Wind-generated ocean waves are generally the most significant physical force involved in the erosion of coasts and experienced by wave-swept plants and animals that live along the shore. The heights of the waves generated by storms over the ocean commonly reach io to 15 meters, with the highest reliably measured wave having achieved 34 meters (measured in 1933 in the South Pacific to the east of the Philippines). The energy carried by the waves as they cross the expanse of the ocean is eventually delivered to the coast, where they break on beaches or crash against cliffs. There the waves expend their energy as they wash across the intertidal plants and animals and, over the long term, etch out the natural weaknesses in the rocks to locally form tidepools.
Waves are generated on the ocean whenever the wind blows across the water's surface, the process involving the transfer of energy from the wind to the waves. The greater the speed of the wind, the greater the heights and energies of the waves that can be formed. The highest waves are generated by major storms, since they generally have the strongest winds and can persist for several days during which energy is transferred to the waves. Therefore the heights of the waves depend on the duration of the storm as well as on the speed of its winds, and also on the storm's fetch: the area or length over which the energy is transferred from the winds to the waves. Techniques have been developed by researchers to predict the heights of waves generated by storms, depending on the wind speeds and on the storm's fetch and duration.
The characteristics of relatively simple ocean waves are depicted in Fig. 1, with the waves consisting of a series of crests and troughs. The wave height, H, is the vertical distance from the trough to its crest, while the wavelength, L, is the horizontal distance between successive crests. The motion of waves is periodic, repeating through a relatively fixed interval of time: the wave period, T. From the geometry of the waves it is apparent that they will move a horizontal distance L during the period T, so their speed, or wave celerity, is calculated as
As the waves pass it can be observed that a floating object (such as a cork) rises and falls as the crests and troughs alternately affect its movement, with some to-and-fro horizontal movement as well; the paths of the floating objects are thereby circular to elliptical. Thispattern of movement within the wave extends downward with depth beneath the water's surface (see Fig. 1), but with the diameters of the circular movements progressively decreasing and flattening, so that at the seafloor the water movement is a simple horizontal orbital motion. The water itself makes no net advance in the direction of wave movement, unless there is also a net ocean current in that direction. The waves thereby transfer energy across the ocean's surface, in the form of the wave itself and its internal orbital water movements, but with a negligible net drift of the water. Having derived their energy from the storm's winds, the waves are capable of carrying that energy for hundreds to thousands of kilometers across the expanse of the sea, ultimately delivering it to the coasts that form the ocean's boundaries.
The energy of the waves, E, is directly related to their heights:
where p is the density of the water and g is the acceleration of gravity. It is seen from this relationship that if the wave height is doubled, the energy is increased by a factor of 4. Important is the rate of transfer of this energy across the ocean to the coasts, which is calculated by multiplying E by the wave celerity ? to obtain the wave power,
The factor n has been included because waves travel across the ocean in groups, those formed by a particular storm. It turns out that although individual waves travel at their celerity, C, in deep water the group as a whole travels at half that speed (that is, n = 1/2). The observation is that an individual wave progressively moves to the front of the group as a whole, where it then disappears, while at the same time a new wave is formed at the back edge of the moving group. Fundamentally this occurs because the individual waves are traveling faster than their energy. This is the condition only in the deep water of the ocean. As the waves reach the shallower depths of the ocean basin and approach the coast, the factor n progressively increases until, in shallow water close to the coast, it achieves its maximum value of n = i, at which point the energy of the waves is advancing at the celerity C of the individual waves.
In deep water the wave celerity depends on its period T according to the relationship
That is, the longer the period, the faster the wave's speed and the more rapidly the waves cross the ocean basin. In shallow water close to the coast C depends instead on the local water depth, causing the waves to slow down as they approach the shore. In intermediate water depths, C depends on both the wave period and water depth. Researchers have developed analyses that employ the mathematical relationships in Eqs. 1-4 for the wave energy and power and for the celerity C and factor n, which make it possible to track the paths and rates of movement of the waves from their area of generation by a storm, across the ocean width, and ultimately to the coast where they deliver their energy.
