Water Waves

Type of physical science: Classical physics

Field of study: Fluids

Waves are undulating forms that move along the surface of a body of water or at the interface between layers of different density within a body of water. While any kind of disturbance in water can generate waves, there are three principal causes: wind, gravitational pull of the Moon and the sun, and earthquakes.

Overview

Waves are disturbances of the surface of a fluid that typically occur on the surface between the atmosphere and a water body but may also occur on the surface that separates two water masses. Wave motion is periodic in that it is repetitive through fixed periods of time. At some stationary position, such as a pile off the seashore, a succession of wave crests (or troughs) pass at certain intervals of time, known as the wave period. The horizontal distance between successive crests is the wavelength. By dividing the wavelength by its period, the speed or velocity of wave propagation can be obtained easily. This relationship holds for all periodic waves. The other parameter used to describe simple periodic wave motion is height, the vertical distance from trough to crest. Wave height is independent of the other wave parameters, which accounts for its change as waves approach the shore.

If one carefully watches a cork floating on the water surface as waves pass, it will be observed that the cork rises and falls and at the same time appears to move back and forth, describing a circular motion whose diameter is the wave height and whose time to complete one cycle is the wave period. The cork makes no net advance in the direction of wave motion. Thus, waves may transfer energy and momentum across the water surface, often for thousands of kilometers, with negligible net drift of water itself. Water particles rotate in orbits, but they do not move along with the wave form.

Wind-generated waves are the most common type of waves on the oceans and large lakes. They are also the most variable. Because winds are, by nature, turbulent and gusty, there are local variations in the velocity and in air pressure on water surfaces. As a result, wavelets of all sizes are created simultaneously. The ultimate size and variety of the waves raised by the wind depend on three factors: the velocity of the wind, the open water distance over which the wind blows (known as fetch), and the duration of time the wind blows.

In deep water, as waves continue to grow, the surface confronting the wind becomes higher and steeper and the process of wave building becomes more efficient up to a certain point.

There is a limit on how steep a wave can be. That limit is reached when the height of a wave is about one-seventh of the wavelength. Thus, a wave that is 7 meters between crests can be no more than 1 meter high. When small, steep waves exceed this limit, they break, forming whitecaps. When the surface of a lake or the sea is covered with such waves, it is said to be choppy. When the wind blows the top off a wave, causing it to break, most of the energy goes into longer, more stable waves. The result is a predominance of longer waves that can accept more energy and rise higher than shorter waves generated by the same wind. At the same time, new ripples and small waves are continually being formed on the slopes of the larger waves.

Therefore, where the wind is moving faster than the waves, there is a wide spectrum of wavelengths.

As waves move away from the winds that generated them, their character changes. The original wind waves decay, in that the crests become lower, more rounded, and more symmetrical. They move in groups of similar period and height. Their form approaches that of a true sine curve; such waves are called swells. In this form, they can travel thousands of kilometers across deep water with little loss of energy.

Shallow-water waves, those that are traveling in water whose depth is less than one half the wavelength, interact with the bottom. As a wave moves into progressively shallower water, the surface orbits of the water particles begin to move faster than the wave propagation velocity, which has been slowed by bottom friction. Eventually, the wave steepens beyond the limit of stability and it breaks. In shallow water, waves can be diffracted, reflected, and refracted.

Diffraction is the bending or spreading of waves around objects such that energy is transmitted behind a barrier. As waves pass a barrier--such as a protruding headlands along a rocky coast or an isolated island in deep water--some of the energy is transmitted sideways along the wave crest and extends into areas that might have been otherwise sheltered by the barrier. Waves that meet vertical walls, such as seawalls or jetties, are reflected back to sea with little energy loss. Waves that have their crests parallel to the reflecting surface may form standing waves, which move up and down but do not progress horizontally. Refraction of waves is the bending of wave fronts caused by the effects of shallow water. When one part of a wave reaches shallower water before another part, the part impeded by the shallow water is slowed relative to the other. When waves approach the coast at an angle, which is the usual case, longshore currents are generated to transport excess water away from the direction of the wave approach.

In the open ocean, the length of wind waves is normally several tens of meters, but storms regularly generate waves more than 100 meters in length. Swell waves tend to be longer, with lengths of 300 meters being common; in the Pacific Ocean, swell waves can exceed a kilometer. Near the shore, wavelengths are typically a few tens of meters. The periods of wind waves range from less than 0.1 second in capillary waves (the tiny ripples first formed by the wind) to the largest wind wave, called gravity waves, which have periods mostly in the 3- to 15-second range but can have periods up to five minutes. Wave heights can vary from minute capillary waves of less than a centimeter to giant storm waves of 35 meters.

