Wave Action - How and Why Waves Behave As They Do
Jim Antrim, 1981

Waves are a delight and a devilment: the roller-coaster thrill of a wild surf; the misery of a cold water trickle down the neck, the satisfaction of a well steered course to windward; the agony of seasickness on the lee rail.

Imagine sailing without waves. A glassy smooth surface unbroken even by our own wake. Pleasant perhaps, but boring. What sense of speed would we have without our breaking bow wave as we crank along at a neck-snapping six or seven knots? It is largely the skill of getting on or around waves that separates the winners from the losers in our sport. A better understanding of their nature can only help to improving our sailing ability or at least our ability to sound more knowledgeable at the yacht club bar.

Even a careful observer of the confused and undisciplined surface of the ocean may find it difficult to believe that waves are controlled by natural laws, but they are. We all know something about the natural laws of waves without having thought much about it.

Long waves (crests far apart) travel faster than short waves (crests close together).
Very long waves called swell come from storms far away but are too long and round to be dangerous to small boats until they reach shallow water.
Waves generated by the wind locally are steeper and shorter; and the stronger and longer the wind blows the bigger and longer the waves get.
The key to further understanding is to look at the sea state as a collection of many simple component waves, all of different sizes and lengths, all running in different directions, and all piled on top of one another.

SIMPLE WAVES
The ocean and the wind are even more complex and ever changing than the stock market, so we almost never get to see a simple single wave system all by itself, but it looks like this:

and is called a sine wave, just like the one in your high school trigonometry book. The wave length (L) is the distance from crest to crest, height (H) is from trough to crest, and wave speed (V) is the velocity of any one crest. We also need to worry about water depth (D) , and gravity (g) which of course is constant at sea level. It turns out that the speed of a wave is an exact function of its wave length as given by the formula:
V2 = gL /2p * tanh (2pD / L)

That tanh part is a nasty hyperbolic tangent, but fortunately we can forget about it in most cases. In deep water tanh (2pD / L) is very nearly equal to one, and in very shallow water tanh (2pD / L) is very nearly equal to 2pD / L. A wave thinks the water is deep when D/L is more than .5 and shallow when less then .05. Therefore:

V2 = gL /2p in deep water

V2 = gD in very shallow water.

This means that waves move slower when they get into shallow water. The chart in figure 1 shows the speed of a wave in deep water as a function of its wavelength.

Obviously a boat won’t surf on a wave that is moving slower than the boat. A boat won’t truly surf on a wave where the boat was more than half the wave’s length either, since the bow would be buried in the back of the preceding wave. Exciting surfing begins with wavelengths four or more times the length of the boat. With these thoughts in mind, the chart is a guide to surfing speeds for various sized boats. It also points out the advantage that a small boat has surfing conditions. If a 20-footer and a 40-footer catch the same wave, they will be going the same speed. Furthermore, within the range of conditions we normally sail, the small boat will find many more waves of surfable size than the large boat. As a guide, in winds above 20 knots where surfing might be expected and less than 40 knots, above which only nuts and innocent victims are out sailing (I include myself in the first category), wave lengths normally range from 50 to 400 feet . Swells range from 200 to 1400 feet in length (most often around 700 feet), but are generally too round for surfing.

Some of these principals are demonstrated by a race I sailed a couple years back on a J-24. We had a broad reach up the coast and a very close reach back in winds of about 25 knots. The wave lengths were about 90 feet (a bit more than four times our waterline) with heights of 6 - 8 feet. At 90 feet the waves were traveling at 12-1/2 knots, so almost every time we’d catch a wave we’d peg our 12 knot speeds. At times we would get up on a plane of perhaps 14 or 15 knots and would overtake the wave ahead. We kept this up for 4 hours of broad reaching for a IO-1/2 knot average on the first leg. That was good enough to beat one Peterson 2 ton boat for boat and to be right on the tail of another. They had been getting a good push from the waves, but we had been doing wild surfing on the same waves and consequently going the same speed. Sweet Okole (a light Farr I tonner) was first to the mark and about 20 minutes ahead of us. With a waterline of about 1/3 the wave length, she had done some surfing, and being light and quick to accelerate, she had caught waves more easily than the two tonners. Right on her tail was a Moore 24 - lighter than the J-24 and a down wind rocket. Then we rounded .... Needless to say, the two tonners had a faster and more comfortable trip back against the same waves. The Moore 24 cashed it in at the first harbor along route and our ride back in the J-24 was a very wet thrash.

