Is the Water at the Foot of a Waterfall Warmer Than at the Top?
― The Scientist Who Took a Thermometer on His Honeymoon
The answer is "yes, a tiny bit warmer." The speed of the falling water seems to vanish when it crashes into the pool. But really, it turns into the invisible jiggling inside the water, which is heat. One scientist took a long thermometer on his honeymoon to check this small difference in temperature.
You have gone to see a waterfall in the mountains. With a roar, water pours off a cliff. The air around the pool is full of white spray, and it feels cool.
After such a rush, the water reaches the pool and flows away as a calm stream. Where did all that force go?
About 180 years ago, someone set out to answer this question in earnest.
Why falling water gets warmer: two steps
In the pool, the falling water breaks into swirls and spray, and these split into ever finer pieces. In the end, the motion becomes tiny jiggling of the water's particles. That jiggling is heat.
Water takes a great deal of heat to warm up. So even for a waterfall with a 100-metre drop, the temperature rises only about 0.2 degrees. You could not feel it with your hand.
This sounds obvious today, but it once was not. Nobody knew that "motion" and "heat" were two forms of the same thing.
Heat was thought to be a "substance"
Until around 1800, many scholars thought of heat as a kind of invisible fluid. The idea was that this fluid flowed from hot things into cold things. It is called the "caloric theory."
The man who cracked this idea was a figure known as Count Rumford. In 1798, at a factory that bored out cannon barrels, he noticed that the metal being cut became hot enough to boil water. As long as the drilling went on, heat kept coming out. If heat were a fluid stored in the metal, it would have run out at some point.
Even so, the caloric theory did not die right away. Nobody had carefully measured the link between the two: how much motion makes how much heat?
A brewer's son found the answer with a thermometer
In Manchester, England, there lived a man named James Prescott Joule. His family business was a brewery. He was not a university professor, and he ran his experiments at home. Temperature control matters in brewing, and he is said to have developed a remarkable skill at reading temperatures very finely.
The experiment Joule devised is on the right side of Figure 1. When a weight falls, a string turns an axle. Paddles on the axle stir the water in a container. When the water is stirred, its temperature rises slightly. By comparing how far the weight fell with how much the water warmed, you can work out the exchange rate between "motion" and "heat."
The result was a very small change in temperature. Joule is said to have read his thermometer scale down to a few hundredths of a degree. In this way he showed that there is always a fixed exchange rate between motion and heat. The unit of energy, the "joule," is named after him.
On honeymoon, with a thermometer at a waterfall
In 1847, Joule married and went on honeymoon to an Alpine valley near the Swiss border. There he ran into a young physicist, William Thomson (later Lord Kelvin). According to Thomson's recollection, Joule was holding a long thermometer. He is said to have planned to measure the water temperature above and below a waterfall, to see whether the fallen water had warmed.
But the measurement seems not to have gone well. At a waterfall, spray meets the air and evaporates, which cools the water. Wind and sunshine also have an effect. A difference of about 0.2 degrees is easily buried in such disturbances.
Even so, the meeting mattered a great deal. Thomson was strongly drawn to Joule's ideas, and the two went on to work together. In this way the idea took shape that "energy may change form, but its total amount does not change."
A German doctor named Mayer set out around 1840 as a ship's doctor on a voyage to the tropics. There, he is said to have noticed that the blood in sailors' veins was redder than he had seen at home. In a hot land, perhaps the body needs to burn less food to stay warm. From this observation, he too came to think, independently of Joule, that heat and motion are the same thing.
Summary
Water that falls over a high waterfall gets a little warmer. The force of the fall breaks into swirls and spray in the pool, and in the end becomes the jiggling of water particles, which is heat. But water is slow to warm, and even a 100-metre drop gives only about 0.2 degrees. The people who chased this small difference found a law of nature: "energy does not disappear."
The roar of a waterfall, the spray, all end up as heat.
The force is not lost. It only changes into invisible jiggling.
For a familiar example of rubbing turning into heat, see "Why do your hands get warm when you rub them together?" For how heat is counted, see "What does the 'calorie' in food actually measure?"
- Put about a cup of dry sand into a thermos. Leave it for a while, then measure the sand's temperature with a cooking thermometer and note it down.
- Close the lid tightly and shake the thermos up and down hard for several minutes. Inside, the sand falls hundreds of times and hits the bottom.
- Open the lid right away and measure the temperature again. Compare it with the first reading and see whether it has gone up a little.
Sand warms up more easily than water, so the difference shows more clearly than with water. The thermos is used to stop the warmth of your hand from getting in. How much it rises depends on how you shake it and how much sand you use, and unless your thermometer reads in steps of 0.1 degrees, you may not see the difference.
For those who want to know more ― terms, formulas, and links to textbooksIt states which level each part is pitched at, from junior high science to university specialist courses
- Junior highCovered in junior high school science
- High schoolCovered in high school physics
- High school+Advanced high school material, or treated as a textbook sidebar
- UniversityNot taught in high school; university specialist courses (thermodynamics, fluid mechanics, history of science)
- ResearchNot taught as settled fact even at university; topics researchers are still investigating
Junior highTerms: this phenomenon has names
- Potential energy: the energy an object has because it is high up. When it falls, it turns into the energy of motion.