Tides are long-period ocean waves, having a period of twelve hours and twenty-five minutes and a wavelength of half the circumference of the earth. The crest and trough of these waves are known as high tide and low tide. The wave height is called the range of tide, but because it is measured at the coasts, where it is influenced by the shape of the shore, it varies from place to place. The gravitational attraction of the Moon and the sun on the sea surface causes tides.

The Moon revolves around Earth once a month, during which there are various spatial relationships between the sun, Moon, and Earth that cause variations in the tides. During full moon and new moon, the sun, Earth, and Moon are aligned, thus producing the maximum distortion of the sea surface and therefore maximum tides. These are known as spring tides.

First-quarter and third-quarter moons have the sun and Moon at right angles to the earth, thus working in opposition to each other producing minimal tidal ranges called neap tides.

Because of the difference in distance between the Moon on one side of Earth as compared to the other, there are unbalanced forces acting on the oceans. The gravitational attraction of the Moon exceeds the centripetal or rotational force on the side closest to the Moon, but the centripetal force exceeds the gravitational force on the far side of Earth. These forces produce the bulges on each side of Earth that are the oceanic tides. A lunar day is 24 hours and 50 minutes in duration, which means that the Moon passes a given point on Earth once in that period of time. Because of the two bulges in the ocean surface, two high tides and two low tides would be expected each day. Landmasses complicate the progression of tidal waves as Earth rotates, causing considerable deviation from this idealized plan at various positions along the world's coasts.

If the surface of an enclosed body of water such as a lake or bay is disturbed, long waves may be set up, which will rhythmically slosh back and forth. These waves, called wind tides, or seiches, have a period that depends on the size and depth of the basin. Tilting of a lake's surface, by wind or atmospheric pressure differences, is analogous to the up and down movements of a teeter-totter while the center remains stable. For example, on Lake Erie, northeast storms have produced a 5-meter difference in the water level at the eastern and western ends of the lake. This forced movement of the lake surface is known as wind tide, and the amount of rise produced is the wind setup. The resulting free oscillation of the lake surface, once the winds have abated, is the seiche. The major seiches on Lake Erie are parallel to the long axis of the lake and have a period of about 12 hours.

Disturbances along the boundary between two layers of different density (for example, cold and warm water masses) are known as internal waves. These waves often have an amplitude much greater than surface waves. Internal wave heights near 100 meters and wavelengths of several hundred meters, with periods of several minutes, are common in the sea. They can be generated by the same phenomena that cause tsunamis: earthquakes, submarine volcanic eruptions, and submarine slides. In the Great Lakes, they can be set in motion by storm surges or rapid barometric pressure changes that pile up water at one end of the lake and lower the level at the other. By way of example, a beaker containing half oil and half water, by virtue of their contrasting densities, will not mix well and will result in two distinct layers of liquid with a sharp boundary. A slight disturbance to the container will cause irregularities to the surface that are analogous to the formation of internal waves at sea.

Earthquakes, submarine slides, and other disturbances to the seafloor may generate large waves such as tsunamis or seismic sea waves. These waves have periods of several minutes and lengths of hundreds of kilometers. In the open ocean, they have wave heights of less than a meter. As they approach the shore, however, their long length causes the waves to steepen as the orbits interact with the seafloor and wave heights can rise to tens of meters with devastating effects as they strike the coast.

Applications

When waves eventually encounter a shore, all of their accumulated energy is released in a matter of minutes as the wave is changed into surf. If the surf is composed of swell that has traveled from distant storms, breakers will develop relatively near shore and the surf will be characterized by parallel lines of relatively uniform breakers. Yet, if the surf is composed of waves that have been generated by local storms, the waves may not have been sorted out into uniform groups, and the surf will be characterized by unstable, steep, high-energy waves. These waves will break shortly after "feeling" the bottom some distance from shore, and the surf will be rough and choppy with an irregular nature.

The breaking experienced in the surf results when the particle motion near the bottom of the wave has been interfered with by contact with the seafloor, and this tends to slow the waveform. Nevertheless, the particles that are orbiting near the ocean surface have not been slowed down as much. Thus, the top of the waveform begins to lean toward the shore, and the wave height increases. This motion can be seen particularly well in plunging breakers, which have a spectacular curling crest because the top has outrun the rest of the wave and there is nothing beneath to support their motion. Plunging breakers generally form on moderately steep beach slopes like those of Waimea Bay on the island of Oahu, Hawaii. The more commonly observed breaker is the spilling breaker that results from a relatively gentle slope of the ocean bottom, which more gradually extracts the energy from the wave, producing a turbulent mass rather than a curl. Spilling breakers are the type formed on the gentle shore of Waikiki, also on the island of Oahu.