The foregoing consideration of regular waves is primarily applicable after they have left their area of generation by the storm. In the storm area itselfthe waves are highly irregular and are termed sea, while the regular waves are referred to as swell. The irregularity of the waves within the fetch of the storm is due to the fact that the winds generate waves having ranges in both periods and heights. When the wind blows over initially still water, it forms small ripples, but with time those ripples grow to waves of progressively longer periods and greater heights, while at the same time new ripples are formed, hence the resulting ranges of periods and heights. With this mix of waves in the storm area, it is difficult to follow the movement of an individual wave for any length of time, as the observed highest crests generally occur as the chance summation of several waves that individually have different heights and periods; in this case the crest exists for only a few seconds during which these several waves have combined. In the extreme this may produce what is termed a rogue wave, one of such exceptional height that it can be a danger to ships; fortunately, this degree of chance summation is rare. As this mixture of waves leaves the storm area, they sort themselves out by period, a process that is termed wave dispersion, since the longer-period waves travel away from the storm faster than the shorter-period waves. This dispersion eventually yields the more regular swell, but even then there will be waves having a narrow range of periods, not yielding waves of perfectly uniform periods and heights.
As a result of these processes, whenever waves are measured in the ocean, one finds ranges of wave heights and periods, distributions that need to be analyzed statistically. One can, for example, define an average wave height from those that have been measured, but it is more difficult to establish the maximum wave height, because this tends to depend on how long the measurements are made (which typically is on the order of 20 minutes to make themeasurements representative). A commonly used measure of waves is the significant wave height, the average of the highest third of the measured waves—"significant" in the sense of having ignored the smallest waves in calculating the average, focusing instead only on those that have the greatest energies. Many measurements of waves under a variety of conditions have shown that the maximum height of the waves is on the order of i.5 to 2.o times greater than the significant wave height. Therefore, when you hear predictions of wave conditions, for example by the Coast Guard, it generally is the significant wave height that is being reported. You therefore need to realize that if you go to the shore, you can expect to actually experience waves that are potentially twice as high as that prediction.
The wave climate for a specific location in the sea refers to its ranges of wave conditions experienced over the years, including the wave heights, periods, and directions of travel. Waves are most commonly measured with buoys that are tethered to the sea floor and contain instruments that continuously record the pitch and heave of the buoy, which can then be analyzed to decipher the waves that produced those motions. The recorded data can then be transmitted to a satellite, which relays it to a laboratory, where the wave data is further analyzed and compiled to form the wave climate. Lacking such direct measurements, the waves at the site can be calculated from the storm systems present on that body of water, using the analysis techniques described in the preceding section, which depend on the wind speeds of the storms and on their fetches and durations. This requires a number of years of wave measurements or calculations to adequately define the wave climate for a particular site, primarily because of the importance of establishing the occurrence of the extreme but rare events, the highest wave conditions, because they are most relevant to ship hazards and to the erosion of coasts.
Figure 2 is an example of the distribution of significant wave heights measured by a buoy in the North Pacific located off the coast of Oregon, representing a partial documentation of the wave climate for that coast. This graph is the product of 30 years of hourly wave measurements by this buoy. It is seen in the bar-histogram for the number of observations that the dominant significant wave height is on the order of 2 meters, with fewer occurrences of large waves because their generation depends on the comparatively rare major storms; those more extreme storm waves are more easily identified where the numbers of observations are plotted using a logarithmic scale, also included in Fig. 2. It is the wave conditions that occur during these major events that are of greatest interest in the wave climate, and it is seen in this example that they commonly exceed io meters, with the highest measured significant wave height having been 15 meters, which occurred during a storm in March 1999 that resulted in considerable erosion along the Oregon coast. Statistical analyses are applied to the extreme-wave conditions in order to project what might be expected to be the most severe storm event and wave heights: the so-called ioo-year storm, which actually is the event that has a i% probability of occurring during any particular year. In this example for the Oregon coast, the ioo-year projected significant wave height is 16 meters. Recalling that the highest waves generated by a storm are i.5 to 2.o times as high as these significant wave heights, the most extreme storms off the Oregon coast can be expected to generate individual waves as high as 25 to 30 meters, roughly the height of a io-story building.
