How High Can an Ocean Wave Get?
― Once a wave is taller than one seventh of its wavelength, it collapses on its own
When a typhoon forecast says "waves of 10 metres", it sounds as if waves could grow without end. But a wave has a upper limit to its height. A wave that passes the limit breaks by itself, right where it is. The wind can drop, and the shore can still be far away.
You are standing on a beach, watching the waves roll in. Out at sea, each wave rises slowly like a smooth hill, then sinks again. It keeps its shape.
But as it nears the shore, the same wave suddenly stands up. Its crest turns white with foam and tips over forward. Nobody pushed it.
Out at sea it held its shape so calmly. Why does it give up partway?
A wave breaks for just two reasons
If the crest gets too sharp, the water at the top moves as fast as the wave itself travels. Then the water can no longer ride along, and it tumbles forward. The tipping point is a height of "about one seventh of the wavelength".
Near the shore, the wave starts to drag on the seabed. It slows down, and its wavelength shrinks. The height drops far less, so the ratio keeps growing. In the end it hits the limit.
The key point is that what limits a wave is not its height alone but the ratio of height to wavelength. A 3-metre wave is perfectly fine if its wavelength is 100 metres, but it cannot survive if its wavelength is 15 metres. Let's take it step by step.
What a wave carries is not water but a "shape"
Watch ocean waves and it looks as if the water is streaming towards the shore. But if you float a ring buoy, it does not travel to the shore. It just bobs up and down, moving round in a loop where it is. What travels is not the water. It is the shape of the wave.
Each bit of water is tracing a small circle on the spot. The higher the wave, the bigger the circle. A bigger circle also means faster motion. The speed at which the wave shape travels, on the other hand, is mostly set by the wavelength, and adding height changes it very little.
So if you keep raising only the height, the "speed of the water at the crest" eventually reaches the "speed of the wave". From that moment, the crest water sticks out beyond the front of the wave and starts to fall forward. This is a breaking wave. In Figure 1, the left side is a smooth wave with room to spare, and the right side is a wave that has hit the limit and is breaking.
In shallow water, depth decides wave height
In deep water, a wave does not feel the seabed at all. The circling motion of the water fades quickly with depth, and by a depth of about half the wavelength it has almost vanished.
But once the water is shallower than half the wavelength, the bottom gets in the way and squashes the circles. The wave's speed is then set almost entirely by the depth. The shallower it gets, the slower the wave, so waves from behind catch up with those in front and the wavelength shrinks. The height, on the other hand, rises slightly.
As a result, the ratio of height to wavelength always heads towards the limit as the wave nears the shore. As a rough rule, a wave is said to break when its height reaches about 80% of the water depth. In water 1 metre deep, only waves of about 80 centimetres can stand. This is why the waves at a swimming beach often look much smaller than the forecast for the open sea.
The wave height in a weather forecast is the "significant wave height". It is found by taking the highest third of the observed waves and averaging them. Higher waves are mixed in too, and the largest wave can be about 1.5 to 2 times the significant wave height. In a sea with "2-metre waves", a wave close to 4 metres can slip in.
On a windy day you can see white streaks dotted across the open sea. These are breaking waves too. As the wind keeps pushing, the waves grow, and once they hit the limit only the crest breaks. White horses are a sign of how far the wind has raised the waves, and they mark the limit as well.
Summary
What stops a wave's growth is neither the shore nor the wind, but the wave's own shape. Once the height passes about one seventh of the wavelength, the crest water moves faster than the wave and spills forward. In shallow places the wavelength shrinks, so even a low wave reaches that limit first.
A wave that grows too tall cannot hold itself together.
Breaking is the sign that it has reached its limit.
How a wave's shape travels is covered in more detail in "Why Do Round Rings Spread When You Throw a Stone into a Pond?", and how long waves raised by a distant storm arrive is covered in "Why Does a Big Wave Suddenly Arrive Even When There Is No Wind?".
- Pour about 5 cm of water into a flat bowl. Touch the edge of your palm to the surface and move it slowly up and down. A smooth wave should run across.
- Keep the same depth, but make your up-and-down movement a little bigger each time. Past a certain size, the crests start to break white. That is the limit.
- Sink a plate at a slant on one side of the bowl to make a shallow slope. You will see the same wave break just as it reaches the slope.
Measure the water depth where the wave starts to break, and the wave height, with a ruler, and compare them. If the height is about 80% of the depth, you are matching the rule of thumb for the real sea.
For those who want to know more ― terms, formulas and links to textbooksFrom junior-high science to university courses: each part says what level it is
- Junior highCovered in junior-high science
- High schoolCovered in high-school "Basic Physics / Physics"
- High school+Advanced high-school material, or textbook sidebars
- UniversityNot taught in high school; university courses (fluid mechanics, coastal engineering)
- ResearchNot yet settled textbook knowledge; what researchers are studying now
Junior highTerms: this phenomenon has names
- Wavelength: the distance from one crest to the next. Written L. The unit is the metre.
- Wave height: the height difference from trough to crest. Written H. The unit is the metre.
- Steepness: the wave height divided by the wavelength. The larger this value, the sharper the wave's shape.