REAL WAVES
The confused sea state we usually see is the sum of many simple wave systems, all of different sizes and lengths, running in different directions. These range from very long swells through a whole range of wave lengths generated by local winds all the way down to ripples. The concept of many waves adding together is an important one and deserves careful thought as it explains many of the phenomenon we see and read about. Figure 2 shows how simple waves add together to form a random sea. First wave "A" and wave "B" form wave system ‘-AB", then a third wave "C" is added to form wave system "ABC." It is easy to see how quickly a wave system takes on a random appearance. Imagine several more wave systems thrown in to the three dimensional view, add some complications like wind blowing the tallest peaks off or waves refracting off a shore, and you get a picture of the mess we sail in. Two wave systems crossing each other at an angle will create the familiar temporary high peaks that seem to rise out of nowhere, then vanish just as quickly. Imagine if the wave crests of all the wave systems in the area were to coincide at the same moment. This doesn’t happen often, but can explain many of the "rogue waves" we hear about.

The next time you are sailing to windward, try to separate in your mind’s eye the various component wave systems. This will often allow you to predict where a high peak or low valley will appear, and assist you in selecting the fastest course through the waves. Developing this skill will help boat speed and crew comfort enormously in heavy weather.

LARGE WAVE SEQUENCES

Sometimes it’s every third wave, sometimes every seventh.Sometimes it’s a repeating set of three large waves in a row. These phenomena of a regularly repeating large wave sequences have been mentioned by surfers, and in reference to the great Fastnet storm. This could be caused by two waves of nearly equal length adding together. Such an occurrence creates a "beat" phenomenon where waves get large when in phase and small when out of phase. Anyone who has tuned a guitar is familiar with this effect. When two strings are plucked at the same note but are not quite in tune, the volume pulsates as the sound waves first add together then cancel each other out, This is demonstrated by the sum of the first two waves in Figure 2 (wave system ‘’AB"). Once can see how this might be interpreted as some regularly sized waves with an occasional group of big ones. Another occasional possible explanation is an impulsively generated wave set, such as you observe when tossing a pebble into a. puddle, where a set of usually three dominant crests moves In a ring pattern. A macro example of this is a tsunami wave set, such as the set of three waves that blasted the coast of Alaska following the 1964 earthquake. (An interesting aside - Oceanographers, frustrated with the inaccuracy of the popular term "tidal wave", decided to adapt the Japanese term "tsunami". They later discovered that tsunami, literally translated, means tidal wave.)

HOW BIG WERE THEY REALLY?

Unfortunately, there is no neat formula to tell us exactly how big a wave will be. Like the volume of a sound wave, the same note or wavelength can range from the quiet strum of a classical guitar to 4,000 watts of electric power jammed through a loud speaker.
As waves get taller, they look less and less like a sine wave and develop a sharp peaked crest. This peak eventually becomes unstable and the wave reaches an upper limit of steepness above which it will fall over under its own weight. The upper limit is a wave with a height/length or "steepness" ratio of .14. This limit is small comfort when waves get into the 300 to 1000 foot length ranges with corresponding maximum heights of 42 to 140 feet! Fortunately, this limit is seldom. reached. post waves fall in the steepness range of .02 to .10 with an average of .03 or .04; and steepness, it turns outs, is related to the age of the wave. As a wave ages, it gradually grows higher, longer and consequently faster (since wave speed is a function of length). The wave will continue to grow in these ways dependent on both duration (length of time the wind has blown on the wave) and fetch (distance over which the wind has blown on the wave), until it reaches its "fully developed" state. Meanwhile, because wave speed is increasing, the ratio of wave speed/wind speed is also increasing. The wave builds in height more quickly than it builds in length, until it reaches its maximum height for a given wind strength. This occurs at a ratio of wave speed/wind speed of .5, and because wave height is maximized and wave length is still growing, the wave is at the steepest point in its development. At this point the wave height averages about 10 percent of the wave length. Still more duration and fetch are required before the wave is fully developed in length. As the wave continues to lengthen and consequently increase its speed, the steepness decreases gradually until the wave speed/wind speed is about 1.3 and steepness averages about .025. 1 suspect that during the Fastnet storm the waves may have been at the age of maximum steepness. Thus the 40 foot waves reported would have been about 400 feet long and would be traveling at 26.8 knots. A wind speed of twice this would be 54 knots, or at the upper end of force 10.