In that the highest waves are generated by extreme storms, their occurrence depends on the earth's climate, its seasonality during a year, and any long-term changes. Here it is also necessary to distinguish between tropical and extratropical storms. Tropical storms (the most severe of which are known as hurricanes and typhoons) develop close to the equator during the summer to fall in the hemispheres they affect, since their energy is derived from the warm surface water in the tropics. Extratropical storms form at higher latitudes by the interactions of cold and warm air masses and are associated with the paths of the jet streams. There have been long-term changes, spanning decades, in the frequencies and magnitudes of tropical storms, but the connection of such changes with the earth's changing climate is not fully understood. It has been suggested that the recent increase in hurricanes in the Atlantic Ocean, associated with warmer water temperatures (particularly in the Gulf of Mexico), is associated with human-induced global warming resulting from our emissions of greenhouse gases. Hurricane Katrina in the summer of 2005, a category 5 hurricane when it was offshore, generated significant wave heights of 19 meters, with its maximum wave heights likely having been on the order of 30 to 35 meters. The strongest extratropical storms occur during the winter, so the highest wave conditions seen in Fig. 2 for the Oregon coast begin in the late fall and end in the early spring. It also has been found that higher waves are formed along the U.S. West Coast by storms during El Niño winters, a change that is most dramatic along the California coast, because the El Niño storms follow tracks that take them further to the south than during normal years. Furthermore, the storm intensities and wave heights in both the North Pacific and North Atlantic have increased markedly during the past 25 to 50 years (Carter and Draper 1988; Allan and Komar 2006); researchers remain uncertain whether this has resulted from a natural cycle in the earth's climate or is associated with global warming.
Wave climates generally are developed for deep-water locations, that is, before the wave heights and directions of travel are modified by crossing the shallower water of continental shelves. In wave analyses the term deep water has a technical definition: the water depth is greater than half the wave length (L). Because L is shorter for lowerperiod waves,
it follows that "deep water" extends to shallower depths closer to shore for the lower-period waves (for example, waves of T = 5 seconds are still in deep water at a depth of 20 meters, while waves of 15 seconds period are in deep water at a depth of 175 meters). It follows that the modifications of these waves begin shoreward of those water depths, seen in their changing heights and directions, but not in their periods, which remain unchanged during shoaling.
Part of the change in the waves results directly from the progressively decreasing water depths, which slows their rate of advance and causes the heights of the waves to generally increase. This transformation results because the power of the waves remains relatively constant during the shoaling process; that is,
so that as C decreases as the water depths decrease, the wave energy E must increase, resulting in an increase in the wave heights H. Actually, the transformation is more complex in that while C decreases with water depth, n increases from 12 in deep water to 1 in shallow water, such that the rate of energy advance (Cn) initially increases slightly, resulting in a small decrease in E and H, but this is followed by a more dramatic decrease in Cn so that E and H undergo significant increases; it is this latter increase that is most easily seen as the waves approach the coast and eventually break on the beaches and cliffs. This transformation of the waves in shallow water is further complicated by the frictional drag of the waves on the sea floor, so that some of their power is lost rather than being constant as previously assumed, and especially by the changes in the wave directions as they approach the coast, which can either concentrate or spread out (defocus) the energy of the waves.