- Breaking wave: the phenomenon where a wave can no longer hold its shape and its crest collapses. The white waves crashing on the shore and the white horses out at sea are both this.
Junior highHigh schoolChecking with a formula: how high can a 100-metre swell get?
The limit on a wave is set not by the height itself but by the ratio of height to wavelength. First we write down the formula for that ratio, then put in numbers.
| In symbols | H ＝ L ÷ 7 (H is wave height, L is wavelength) |
| In words | Wave height just before breaking ＝ wavelength ÷ 7 |
| Where the formula comes from | It comes from a balance condition: the "speed of the water circling at the crest" catches up with the "speed at which the wave shape travels". A wave shape that meets this condition has a crest with a sharp 120-degree angle, and a ratio of wave height to wavelength of about 0.142. One seventh (0.143) is an easy-to-remember version of that value. |
| Wavelength of the swell at sea | 100 metres |
| Ratio just before breaking | 1/7 |
| Rule of thumb in shallow water (wave height ÷ depth) | said to be 0.78 |
| Height of a person | 1.6 metres |
| Wave height just before breaking at sea | 100 ÷ 7 ≒ 14.3 |
| How many person-heights | 14.3 ÷ 1.6 ≒ 8.9 |
| Limit in shallow water 3 metres deep | 3 × 0.78 ≒ 2.3 |
A swell with a 100-metre wavelength can keep its shape up to 14.3 metres. That is about 9 times a person's height, or the height of a four-storey building. But when the same wave enters shallows 3 metres deep, the limit drops to 2.3 metres. The height lost there turns into foam, sound and heat.
High schoolHigh school+What decides a wave's speed?
High schoolIn high-school Basic Physics you learn that wave speed ＝ frequency × wavelength. The same is true of ocean waves. The frequency of a wave coming from the open sea does not change as it nears the shore, so when its speed drops, its wavelength shrinks by the same proportion.
High school+The speed of a water-surface wave changes with wavelength. This is called dispersion, and for deep-water waves it is described by a formula known as the "dispersion relation of surface gravity waves". In deep water, longer waves travel faster, and in shallow water the speed is set almost entirely by the depth. This difference is also why the long waves raised by a distant storm arrive first.
UniversityThe people who first calculated the limiting shape
In university fluid mechanics you learn about "Stokes waves", which treat surface waves as nonlinear periodic waves. In 1880, Stokes showed that the crest of a wave just before breaking should have a sharp 120-degree angle. The specific ratio (wave height to wavelength of about 0.142) is said to have been found by Michell in 1893 through numerical calculation. Rules of thumb for breaking in shallow water have been organised in coastal engineering, and are used in breakwater design in forms such as the "Miche breaking criterion", which uses the ratio of wave height to depth.
📖 Derivations and further reading: Wikipedia (English) "Stokes wave" ／ Wikipedia (English) "Breaking wave"
ResearchWhat is still not well understood
- Rare giant waves. Abnormally high waves, several times the height of the waves around them, are occasionally seen in the open ocean. Whether chance overlap of several waves is enough to explain them, or whether a mechanism by which waves feed energy to one another is at work, is still debated.
- Where a broken wave's energy goes. When a wave breaks, the energy it carried splits into foam, eddies and heat. Measuring that split in the real ocean is hard, and numerical forecasts still replace it with empirical formulas.
- How much gas the bubbles carry. The bubbles drawn in by white horses act as a window for passing carbon dioxide and other gases between the air and the sea. How much moves worldwide is still uncertain, with a wide range of estimates.
In other words, this article too is "an explanation within what is known today". The figure of one seventh is also an ideal value for a single clean wave. On a real sea surface where many waves mix, waves can break at even lower heights.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Junior high | Science / Sound and light, properties of waves | The words wavelength and wave height; the idea that a wave carries a shape, not water |
| High school | Basic Physics / Physics: wave equations | Wave speed ＝ frequency × wavelength; why the wavelength shrinks in shallower water |
| High school+ | Physics / Dispersion of water waves | Speed changing with wavelength; deep versus shallow water |
| University | Fluid mechanics, coastal engineering | Stokes waves, the limiting shape with a 120-degree crest angle, breaking criteria |
| Research | Physical oceanography | How giant waves form; energy and gas exchange in breaking waves |
| ― | Links to daily life | How to read forecast wave heights; estimating the size of waves at a shallow beach |
- Japan Meteorological Agency (気象庁), explanatory materials on "Wave observation and forecasting" (definition of significant wave height, approach to wave forecasting)
- Yoshimi Goda (合田良実), Design of Port Structures against Waves (港湾構造物の耐波設計), Kajima Institute Publishing (鹿島出版会) (limiting breaking wave height, the idea of design waves)
- J. H. Michell, "On the highest waves in water", Philosophical Magazine, 1893 (numerical calculation of the limiting wave ratio)
- O. M. Phillips, "The Dynamics of the Upper Ocean", Cambridge University Press (theory of surface waves and wave breaking)
※This article is a general-audience science explainer. The figures given are rough guides meant to help you understand the mechanism. At the seaside, always follow the Japan Meteorological Agency's wave information and the instructions of the fire service, local authorities and Coast Guard.