent mixing, prevents the water close to the seafloor from stagnating.15


A tsunami is a group of enormous waves set in motion by a sudden disturbance on the seafloor. The disturbance is usually an earthquake, but other causes are possible, such as a submarine landslide, a slumping of the seafloor, or a submarine volcano. Probably tsunamis are often caused by an earthquake plus a submarine landslide triggered by it. A meteorite falling into the ocean can cause a tsunami, as can a vast rockfall from cliffs bordering the sea; these last two causes do not, of course, originate on the seafloor, so perhaps a tsunami could be more exactly defined—though it never is—as a group of enormous waves caused by a sudden, unpredictable, short-lived natural calamity.

It is also necessary to say what a tsunami is not. It is not a tidal wave; tsunamis have nothing whatever to do with the tides. The misuse of the term "tidal wave" to mean a tsunami may have arisen because a tsunami wave sometimes looks like an unusually high tide that has risen exceptionally fast at an unexpected time; primitive people seeing a tsunami for the first time would have been puzzled and may have misidentified the cause.

The most noteworthy facts about tsunamis are, first, that they are nearly always caused by earthquakes and therefore derive their energy from the earth's internal energy, and second, their awe-inspiring size when they reach shore. Presumably little tsunamis, triggered by small earthquakes or landslides, are common, but they go undetected because they are masked by the sea's continual movement. A tsunami has to be big to be recognized; a group of waves caused by an earthquake weaker than about 6.5 on the Richter scale could go unnoticed. Big tsunamis are the product of truly energetic earthquakes: it has been conjectured that only 1 percent of a submarine earthquake's energy becomes converted to wave energy.16

Although the waves of a typical tsunami are "big," this does not mean they are high before they reach shore; out in the open ocean far from land they are seldom more than a meter high, and they attract no attention from the passengers and crew of a ship whose path they cross. In the open ocean their bigness consists in their exceedingly long wavelengths and periods compared with those of swells. Tsunamis have wavelengths of hundreds, sometimes thousands, of kilometers and periods several hours long; their energy is therefore thinly spread until they reach shore; but when they do come ashore, they are brought to an abrupt stop from a speed that may exceed 900 km/h: the collision turns an unremarkable wave into a killer.

About two-thirds of all tsunamis happen in the Pacific Ocean, because earthquakes are so numerous in the "ring of fire" around the Pacific. The earthquakes most often responsible are those caused by tectonic plates grinding against each other, as one plate is subducted under another (see chapter 15). Tsunamis often travel huge distances; for example, the tsunami generated by the 1960 earthquake off the coast of Chile traveled across the whole width of the Pacific to Japan; the distance was 17,000 km, the time taken twenty-two hours, and the average speed 773 km/h.17

In spite of crossing the Pacific, this tsunami, like all tsunamis, was technically speaking a shallow-water wave. The term applies to any wave having a wavelength greater than twenty times the depth of the water. The average depth of the ocean is less than 4 km, and the wavelengths of tsunamis are always hundreds of kilometers, which means they all rank as shallow-water waves. The defining characteristic of a shallow-water wave is that its speed is governed by the depth of the water. The wave is said to "feel the bottom," and it experiences appreciable drag all along its path. The speed of a shallow-water wave is given by the formula

here C is the wave's speed, d is the depth of the water in meters, and g is 9.81 m s-2, the acceleration caused by gravity. A tsunami traveling through water 500 m deep, for example, will have a speed of about 3.6 x V(9.81 x 500) km/h = 252 km/h.

The waves of a tsunami change their speed as they travel, slowing down where the water becomes shallower and speeding up if it deepens again; the separate waves of a single tsunami don't maintain the same speed: each goes at the speed appropriate to the depth of the water below it, which is continually changing as the wave travels over the surface above an uneven ocean bottom.

Tsunamis dissipate energy while they travel, but in a somewhat different way than swells do. Two of the differences are noteworthy.

First, because of their enormous wavelengths, tsunamis experience the drag of the seafloor wherever they are, whereas swells feel the bottom only when they are comparatively close to shore. Figure 7.7 shows why this is so; it illustrates how the water moves within a shallow-water wave and should be compared with figure 7.3, which shows a deepwater wave. In the deepwa-ter wave, the circular orbits of individual molecules of water shrink to nothing at some distance above the seafloor. By contrast, in a shallow-water wave the water moves in elliptical orbits that get progressively flatter the greater the depth, with the water at the bottom simply moving back and forth. The back-and-forth swishing drags constantly on the seafloor, dissipating the wave's energy.