This latter process is termed wave refraction, seen in the photograph in Fig. 3, where the crests of the waves become more closely parallel to the shoreline as they progressively reach shallower water depths (the increase in the wave heights is also apparent in this photograph). This change in crest orientation and direction of wave advance also results from the dependence of the celerity C on the water depths in shallow water. As illustrated in the diagram in Fig. 3, this results in a rotation of the wave crests with respect to the depth contours and shoreline. The portion of the wave crest at B is in deeper water than at A, and accordingly moves faster and therefore reaches B', which represents a greater shift in position than the movement from A to A', where the wave celerity is slower. This trend continues such that the wave crest progressively rotates and becomes more nearly parallel with the shore, just as seen in the photograph.
The refraction of water waves in the ocean is analogous to the bending of light rays when they pass through glass, as occurs when a convex lens focuses the energy of the light to a degree that it can cause paper to burst into flame. In contrast, a concave lens spreads the energy of the light passing through it. The similar patterns for the refraction of ocean waves are depicted in Fig. 4 for the concentration of wave energy along a rocky headland on the coast or its spreading due to refraction over the deeper water of a submarine canyon. In this diagram, rather than considering the bending of the wave crests due to their refraction, the analysis is presented as the changing directions of the wave energy, the wave rays that are perpendicular to the wave crests (the directions AA' and BB' shown in Fig. 3). In the case of the waves approaching a headland, the rays are bent by refraction so that they become concentrated along the headland, the shallow water depths offshore from the headland having much the same effect as the convex lens in focusing the energy of light rays. Ignoring wave energy losses due to friction, the wave power between adjacent wave rays in Fig. 4 is constant, so it becomes more concentrated as the rays converge on the headland, locally 1 ncreasing the energy and wave heights. This would tend to increase the rates of erosion of the headlands, compared with the shores of the adjacent bays, except for the fact that the headlands likely exist because of the greater resistance of their rocks compared with those backing the shores to the side of the headland. It is apparent that the refraction of the waves must exert a strong control on the wave intensities and hydrodynamic forces experienced by plants and animals living in tidepools along the shores of rocky headlands.
Any irregularities in the bottom topography and water depths on the continental shelf can affect the refraction of the waves, resulting in significant variations in wave energies and heights along the shore. This is seen in Fig. 5 for the shoreline at La Jolla (San Diego), California, where both the wave crests and wave rays are included in the refraction diagram, and the water depths (in fathoms) are shown by the dashed contours. The pronounced spreading of the rays where the waves cross the deeper water of the two submarine canyons is readily apparent, resulting in reduced wave energies and heights along their lee shores, while there is a degree of concentration or focusing of the wave energy between the submarine canyons, and also along the La Jolla headland. These longshore variations in the wave conditions are well known to surfers who frequent this beach.
The heights and energies of the waves that reach a coast depend on the severity of the storms that generated those waves. This determines the coast's wave climate, which can vary from site to site depending on the location relative to the occurrence of the most severe tropical or extratropical storms and on the subsequent paths of those generated waves as they cross the ocean. The waves in deep water can be considerably modified as they cross the shallower water depths of the continental shelf, generally increasing in their heights compared with deep water, but altered in complex ways by wave refraction that can either concentrate (focus) the wave energy or defocus it. All of these processes ultimately determine the energy and power of the waves once they reach the shore and their capacity to erode rocky cliffs to form tidepools and to disturb the plants and animals living there.
Climate Change / Hydrodynamic Forces / Rogue Waves / Storm Intensity and Wave Height / Tidepools, Formation and Rock Types / Wave Forces, Measurement of
- Climate controls on U. S. West Coast erosion processes. Journal of Coastal Research 22, 511-529. , and . 2006.
- Waves and beaches. Garden City, NY: Anchor Books. 1980.
- Has the northeast Atlantic become rougher? Nature 332: 494. , and . 1988.
- Water wave mechanics for engineers and scientists. Englewood Cliffs, NJ: Prentice-Hall. , and . 1984.
- Beach processes and sedimentation, 2nd ed. Upper Saddle River, NJ: Prentice-Hall. 1998.
- Refraction of ocean waves: a process linking underwater topography to beach erosion. Journal of Geology 55: 1-26. , and . 1947.
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