Why Does a Wave That Keeps Its Shape Never Fall Apart?
— An Engineer Once Chased One Along a Canal on Horseback
Ordinary waves either spread out and flatten as they travel, or steepen at the front and break. Yet sometimes a single hump of water can travel for miles with its shape and speed unchanged. In 1834 an engineer chased one on horseback, and it took 130 years to work out what this "impossible wave" really was.
It is August 1834, on a canal near Edinburgh in Scotland. Two horses are pulling a long, narrow boat along from the bank. A young naval engineer, John Scott Russell, is watching. He is studying how the shape of a boat affects the resistance of the water.
Suddenly the boat stops. The water that had been pushed up in front of the bow rolls forward as a single rounded hump. It leaves the boat behind and glides straight down the canal.
Russell leapt onto a horse and gave chase. The hump moved at roughly 13 to 14 km per hour. After following it for about 1.5 to 3 km, he wrote, he lost it among the bends of the canal. All that time, he said, its shape barely changed.
Why could this wave keep its shape? Just two reasons
A wave in shallow water travels faster where the water is deeper. The crest of a hump is, in effect, sitting in deeper water than its foot. So the crest tries to catch up with the front, and the front slope grows steeper and steeper. Left alone, this would end like a wave on a beach, breaking forward.
On the other hand, a narrow hump left to itself spreads sideways and gets lower. This is because finer ripples travel at different speeds. This tendency works the other way, smoothing the hump out.
"Steepening at the front" and "spreading out sideways" are opposite tendencies. When they balance exactly, the hump travels on without breaking and without spreading. What Russell saw was a wave in which that balance happened to hold.
Taller humps are faster, and narrower
Russell could not forget the wave he had seen. It is said that he built a tank about 9 m long in his garden and recreated the hump many times. From those experiments he worked out that the speed of the hump is set by "water depth + hump height". The taller the hump, the faster it goes.
Look at Figure 1. The water depth is fixed at 1.2 m, and you can change the height of the hump with the slider. Make it taller and the speed goes up, while the hump becomes narrower. Height, width and speed cannot be chosen separately. A solitary wave of a given height has one particular width and speed, as a set.
Confirming what it was took 130 years
In 1844 Russell reported this wave and called it the "great wave of translation". But the leading scholars of the day did not accept it at once. It was common knowledge that waves change shape as they travel.
The tide turned in 1895. The Dutch scientists Korteweg and de Vries wrote down an equation for waves in shallow water. Built into it was a solution for a hump that travels without changing shape. Russell's observation was right in theory too. But Russell had died 13 years earlier.
Another 70 years on, in 1965, the Americans Zabusky and Kruskal solved this equation on a computer. They found that when two humps collide, they pass through each other and each carries on with its original shape. Because they behaved like particles, the two named them "solitons".
Figure 2 is the heart of this article. With only the left tendency, the wave would break. With only the middle one, it would flatten and fade away. Only for the right combination of height and width do the two cancel, as on the right. When you changed the height in Figure 1 and the width changed too, it was to keep this balance.
In northern Australia, around September and October each year, a tube-shaped cloud hundreds of kilometres long can roll across the morning sky in a single line. Called the "Morning Glory", this cloud is thought to be a solitary wave travelling through the air, made visible as a cloud. Under the sea, too, giant solitary waves are known to travel along the boundary between layers of water of different density. Their streaks can sometimes be seen on the sea surface in photographs taken from space.
Light signals also spread and blur as they travel along a thin fibre. In 1973, Akira Hasegawa and a colleague in Japan showed in theory that solitons can be made from light as well. As with the water hump in the canal, the idea is to balance a "tendency to blur" against a "tendency to shrink".
Summary
Ordinary waves either steepen at the front and break, or spread sideways and fade. In shallow water these two tendencies work in opposite directions. When height and width are matched just right, the two cancel, and a single hump keeps its shape as it travels on. Russell's observation of a canal wave, chased on horseback, became the starting point of a new branch of physics 130 years later.
An unbreaking wave is not strong.
The force that breaks it and the force that blurs it are simply held in a draw.
The side of the story where waves break forward is covered in more detail in How High Can an Ocean Wave Get?, and how water depth changes wave speed in Why Do Ocean Waves Almost Always Arrive Parallel to the Shore?
- Fill a long, narrow container (a storage box, or the long side of a bath) with water about 3 to 5 cm deep.
- Stand a plastic sheet or board at one end, push it forward quickly just once, and stop. You will see a single bump travel toward the other end.
- Next, push harder to make a taller hump. Compare how long each takes to reach the far wall. The taller hump should arrive a little sooner.
If the water is deeper, the hump goes faster and the difference is harder to see. The shallower the water, the easier it is to observe. To keep the floor dry, we recommend trying this in the bathroom.
For those who want more — terms, equations and links to textbooksEach part is labelled by level, from middle school science to university courses
- Middle schoolCovered in middle school science
- High schoolCovered in high school "Physics"
- High school+Advanced high school material, or textbook sidebar content
- UniversityUniversity-level courses not taught in high school (fluid dynamics, nonlinear physics)
- ResearchNot yet taught even at university as settled fact; researchers are studying it right now
Middle schoolTerms: this phenomenon has names
- Solitary wave: a wave that travels as a single hump, with no other waves before or after it.
- Soliton: a solitary wave that keeps its shape as it travels and regains it after colliding with other solitons. The name was coined in 1965.
