Energy on the move

Perhaps no other aspect of the sea is as fascinating or inspires as much dread as ocean waves. Waves, or sea state, the often tumultuous sum of the height, frequency and direction of waves, determines whether a voyage is possible or advisable, even more so than the strength and direction of the wind. Consider whether you would sail in 40 knots of wind if, by fiat, the seas were kept calm. Any blue-water sailor who has enviously longed to try ice-boating has answered this in the affirmative. Despite their importance, waves remain the less-understood sibling of the wind.

Describing (the first step in understanding) the wind is relatively easytwo numbers give speed and direction. A typical description for real waves in the real sea utilizes 15,000 numbers. At any time, a patch of sea is crossed by waves moving in different directions, at different speeds, and with different heights. Understanding waves and their potential effect on a vessel takes far more information than just the height of the average wave. Waves of 20 feet every 30 seconds are far less dangerous to a 32-foot sloop than six-footers every few seconds.

Fortunately, the Army Corps of Engineers and the National Oceanic and Atmospheric Administration (NOAA) have recognized this for years. These organizations currently have 70 wave-gauging stations placed around the coastline of the U.S. (including the Great Lakes) collecting data on wave height and direction in near-shore areas. Within a few years, these data will be availablein near-real timeto sailors throughout the country. Using the data requires changing the way one understands waves.

Waves in the sea

In the simplest sense, waves are the response of the air/sea interface to perturbations by an energetic process. These are not piles of water moving ever onward; they are piles of energy moving along the interface. Non-breaking waves produce little net movement of the water, as one can confirm with a wood chip thrown onto the surface. The chip (and the water on which it rests) is carried up and backward as the wave approaches, then down and forward as the wave passes by.

Sailors experience an enormous range of waves at sea, from the tides with heights of 50 feet to the smallest capillary waves rippling the surface. Regardless of size, all waves share characteristic features. Wavelength is the distance from a specific part of one wave (the crest, perhaps) to the same part of the next wave. Wave period, often the easiest feature of a wave to measure from a moving vessel, is the temporal equivalent to wavelengththe time it takes for consecutive crests to pass a stationary observer. Wave height is measured from peak to trough, while water depth is measured from some average water level to the sea floor. Wave velocity (or "celerity," from the Latin for "quickness") and direction complete the picture of ideal (as opposed to real) wave characteristics.

Armed with a handful of rules, a sailor can easily determine the likely sea state from two or three properties of waves. Clearly the height and period of waves are important, but alone they fail to adequately predict the sea state. One needs to gauge the steepness of the sea, and the amount of energy contained within it. These allow the sailor to predict how his or her crew and boat will fare on the water. A fundamental relationship is that wavelength equals the product of wave period and velocity. Even as a wave approaches the shore and begins to break upon the beach this relationship holds. The energy within a wave is proportional to the square of the wave’s height. This has profound implications for choosing acceptable sailing conditions: four-foot seas are 16 times as energetic as one-foot seas, all else being equal.

But things are rarely equal. A long, slow, four-foot sea can be quite pleasant, even in an open boat. What matters is how close togetherhow steepthose waves are. A good parameter for steepness is the wave height divided by the square of the period, and approximates the acceleration one experiences riding up the face of the wave. Halving the period of a sea quadruples the acceleration, and more than likely quadruples the mal de mer on board the boat. Severity indicates the amount of energy carried by each bit of wave and is proportional to the energy of a wave divided by its wavelength.

These parameters allow one to judge the quality and character of a sea by the parameters most likely to be available on future wave data sources. The key to using wave data is to record ship and crew performance relative to wave height, period, steepness, and so on, and then use real-time wave data to make navigation decisions. Experienced sailors do this with winds and weather all the time.

Deep, intermediate, and shallow

Within the Earth’s oceans, a typical wind-generated wave is less than 10 feet high and has a wavelength of 200 to 500 feet traveling at 50 knots. Despite appearances on the surface, waves have little effect on the water column below a depth of approximately half a wavelength (not height). A diver only 100 feet below the surface would never feel the typical ocean wave.

This allows division of ocean waves into three domains. In deep water, waves travel through water greater than half a wavelength deep. In shallow water, waves travel through water shallower than one twentieth a wavelength. Between the two lies the domain of intermediate waves. At the two extremes, very simple relationships exist between wave parameters, simple enough to be useful to the mariner, and easily summarized on a graph (see figure 1). As might be expected, the water depth controls the behavior of shallow waves; wave period plays the same role in deep water.