Surfboards, small craft, and mammals, including porpoises and bodysurfers, can take energy out of the waves to propel themselves by sliding down the forward surface of an advancing wave. A surfboard is thrust forward by a downhill or slope force, which is balanced between gravity and buoyancy forces. The slope force is greater than the hydrodynamic drag, or water resistance, and the surfboard moves at wave-crest speed. The trick of surfing is to get the board moving and the weight properly balanced so that the slope force can take over the work of propulsion at the moment the wave passes beneath. Porpoises are neutrally buoyant and have learned to tilt themselves at the proper angle to take advantage of the slope force of an underwater constant-pressure surface. These mammals can ride beneath the bow wave of a ship indefinitely without appearing to exert any effort at all.

Any disturbance of the water surface, including the passage of ships, creates waves, which are sometimes called stern waves and bow waves. Much of the power expended in propelling a ship is converted to waves, and anything that can be done to reduce this power results in more efficiency. A ship moving through the water is accompanied by at least three pressure disturbances on each side, which produce several trains of waves. In order to improve ship designs and create hulls with a minimum loss of energy, model ships are tested by towing them through long tanks. Fortunately, the waves made by ship models are accurate predictions of those that will be created by the full-size ship they represent.

Extracting power from waves as they approach the shore is a possibility if significant problems can be overcome. Using the phenomena of wave diffraction and refraction, a large half-cone could be constructed to focus the energy of waves to a turbine at the apex of the structure. Such a device might extract up to 10 megawatts per kilometer of shore, but could produce significant power only when large storm waves existed at the structure. Thus, this system would operate best as a power supplement. Tides have proved to be a more effective use of wave energy. The world's first major tidal power system is located on the Rance River in France. Tides in this region have an amplitude of 13.5 meters. Yet, using tides for energy has some inherent difficulties, especially the fact that tides do not flow continuously but change directions several times a day and that they vary in strength from day to day on a two-week cycle. It has been possible to alleviate the problem to some degree by storing water in a reservoir at high tides, as is done at the Rance power plant, and letting it out slowly to turn turbines on a more continuous basis. Energy production at the Rance plant is 240 megawatts.

Context

Waves have attracted attention for untold centuries. Perhaps the first people to formulate ideas about wave mechanics were the ancient sailors of the Mediterranean Sea. They observed that a float tossed into the sea would rise and fall, move back and forth, as waves passed under it, but it did not move shoreward with the waves. With these simple observations, oscillatory waves were discovered. In the eighteenth and early nineteenth centuries, Daniel Bernoulli, often called the "father of modern fluid mechanics," and Franz Gerstner began the mathematical study of waves and produced the first wave theories. Later, Lord Kelvin, Sir George B. Airy, George G. Stokes, and William John Macquorn Rankine added to the theoretical understanding of waves. It is only within the later twentieth century, however, that the application of wave theory came to extensive, practical fruition.

The modern description of how the wind transfers its energy to the waves derives from the research of Harald Ulrik Sverdrup and Walter Munk of the Scripps Institution of Oceanography in La Jolla, California. During World War II, they worked on the problem of predicting the waves and surf that would exist on an enemy-held beach during amphibious landing operations. They were able to give the first reasonably quantitative description of how waves are generated, become swells, and move across the ocean to a distant shore.

In the past several decades, further studies of wave characteristics in the open sea and improved forecasting techniques have found numerous applications to the selection of shipping routes, as well as in the design of ships and coastal structures, such as piers, breakwaters, and jetties. Even today, however, it is recognized that substantial gaps exist between mathematical wave theories and actual waves observed at sea or in the laboratory, particularly under extreme conditions. The devastations that can be caused by such conditions illustrate the need for continuing research on water waves.

Storm waves are capable of great damage when they reach the shore. Concrete blocks weighing 65 tons have been torn loose from breakwaters and jetties by such waves. One of the most destructive waves is the tsunami. Most frequent in the Pacific Ocean, this type of wave is almost imperceptible in the open sea but can engulf boats, buildings, and people when they reach shore at heights of 30 meters or more. An added danger is that these seismic sea waves have a series of crests, which arrive about 20 minutes apart, with the third or fourth crest to arrive being the largest. Storm surges, usually accompanying hurricanes, are another dangerous wave type.

