Do Freak Waves Really Exist?
— A Giant Wave Once Dismissed as a Sailor's Tall Tale Was Measured by an Offshore Platform in 1995
On a rough sea, a wall of water twice as tall as the waves around it suddenly appears. For a long time, people brushed off such stories as sailors' exaggeration. Then, on New Year's Day 1995, an instrument on a North Sea platform clearly recorded this "impossible wave". In fact, a giant wave needs no special cause. Ordinary waves simply piling on top of one another can create one.
The day after a typhoon has passed, you are watching the sea from a hill. White waves roll toward the shore one after another, and they all look about the same height.
After a while, a slightly bigger wave turns up now and then. Even so, after ten minutes you feel you have the measure of the sea: "It's about this rough."
But is that estimate right? If you kept watching for hours, wouldn't a wave far higher than the biggest one you have seen so far eventually arrive?
Giant waves come from just two things
Ocean waves are made of many waves, each with a different length and speed, laid on top of one another. Usually crests and troughs cancel out. Sometimes, though, crests meet at the same spot and their heights add up.
A big alignment of crests is a rare event, perhaps one wave in several thousand. But in a storm, a wave arrives every ten seconds. A few hours is enough to reach several thousand.
The reason it sounded like a tall tale lies in the second point. A person sees only a few dozen waves at a time. A one-in-several-thousand wave is very unlikely to be among them. The less often someone had seen one, the more likely they were to assume such waves did not exist.
New Year's Day 1995: what the platform saw
In the North Sea off Norway stands a platform for offshore gas production called Draupner. It carried an instrument that measured, using light, the distance down to the sea surface directly below. On 1 January 1995, this instrument recorded a wave about 26 metres from trough to crest.
At the time, the "ordinary big waves" in the surrounding sea were about 12 metres high. In other words, a single wave more than twice that height had appeared. The waves before and after it did not seem to include anything nearly as large.
Until then, ship design and marine engineering had widely relied on the idea that wave heights stay within a certain range. Once a reliable instrument record existed, the freak wave stopped being folklore and became a subject of research. Later, when several weeks of satellite radar images were examined, more than ten waves over 25 metres high were reported across the world's oceans.
Reason 1: ordinary waves just add up
Look at Figure 1. The three thin lines at the top are three waves of different lengths. All have the same height, and none is special. The thick line at the bottom is the sea surface you get by simply adding the three together.
Normally, the crest of one wave lands on the trough of another, and the sea surface stays low. But where crests line up, their heights stack directly. Waves of different lengths also travel at different speeds. So a lined-up set of crests soon drifts apart again. This is why a freak wave is said to "appear suddenly and vanish quickly".
A real ocean has not three but countless waves overlapping. So the degree of alignment differs every time. A little alignment gives a "somewhat big wave", and a lot gives a freak wave.
Reason 2: the height ratio decides "one in how many"
We know that statistical formulas describe well how often ocean waves reach a given height. The yardstick is the "ordinary big-wave height". To get it, line up the waves from highest to lowest and average the top third. This is also what a weather forecast means by "wave height".
Use the slider in Figure 2 to choose how many times the yardstick height you want to consider. A wave equal to the yardstick comes about once in 7 waves. At 1.5 times, it is once in 90. At 2 times, it jumps to about once in 3,000.
What matters is the time in the box on the right. In a stormy sea with a wave every ten seconds, a wave twice the yardstick height works out to once every 8 hours or so. In a storm that lasts several days, it would not be strange for one to come many times. The old stories were not exaggerated. People simply watch for too short a time.
By the same reasoning, the biggest wave seen in about ten minutes from a hill is only about 1.4 times the ordinary big wave. Watch for several hours, though, and the biggest wave grows to about 1.9 times. "It was fine for a while" is no promise of safety hours later.
The curve in Figure 2 assumes that waves do not affect one another. In the real sea, a wave can draw energy from its neighbours and grow taller by itself. For this reason, under some conditions freak waves are thought to occur several times more often than the calculation says.
It is known that big waves tend to arrive in groups, but there is no fixed number such as "always the seventh". The length of a group depends on the wave components of the sea that day. Counting will not keep you safe. Watching from high ground will.
Summary
A freak wave needs no special cause. When ordinary waves overlap and their crests happen to line up, a wave more than twice the height of its surroundings appears. For any single wave this is a rare event, one in several thousand, but in a storm it works out to once every few hours. It sounded like a tall tale only because people see too few waves at a time.
Even a very rare event
will happen eventually if you repeat it thousands of times.
For how big waves arrive in groups, see "Why do expert rock fishers watch the sea for a while before casting?". For the limit on how tall waves can get before they break, see "How high can ocean waves get?". For waves that reach us from distant storms, see "Why does a tall wave suddenly arrive on a windless day?".
- Get three dice. Treat each die as one wave, and the number it shows as "the wave's height at that moment".
- Roll all three together and record the total. Repeat 50 times. The totals should mostly cluster around 8 to 13.
- A total of 18, where all three show a 6, comes up only once in 216 rolls. You probably will not see it in 50 rolls. But if the whole class rolls, someone might.
A result that seems impossible to one person will certainly show up if you roll often enough. Ocean waves are the same. When you watch the real sea, stay away from the water's edge and watch from high ground.
For those who want to know more — terms, formulas and links to textbooksEach part is labelled by level, from junior-high science to university courses
- Junior highCovered in junior-high science
- High schoolCovered in high-school physics and maths
- HS+Advanced high-school material, or textbook sidebars
- UniversityNot taught in high school; university courses (ocean wave theory, nonlinear waves)
- ResearchNot yet settled textbook knowledge; researchers are studying it now
Junior highTerms: this phenomenon has names
- Freak wave (giant wave): a wave more than twice the height of the ordinary big waves around it. Also called a "rogue wave".