This concept of wave aging and maturity is a large part of what makes one sailing area different from another. The key here is fetch. In a small lake, waves lead a brief life. They are born on one side of the lake and die on the other side before ever getting a chance to grow. Waves in areas like the Great Lakes, Long Island Sound, and the Chesapeake are still fetch limited and tend to be in that age where height is fully developed but length is still growing. Thus peaks are very steep and close together; and wave speed is still a good deal less than wind speed.

A good way to remember how waves age is to think of them as people . They start out small and inconsequential, eventually reaching teenage where they are cocky and fully grown in height, but still short on experience and not yet up to stride. Finally with time they reach adulthood. Logically, sailing in waves is like dealing with people. When they are adult, you’ve got to duck and slide around, over, or between them, but when they are teenagers, often the only solution is to punch right through.

Numerous studies have attempted through experimental or theoretical means to predict wave heights as a function of wind speed. These are usually of a probabilistic nature as even the casual observer can see that there is no exact wave height corresponding to a given wave speed, but rather a range of wave heights with some height being average or most frequent. In comparing these studies one often finds an alarming lack of agreement between theory and experiment, particularly in the high wind speed ranges. Much of this can be explained by the fact that with winds in excess of 25 or certainly 40 knots, winds very rarely have sufficient duration or fetch to generate fully developed seas, add with winds less than 10 knots, there is almost always leftover chop that shows up in the experiments. Rather than go into complex wave height theory or review the history of wave height experimentation, I have added a range of wave heights to the Beaufort wind scale as a guide to conditions normally encountered. This brackets both theory and experiment up to 25 knots of wind (neglecting leftover slop), then follows experimental results more closely above this speed. Conceivably, one could encounter larger waves at these higher wind speeds, but only in a storm of unusual duration and fetch, and this is most unlikely except perhaps in the southern oceans where storms can circle the globe unhindered by land. I wish to emphasize that these limits, while often coinciding with those of the International scale seas state, are the author’s own invention, and are not an official part of the Beaufort scale. As another rule of thumb, average wave heights is .6 times wind speed minus 5 feet. This is reasonably accurate above 12 knots of wind. For example, if wind speed is 30 knots: .6 x 30 is 18. Subtract 5 gives you a 13 foot wave height.

COCKTAIL HOUR
Now that your cranium is packed with all there is to know about wave theory, you are ready to gather a gullible group by your side, plunk an olive in your martini, and launch into a dissertation on the day’s race. "Ya see, guys, it went like this. I woke up this morning and sniffed the breeze. It was blowin’ 30 and the wave crests were about 125 ft. apart. My chart told me they were moving at 15 knots. Since that was half the wind speed, see, and the peaks were real steep, I knew they were young guns and by the time we rounded the weather mark that afternoon they’d be long and moving fast, probably still about 13 feet tall. Sure enough, when we rounded they were 200 feet apart and traveling 19. Just about right for my light weight 50 footer to get up and ride! The straight course home was right across the bar where the water is 20 feet deep. I knew it’d be rough in there but I took out my calculator and figured those waves would be speeding up in the shallow stuff to 25 knots. I was laughing at you clowns surfing at only 20 knots as I shot away at 25. It was then that the rudder went .....