From a typical wave’s point of view, the Earth’s oceans are pretty shallow. Tides, with a wavelength of 12,000 miles, are certainly shallow-water waves, but so too are tsunamis and most mature seas with periods above 60 seconds. At depths typical of the shallow shelves surrounding the continents (less than 50 fathoms), all but the smallest wind waves are intermediate waves. Only off the continental shelves, with water depth exceeding 50 fathoms, do the typical waves created in a storm reach deep water. In either case, a watch and a chart allow the mariner to determine all of the important wave characteristics, from velocity to steepness.

Deep. Waves in deep water are characterized by their period, easily measurable from a boat. It is more accurate to time a number of waves, perhaps 10, and take the average. Remember to count all the waves, even the smaller ones, when doing this. (One needs to consider the boat’s speed through the water as well, but for most deep waves, and most sailboats, this is a small correction, and will be ignored here.) With the period in hand, one can find the wave speed and length quickly and with reasonable accuracy.

Wave velocity (knots) = 3T (sec)

Wavelength (feet) = 3T2 (sec)

For example (see point "D" on figure 1), while sailing in four foot seas with 500 fathoms of water, one might measure 120 seconds during the passage of 10 waves. These waves have an average period of 12 seconds, a speed of 36 knots, and a wavelength of 430 feet. (All values are rounded to whole numbers.) The graph gives the same results. One could also determine the steepness and severity (0.03 and 0.04 respectively), indicating that the seas are comfortable. One can do the same analysis with shorter-period wavessay seven secondsand find much less comfortable conditions.

Shallow. In shallow water, wave behavior is determined primarily by water depth, and again a chart and a watch are the most helpful tools to have handy. Wave speed is a function of the square root of the depth:

Wave velocity (knots) = 3.3 x square root of (depth in feet)

while wavelength is just the speed multiplied by the period:

Wavelength (feet) = V (knots) x T (sec)

The pesky square root makes the math more difficult, and often it is easier to simply use a diagram like figure 1 to find the parameters. For example (see point "S" on figure 1), sailing in 30 feet of water, one might time the passage of six waves in 118 seconds. These are shallow waves with an average period of about 20 seconds. The waves are traveling at about 18 knots, with a wavelength of 350 feet. At the slow speeds of shallow waves, a fast-moving boat can easily match, or even overtake, a wave. Remember this when timing the period of a wave: one must account for the boat’s speed! A chip log, immobile in the water, provides a useful benchmark in this situation.

Non-wind waves

The longest waves in the oceanswith a wavelength of half the Earth’s circumferenceare the tides, which sweep across the Earth with periods of approximately 12 and 24 hours. While rarely threatening to a vessel, they are the most powerful waves in the oceans, containing well over 80% of the energy flowing as waves. Over the past billion years, the day has lengthened from 19 to 24 hours due to the friction of the tides on the ocean floor. At shorter periods lie seismic sea waves, or tsunamis. Traveling at speeds of 400 knots, tsunamis form in response to sudden changes in the shape and location of the sea floor, often due to landslides or earthquakes. Although devastating to affected areas, they are a small portion of the wave energy in the sea.

Between tsunamis and the true wind waves lies a netherworld of long-period (one to 10 minute) but little-studied infra-gravity waves. Sailors and surfers are all familiar with wave grouping, the habit of taller waves to come in packages, each package separated by a few minutes. These are a manifestation of infra-gravity waves that modulate the sea surface over scales of a few miles. They remain somewhat mysterious: it is difficult to find forces in nature that vary on these time scales and that could drive these long-period waves.

Wind waves

True wind waves have periods of (roughly) 60 seconds or less. While they carry only a small proportion of the energy in the oceans, they carry it in a dangerous manner. Boats, like waves, have natural periods, as anyone who has successfully rocked a boat over can attest. Waves and boats have similar periods: because of this waves have the potential to change the motion, perhaps catastrophically, of boats. Thanks to the Corps of Engineers and NOAA, mariners will soon have the data necessary to estimate wave threats.

Winds obviously generate waves, as even a casual observer can see. But even an expert would have a hard time explaining what happens. "We have a lot to learn about exactly how wind energy turns into wave energy, " said David McGehee, manager of the Army Corp of Engineers’ Field Wave Gauging Program. The physics of wave generation are hard to pin down precisely, with processes from the molecular scale to the macroscopic scale all contributing equally to the production.

A consensus model for the generation of wind waves begins with a calm sea, a uniform wind above it, and some assumptions. The first is that waves are generated by forces pushing vertically down on the sea.