- Significant wave height: the average height of the top third of waves when ranked from highest to lowest. It is the "ordinary big-wave height" in this article, and the wave height in weather forecasts.
- Wave height: the height of a single wave from the bottom of the trough to the top of the crest.
Junior highHigh schoolCheck with a formula: one in how many waves, and once in how many hours, for a wave twice the yardstick?
We put the height ratio of 2 into the formula for how wave heights are distributed, and work out the interval between freak waves.
| In symbols | P ＝ e−2 × (H ÷ Hs)² N ＝ 1 ÷ P t ＝ N × T |
| In words | The fraction of waves exceeding height H = e raised to the power of "−2 × (height ratio squared)". One in how many = its reciprocal. Interval between arrivals = number of waves × period of one wave |
| Where it comes from | When countless small waves overlap independently, the sea surface height scatters in a bell shape (a normal distribution). Taking the height of each individual wave from that gives this form (the Rayleigh distribution) |
| Symbol | Meaning and unit |
| H | Height of the wave considered (m) |
| Hs | Significant wave height = ordinary big-wave height (m) |
| P | Fraction of waves exceeding height H (no unit) |
| N | One wave in how many (waves) |
| T | Time for one wave to pass = period (seconds) |
| Freak-wave mark (height ratio) | 2 times |
| Wave period in a stormy sea | About 10 seconds |
| e to the power −8 | About 0.000335 |
| Height of the Draupner wave | About 25.6 metres |
| Significant wave height at the time | About 12 metres |
| Height ratio squared | 2 × 2 ＝ 4 |
| Size of the exponent | 2 × 4 ＝ 8 |
| One wave in how many | 1 ÷ 0.000335 ≒ 2985 waves |
| Interval (seconds) | 2985 × 10 ＝ 29850 seconds |
| Interval (hours) | 29850 ÷ 3600 ≒ 8.3 hours |
| Draupner wave ratio | 25.6 ÷ 12 ≒ 2.1 times |
A wave of twice the yardstick height comes about once in 3,000 waves, or about once every 8 hours in a sea with a wave every 10 seconds. The 1995 wave was about 2.1 times, clearly above the freak-wave mark.
High schoolHS+Superposition, and how the biggest wave grows
High schoolFigure 1 is exactly the principle of superposition of waves from high-school physics. When several waves arrive at the same place, their displacements add. Crest on crest reinforces, and crest on trough weakens. Ocean waves travel at different speeds depending on their length (they are dispersive), so the reinforcing spots keep moving, appearing and disappearing.
HS+From the formula above, the highest wave among N waves is roughly "Hs × the square root of (natural log of N ÷ 2)". For 60 waves (about 10 minutes) that is about 1.4 times, and for 1,000 waves (about 3 hours) about 1.9 times. The longer you watch, the higher the biggest wave creeps.
UniversityWaves that feed waves: modulational instability
Waves on deep water are not strictly just a matter of addition. When the wave shape becomes slightly uneven, the taller parts gather energy from their surroundings and grow taller still. This is called the Benjamin–Feir instability (modulational instability), and the change in the wave envelope is described by the nonlinear Schrödinger equation. Among its solutions is one called the "Peregrine soliton", in which a wave nearly three times as tall suddenly appears from nothing and then vanishes. It is studied as a template for freak waves.
📖 Derivations and further reading: Rayleigh distribution (Wikipedia) ／ Freak wave (Wikipedia)
ResearchWhat is still not clear
- How "too many" are there? In measurements, some sea areas have more freak waves than the Rayleigh distribution predicts, while others match it closely. Which conditions increase them is still being sorted out.
- The role of crossing waves. One idea is that freak waves are more likely where two groups of waves from different directions meet. It has been suggested that the Draupner wave may have been like this.
- Forecasts minutes ahead. Attempts continue to use ship radar to measure the surrounding waves and warn of a giant wave tens of seconds to minutes ahead, but practical use still faces challenges.
So the content of this article, too, is "an explanation within what is known today". The focus of current research is not "whether they happen" but "how often they happen".
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Junior high | Science Field 1 "Sound and waves", maths "Probability" | Crests and troughs, the dice experiment |
| High school | Physics "Superposition of waves", maths "Exponential functions" | Figure 1, checking with a formula |
| HS+ | Maths "Statistical inference" | How the biggest wave grows |
| University | Ocean wave theory, nonlinear waves | Modulational instability, nonlinear Schrödinger equation |
| Research | Ocean engineering, wave forecasting | Frequency of occurrence, crossing waves, short-term forecasting |
| ― | Link to daily life | When watching waves from the shore, don't judge by a short-time impression |
- Dysthe, K., Krogstad, H. E., Müller, P. (2008) Oceanic Rogue Waves. Annual Review of Fluid Mechanics 40
- Haver, S. (2004) A possible freak wave event measured at the Draupner jacket January 1 1995. Rogue Waves 2004 Workshop, Brest
- Kharif, C., Pelinovsky, E., Slunyaev, A. (2009) Rogue Waves in the Ocean. Springer
- ESA (2004) Ship-sinking monster waves revealed by ESA satellites (press material from the European Space Agency)
- Yoshimi Goda (合田良実), 『港湾構造物の耐波設計』 (Wave-Resistant Design of Port Structures), Kajima Institute Publishing (鹿島出版会)
*This article is a popular-science explanation for general readers. The figures given are rough guides to help you understand how things work. When observing waves at the coast, stay away from the water's edge and follow weather information and instructions from local authorities, the Japan Coast Guard and others.