They periodically occur along the Atlantic and Gulf coasts of the United States and have been known to carry oceangoing vessels a kilometer or more inland. Six thousand people lost their lives in one such storm surge in Galveston, Texas, in 1900. Lakes are not immune from such disasters. In 1928, a hurricane-created seiche overflowed the dike surrounding Florida's Lake Okeechobee and drowned two thousand people.

Principal terms

DEEP-WATER WAVE: a wave traveling in water with a depth greater than one-half the wavelength so that it is little affected by the bottom

ORBIT: the path followed by a water particle affected by wave motion; deep-water orbits are nearly circular, and in shallow water they are flattened

REFRACTION: the bending process when a wave approaches the shore at an angle so that the part advancing in shallower water moves more slowly than that part in deeper water

SEICHE: a standing wave in an enclosed water body that does not progress forward but continues as a pendulum-like oscillation after the originating force has ceased

SHALLOW-WATER WAVE: a wave that is noticeably affected by bottom topography

SWELL: wind-generated waves that have traveled out of their generating area

TIDAL WAVE: the wave motion of the tides; periodic rising and falling of the sea that results from gravitational attraction of the Moon and the sun

TSUNAMI: a long-period sea wave produced by a submarine earthquake or volcanic eruption

WAVE HEIGHT: the vertical distance between a crest and the preceding trough

WAVE PERIOD: the time required for two adjacent wave crests or troughs to pass a stationary point

WAVELENGTH: the horizontal distance between two successive wave crests or troughs

Bibliography

Bascom, Willard. WAVES AND BEACHES: THE DYNAMICS OF THE OCEAN SURFACE. 2d ed. Garden City, N.Y.: Doubleday, 1980. This is a thorough, but nontechnical treatment of the nature of waves and their interaction with the shore. An excellent source of information for nonscientists and a good review for professional oceanographers. Contains many helpful diagrams and tables, as well as numerous black-and-white photographs. Suitable for general audiences.

Davis, Richard A., Jr. OCEANOGRAPHY: AN INTRODUCTION TO THE MARINE ENVIRONMENT. Dubuque, Iowa: Wm. C. Brown, 1987. Chapter 6 provides a comprehensive overview of ocean waves, and chapter 7 describes tides. Contains many excellent diagrams and photographs of all aspects of deep-sea and shallow-water waves. A sidebar on measuring waves describes modern techniques for obtaining wave data. Suitable for college-level readers.

Fairbridge, Rhodes W., ed. THE ENCYCLOPEDIA OF OCEANOGRAPHY. New York: Reinhold, 1966. This is an indispensable oceanography source book for the student and professional. Contains excellent sections on waves and the related topics of fetch, tsunamis, wave energy, wave refraction, and wave theory. A well-illustrated and carefully cross-referenced volume, which is aimed at readers with some technical background.

Gross, Grant M. OCEANOGRAPHY: A VIEW OF THE EARTH. 4th ed. Englewood Cliffs, N.J.: Prentice-Hall, 1987. A well-written and well-illustrated text that has been kept current with frequent revisions. Chapter 9 provides a comprehensive overview of all aspects of ocean waves, with good diagrams, tables, and photographs. The appendix includes a table of conversion factors and an excellent glossary. Suitable for general audiences.

Parker, Henry S. EXPLORING THE OCEANS: AN INTRODUCTION FOR THE TRAVELER AND AMATEUR NATURALIST. Englewood Cliffs, N.J.: Prentice-Hall, 1985. This book is designed to impart basic ocean science information in a clear and readable fashion without sacrificing the technical substance required for the serious student. Chapter 8 covers waves and tides with an exploratory approach. The text is well-supplemented, with good diagrams and photographs.

Trefil, James S. A SCIENTIST AT THE SEASHORE. New York: Collier Books, 1984. Trefil invites the reader to walk along a favorite beach and deepen the appreciation of the experience by seeing it through his eyes. Chapters 6 through 9 contain fascinating accounts of waves and related features. The book is well illustrated with diagrams and photographs. Readers with some technical background will enjoy it most.

Tricker, R. A. R. BORES, BREAKERS, WAVES, AND WAKES: AN INTRODUCTION TO THE STUDY OF WAVES ON WATER. New York: Elsevier, 1964. This is a comprehensive treatment of water-wave behavior, written with the aim of simplifying the mathematics so as to make the discussion understandable to the general reader. Chapter 17, "Ships' Wakes," provides an excellent coverage of the topic. There are many black-and-white and color plates illustrating various wave types. Suitable for college-level readers.

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