Sailors are familiar with the atmospheric shear zone at the sea surface. Winds are stronger above the sea surface than close to it, one of the reasons for carrying a tall rig on a boat. This shear zone is inhabited by a range of eddies, which, like tumbleweeds, spin as they blow across the sea. Larger eddies move faster and exert more pressure on the sea surface than smaller ones. In response, the sea surface is subtlety lowered beneath the larger eddies and raised below the small ones.

These hills and valleys in the sea surface quickly die out unless the speed with which they move through the water is the same as the speed of the eddies through the air. Such a resonance drives these so-called capillary waves forward, and causes them to grow higher. Still diminutive, with wavelengths of less than an inch, these capillary waves give rise to all other wind waves. One can easily create capillary waves by blowing across a pan of water.

In low winds, capillary waves form in response to eddies in the air; however, in winds greater than six knots this reverses, and the waves themselves produce the eddies. At six or seven knots of wind, waves are three inches high. Relative to these moving waves, the air within a quarter inch or so of the surface flows backward, while the air above this level flows forward. This is an eddy, produced by the wave itself, and these wave-made eddies are bigger on the downwind side of the small wave. These larger eddies push the wave forward, transferring energy from the air into the sea. Larger waves produce stronger eddies, and so grow more quickly and efficiently. This feedback mechanism operates whenever the wind is stronger than Force 3, and explains, if only partially, why seas can grow rapidly in a fresh breeze.

The sea that results from all this is much more complicated than the rather dry theories erected to explain it. The waves present at a given point arrive from many directions, at many speeds, and with different periods and steepnesses.

Defining height

An average wave height can be defined in a number of ways. Experienced mariners are often quite good at determining average height by observation. It is also helpful to have a way of determining the entire spectrum of wave heights and periods, allowing one to accurately judge the sea state. Humans lack the ability to do this easily on their own, but computers, hooked up to a wave gauge, excel at it.

A graphic record is shown in figures 2 and 3. These diagrams show wave records for 00:00 hours on August 28, 1998, from the Westhampton, N.Y. (Long Island), wave gauge maintained by the Corp of Engineers. The "wave record" line in figure 2 is simply the pressure recorded by a pressure gauge lying on the sea floor, converted into water depth. The total record is 1,024 seconds (about 17 minutes). The significant waveto the human eyehas a period of around 13 seconds, and a height of perhaps 6.5 feet. But this is an oversimplification; the wave record is rich with additional information that can be teased out with a numerical analysis.

The analysis begins by assuming that waves shown on the record are simply the sum of an infinite number of regular (sinusoidal) waves, each with a different height and period. The result of thiscalled a Fourier Transformis a model of the spectrum of waves at sea. The spectrum line is the lower line in figure 2. The wave gauge also returns the wave direction (not shown here). The result (available online, see sidebar) is the frequency (the inverse of the period), energy (as a function of frequency) and directions of the waves forming the sea at that time.

The analysis results for the August 28 data is shown in figure 3. With additional manipulation, one can derive the triumvirate of wave analysis: the peak height, frequency, and direction of the "significant wave."

The largest peak (the red dot on both figures), with a frequency of 0.08 Hz and a height of 6.8 feet, is the significant wave, and it agrees well with our estimate from the wave record (a typical result, in seas that aren’t overly confused.) But the numerical analysis shows another important wave set, with a frequency of 0.1 Hz and a height of 5.5 feet. With this insight, we can see on the wave record the interaction produced by these two similar waves, such as the double peak at about 50 seconds (on figure 2).

The significant wave returned by this analysis is taller than approximately 87% of the waves in the record. This agrees well with the sea state an experienced observer would give for the same sea. Our discussion of significant sea state has avoided an importanteven crucialaspect. About a third of the waves in a given sea will be higher than the significant height. One can use the wave spectrum (or, more precisely, a mathematical model of the spectrum) to predict the probability of extreme waves, even ones higher than any recorded on the wave record.

The extremes of wave height aren’t all that uncommon, although their probability decreases very sharply (more than exponentially) with height. One can expect a wave twice the height of the significant one every 80 waves! For the 12.8-second waves we’ve been discussing, that is one every 17 minutes. (It is a testament to the random nature of statistics that the Westhampton record does not, in fact, display such a wave.)

A caution here: "There is a real need for standardization in the wave community," the Corps’ McGehee warned, "even for the definition of wave height." Different groups use different definitions of significant height, and one should always err on the cautious side when using the data.