- Shallow-water wave: a wave in water that is shallow compared with its wavelength. Its speed depends only on depth, and it goes faster the deeper the water.
Middle schoolHigh schoolChecking with an equation: how fast was Russell's wave?
We use the formula for the speed of a solitary wave that Russell found from his tank experiments. We do not know the canal's depth exactly, so here we assume 1.2 m. We put in the recorded hump height (about 0.4 m) and check whether it matches the speed of the horseback chase.
| In symbols | c ＝ √( g × ( h ＋ a ) ) |
| In words | Speed of a solitary wave = square root of (gravitational acceleration × (water depth + hump height)) |
| Where it comes from | An empirical formula Russell found in his tank experiments. It takes the speed of a shallow-water wave, √( g × h ), and replaces the "water depth" with the "depth up to the top of the hump". It was later also derived from the KdV equation as an approximation for low humps. |
| Symbol | Meaning and unit |
| c | Speed of the solitary wave (m/s) |
| g | Gravitational acceleration (m/s²) |
| h | Water depth without the hump (m) |
| a | Hump height measured from the original surface (m) |
| Gravitational acceleration g | 9.8 m/s² |
| Water depth h (assumed) | 1.2 m |
| Hump height a | About 0.4 m (recorded as 30 to 45 cm) |
| Speed Russell chased | Said to be about 13 to 14 km per hour |
| Depth up to the top of the hump | 1.2 ＋ 0.4 ＝ 1.6 m |
| Multiply by gravitational acceleration | 9.8 × 1.6 ＝ 15.68 |
| Its square root is the solitary wave's speed | The square root of 15.68 is about 3.96 m/s |
| Convert to km per hour | 3.96 × 3.6 ≒ 14.3 km |
| For comparison: a very low wave | 9.8 × 1.2 ＝ 11.76 |
| Its square root | The square root of 11.76 is about 3.43 m/s |
| How much speed rises with height | 3.96 ÷ 3.43 ≒ 1.15 |
The solitary wave's speed comes out at about 14 km per hour, in good agreement with the 13 to 14 km per hour Russell chased on horseback. In the same channel, a hump 0.4 m high travels about 15% faster than a low wave. This "taller means faster" property is the source of ① in Figure 2, the tendency of the front to steepen.
High schoolHigh school+What decides the speed of a wave?
High schoolIn high school physics you learn that wave speed = frequency × wavelength. For ordinary waves, the height of the hump (the amplitude) is treated as having no effect on speed. A solitary wave is a slightly special wave for which that assumption does not hold.
High school+The property that speed changes with amplitude is called "nonlinearity", and the property that it changes with wavelength is called "dispersion". Nonlinearity steepens the hump, and dispersion spreads it. A solitary wave is one in which these two balance. For that reason, the height a and a measure of the width L are linked by L ＝ √( 4h³ ÷ 3a ). The slider in Figure 1 draws the width using this relation.
UniversityThe KdV equation and the inverse scattering method
Weakly nonlinear waves in shallow water are described by the KdV equation (the Korteweg–de Vries equation). It is a partial differential equation with one nonlinear term and one dispersive term, and its solitary-wave solution has a sech² shape built from hyperbolic functions. In 1967, Gardner and colleagues found the inverse scattering method, which solves this equation exactly, and the field known as integrable systems expanded greatly. Solitons of light in optical fibres are treated by the nonlinear Schrödinger equation.
📖 Derivations and further reading: Wikipedia, "KdV equation" (Japanese) ／ Wikipedia, "Soliton" (Japanese)
ResearchWhat is still not clear
- The role of giant solitary waves in the ocean. Solitary waves travelling along the boundary between water layers are thought to mix deep and shallow water and carry nutrients. How much they affect the mixing of the ocean as a whole is still being estimated.
- How the Morning Glory forms. The leading idea is that it arises when sea breezes collide. It occurs in the same places every year, but the conditions that separate days when it appears from days when it does not are still not fully understood.
- Solitons and sudden huge waves. Some research tries to explain part of the abnormally high waves that appear suddenly in the open ocean as a gathering of nonlinear waves. How far this applies to the real ocean is still debated.
In other words, the content of this article too is "an explanation of what is understood so far". The question that began with one wave on a canal is still being pursued today, in the sea, the sky and in light.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Middle school | Science: how sound and waves travel | The observation that a hump travels without changing shape |
| High school | Physics: properties of waves; square-root calculations | The speed calculation in "Checking with an equation" |
| High school+ | Physics: advanced waves (dispersion and nonlinearity) | The steepening and spreading tendencies in Figure 2 |
| University | Fluid dynamics; nonlinear physics | The KdV equation and inverse scattering |
| Research | Physical oceanography; meteorology | Solitary waves in the ocean; the Morning Glory |
| ― | Links to daily life | Optical-fibre communication; watching tidal bores travel up rivers |
- J. S. Russell, Report on Waves, Report of the 14th Meeting of the British Association for the Advancement of Science (1844)
- D. J. Korteweg & G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine (1895)
- N. J. Zabusky & M. D. Kruskal, Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States, Physical Review Letters (1965)
- Wikipedia, "Soliton" (Japanese)
- Mikio Wadati, Nonlinear Waves (和達三樹『非線形波動』), Iwanami Shoten (岩波書店)
※This article is a general-audience science explainer. The figures given are guides and rough estimates meant to help you grasp the mechanism. The numbers from Russell's observations are round figures based on his own records, and the canal's depth is an assumed value.