The second difference in the way tsunamis and swells dissipate their energy is this. Tsunamis are generated by the jolt of an earthquake at a single spot on the ocean floor, and the resultant waves spread out in expanding circles; apart from the difference in scale, they resemble the rings of waves spreading outward when a rock is dropped in a pond. This means that the energy is spread along the circumference of an ever expanding circle; the total amount of energy is unaltered, but it becomes more and more thinly spread.18

Figure 7.7. Movement, from left to right, of the water in a shallow-water wave. This figure shows the wave in figure 7.3 closer to the shore, where the water is shallow. Note the elliptical orbits of individual water molecules narrowing at progressively greater depths. At the seafloor, the water shifts back and forth in a horizontal plane.

Figure 7.7. Movement, from left to right, of the water in a shallow-water wave. This figure shows the wave in figure 7.3 closer to the shore, where the water is shallow. Note the elliptical orbits of individual water molecules narrowing at progressively greater depths. At the seafloor, the water shifts back and forth in a horizontal plane.

By contrast, swells spread much less (fig. 7.8). As they leave the storm area where their parent wind waves were generated, they all advance parallel with the wind; once they are beyond the wind, they diverge to some extent, but rarely more than 45° to left and right of their original direction.19

In spite of spreading and "friction" with the bottom, powerful tsunamis often wreak enormous damage when they reach the land. Huge waves surge up the shore with tremendous force and inundate low-lying coastland; individual waves come at intervals that are sometimes an hour long. The higher the tide at the time each wave arrives, the farther inland it can go, destroying much that lies in its path. The tsunami in Chile in 1960 destroyed villages along an 800 km stretch of coastline before traveling to Japan, where it was still able to do much damage. The 1964 tsunami originating near Anchorage, Alaska, killed 107 people there before traveling outward over the Pacific and southward along the North American coastline; it killed 61 people in Hawaii and 12 in Crescent City, California.

1,000 km

Figure 7.8. The spreading waves of (a) a tsunami and (b) a swell. Note the different scales of the two maps. In (a) the X is over the site of the earthquake that caused the tsunami, which is a group of four waves; the wave with the longest wavelength spreads the fastest. In (b) the choppy patch of sea is the location of the storm where the wind waves giving rise to the swell originated.

Not all killer tsunamis come from afar. If an earthquake, even a comparatively minor one, shakes the seafloor where thick accumulations of sediment are poised ready to slump down the continental slope (the steep submarine slope between the continental shelf and the deep sea), the disturbance is likely to cause a disastrous tsunami on the nearby coast. Probable examples caused in this way are a 1992 tsunami on the shores of Nicaragua that killed 170 people; a 1993 tsunami on the shore of Okushiri Island, Japan, that killed more than 200; and the catastrophic tsunami that struck Papua New Guinea in 1998, sweeping away several villages and killing more than 2,500.

Solitary Waves

The waves we have considered so far have all been periodic, that is, repetitive, in the sense that one wave follows another, on and on and on, theoretically ad infinitum. Solitary waves, by contrast, occur singly, as their name implies. If that were all, there would be little more to say. But it is not all.A solitary wave is so unlike a periodic wave that it seems inappropriate to call it a "wave." Indeed, because of their extraordinary behavior, another name has been devised: they are now sometimes called solitons, a name used most often for solitary electromagnetic waves (see chapter 18). Before describing their weird behavior, it is worth giving an outline of their discovery.

The first person known to have seen and described a solitary wave was the Scottish naval architect John Scott Russell.20 He described what he saw to a meeting of the British Association for the Advancement of Science in 1844. He had been watching a horse-drawn boat being towed along a narrow canal when "the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed." Russell goes on to say that he followed the wave on horseback as it traveled at eight or nine miles an hour. In form it was "some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel."

This first reported example of a soliton was in a channel. Solitary waves have more recently been discovered in the open ocean, specifically in the Andaman Sea near Thailand.21 They also occur, invisibly, in the atmosphere.22 Their salient characteristic is that, unlike periodic waves, they do not disperse. Rather than changing into a dwindling series of smaller waves as each member of a group of periodic waves does (see fig. 7.5), a solitary wave, once started, becomes more and more distinct—steeper and higher. It proceeds in solitary state at undiminished speed.

The energy in solitary waves is considerable. The solitary waves in the Andaman Sea (which were solitary internal waves) were strong enough to push an oil rig nearly 30 m and spin it through a right angle; solitary waves in the atmosphere have been found to cause a rise in air pressure, gusty winds, and rain. These are occasions when solitary waves are seen to dissipate energy by doing work, in the scientific sense. Their energy is also dissipated, slowly but inevitably, by viscous shearing and conversion to entropy. If it were not for viscous shearing, they would remain unchanged forever, because, as already noted, they do not disperse.

This amounts to saying that the laws of physics describing ordinary periodic waves do not apply to solitary waves. The two kinds of waves differ fundamentally: solitary waves have their own physical laws, and unraveling them is now a fast-growing branch of science.23

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