One interesting feature of the Westhampton data remains unexplained. A long-period signal, at about 85 seconds, is also present. This wave slowly (and imperfectly) modulates the wave heights within the sampled interval. Interaction (known as "beat") between the two dominant waves at 12.8 and 10 seconds might account for this, but McGehee regards this as a "classic, but simplistic" explanation. (Wave theory predicts that the beat period is approximately 40 seconds.) Interference from waves reflected from the shore, or "infra-gravity waves" trapped within the continental shelf, are likely candidates, but no one is quite sure. The source and importance of these waves remains an active area of research.

The fate of wind waves

Seas tend to be quite confused within the area where waves are generated, partly due to the geometry of storms. Most storms are low pressure systems, with cyclonic winds orbiting the storm center. The cyclonic wind field creates waves moving in every direction. These newly formed waves tend to be steeper and more severe and have shorter periods than waves outside of a generation area.

After wind waves form, their growth is limited by the duration of energy input. Energy input is determined by the velocity, duration, and fetch of the wind field. Waves reach an equilibrium state (the "fully developed sea") if the all three of these variables remain constant for sufficient duration. For higher velocities, these conditions are rarely realized, because the waves tend to run out of the area with high winds. Even in a hurricane 300 miles across, a typical large wave will move out of the source area within half a day, far too little time to allow development of a full sea.

Once waves do leave the generation area they become sorted, due to their dispersive nature. In deep water, wave velocity is a function of period; longer (not higher) waves travel faster than shorter. This merely explains what mariners have known for centuries: long-wavelength, often low-height "forerunners" (also called long swell) can arrive on a distant shore days before the storm that generated them.

As the waves propagate away from the generation area they also spread out across the sea surface, much like a ripple in a pond. (Wind waves are focused somewhat better than a ripple, though. Most of the energy in a wave set travels within 45° on either side of the wind direction.) This process too is period dependent. As the energy in the wave spreads out, the wave height decreases in turn, at roughly the square root of the distance traveled.

Inevitably, waves decay as they propagate, due to friction with the atmosphere, the water’s omnipresent surface film, and even ocean currents. Taken together, waves tend to sort out, spread laterally, and lose height and energy as they mature in their travel across deep water.

The differences between mature and young seas show up in synoptic wave records. Recall how the actual wave record from Westhampton was converted into a spectrum, and then a significant wave, with a height and period. A graph of these two parameters over time gives a very simplified history of the sea state for a given place.

The second panel of figure 2 shows the hourly significant wave height and period for August 1998. (These plots are available from the Wave Gauge Program website listed in the sidebar.) Most of August was quite calm, with the significant seas below two feet. Around midnight on the 24th, significant wave height increases, but the period drops rapidly. This indicates a short, steep sea and the onset of local, strong winds. The next day, the wave height stabilizes (the small, bumpy hump), while the period increases, indicating the maturation of sea state during the storm.

A McGehee noted, "this means local winds [were] at work here, and probably not much effect from Hurricane Bonnie yet." The reverse would happen as energy from Bonnietraveling at group velocityreached the sensor offshore Long Island. Sorted, spread, and decayed by their journey, the longest-period low waves will reach Westhampton first, followed by shorter-period, higher waves later. As the 25th comes to a close, the height jumps to eight feet, without much increase in period. McGehee notes that this might be the "arrival of swell from Bonnie, though a more detailed look at the spectra and other gauges would be [needed] to be sure." Bonnie’s energy found the shore of Long Island on the 29th of August. Note that the change in significant height (to 10 feet) was matched perfectly by a change in period.

This analysis provides a powerful tool for analyzing the sea state before one even heads for the dock, whether one is at home or at sea. The U.S. Navy and the Corp of Engineers have for the past few years been engaged in using this sort of information for real-time navigation. King’s Bay, Ga., is the Atlantic port of the Navy’s Trident ballistic missile submarine fleet. Boats enter through St. Mary’s channel, which is shallow even for a Trident on the surface. A shore station downloads a real-time, fully directional wave spectrum from a platform in the entrance to the channel. The model predicts the likely boat behavior in the channel, giving the Trident’s captain and pilot crucial information long before they reach the channel. If the model predicts unacceptable water below the keel, the sub can loiter offshore until conditions improve.

The implications of this technology are profound: just imagine how shipping firms might reduce overboard loss with this sort of en-route information. So, too, might the sailor with a tender crew profit by steering around the worst of the waves.

Larry McKenna is a sailor, author, and teacher living in Overland Park, Kansas. He gratefully acknowledges the expert help of David McGehee, now President of Emerald Ocean Engineering, in Pensacola, Fla., in preparation of this article.

By Ocean Navigator