- Mechanical equivalent of heat: the exchange rate between motion (work) and heat. One calorie of heat equals about 4.2 joules of work.
- Law of conservation of energy: the law that energy may change form, but the total amount neither increases nor decreases.
Junior highHigh schoolChecking with a formula: how much warmer does the water get at Kegon Falls?
We calculate on the assumption that all the potential energy of the falling water turns into heat in the water itself. Energy is in joules, height in metres, and temperature in degrees. In reality, spray evaporates and heat is exchanged with the air, so this is a "rough upper limit."
| In symbols | m × g × h ＝ m × c × ΔT so ΔT ＝ g × h ÷ c |
| In words | Temperature rise ＝ gravitational acceleration × drop ÷ specific heat of water (the amount of water doesn't matter) |
| Where it comes from | Conservation of energy (the first law of thermodynamics). It is a balance in which all the potential energy before the fall becomes heat in the water |
| Energy when 1 kilogram of water falls 1 metre | 9.8 joules |
| Heat needed to raise 1 kilogram of water by 1 degree | 4186 joules |
| Drop of Kegon Falls (Tochigi Prefecture) | 97 metres, as commonly given |
| Temperature rise per metre of fall | 9.8 ÷ 4186 ≒ 0.0023 degrees |
| Temperature rise after falling over Kegon Falls | 0.0023 × 97 ≒ 0.22 degrees |
| Drop needed to warm water by 1 degree | 4186 ÷ 9.8 ≒ 427 metres |
To warm water by 1 degree, you would have to drop it from more than 400 metres up. That is higher than the top of Tokyo Tower. Even when the force of the fall turns into heat, the temperature barely rises. Joule measured this tiny change with a paddle wheel and a thermometer. The number "mechanical equivalent of heat, 427" found in old textbooks means exactly the same thing as this calculation.
High schoolHigh school+The first law of thermodynamics, and the large specific heat of water
High schoolThe heat added to an object plus the work done on it from outside equals the increase in its internal energy. This is the first law of thermodynamics. In a waterfall, the work done by gravity goes into raising the water's internal energy. The specific heat of water is large, about 4.2 joules per gram per degree, so the temperature rises only a little.
High school+In Joule's paddle-wheel experiment, baffles are said to have been fitted inside the container so that the water would not just keep spinning as one body. If the water only goes round and round, the energy of motion is not fully turned into heat.
UniversityHow does the force become heat?
The big swirls of water that fall into the pool split into smaller and smaller swirls. In fluid mechanics, this is called the energy cascade of turbulence. At the end, when the swirls become very small, the stickiness of the water (viscosity) smooths out the motion, and it becomes the random motion of molecules, which is heat. This process is called viscous dissipation. In the history of science, the fact that Mayer, Joule, Helmholtz and others arrived at the idea of energy conservation separately at almost the same time is discussed as a leading example of "simultaneous discovery."
📖 Derivations and further reading: First law of thermodynamics (Wikipedia) ／ Turbulence (Wikipedia)
ResearchWhat is still not clear
- Can the warming from the fall alone be isolated in a real waterfall? In a real waterfall, evaporation, sunshine and heat exchange with the air all happen at once. It is said to be hard to separate the warming caused by the fall from these and measure it cleanly.
- Where does the energy of ocean tidal currents turn into heat? The tidal flows raised by the Moon and Sun also end up as swirls and then heat. Exactly where in the ocean this happens, and how much, is still being observed and studied.
- How should we think about "who discovered it first"? The law of conservation of energy was reached by several people, each by a different route. How to judge the credit for the discovery is debated even among historians of science.
In other words, this article too is "an explanation within what is known today." The historical episodes in particular rely partly on the recollections of the people involved and those around them.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Junior high | Science: "Changes of energy" | The main text: the force of the fall turns into heat |
| High school | Physics: "Heat and energy" | Checking with a formula; the first law of thermodynamics |
| High school+ | Physics sidebar: "Mechanical equivalent of heat" | Joule's paddle wheel and baffles |
| University | Fluid mechanics, thermodynamics, history of science | Energy cascade, viscous dissipation, simultaneous discovery |
| Research | Ocean physics, history of science | Where tidal energy goes; assessing credit for discovery |
| ― | Links to daily life | The same mechanism turns the motion of a bicycle or car stopped by its brakes into heat |
- J. P. Joule, "On the Mechanical Equivalent of Heat", Philosophical Transactions of the Royal Society of London, 1850
- D. S. L. Cardwell, "James Joule: A Biography", Manchester University Press, 1989
- C. Smith and M. N. Wise, "Energy and Empire: A Biographical Study of Lord Kelvin", Cambridge University Press, 1989
- T. S. Kuhn, "Energy Conservation as an Example of Simultaneous Discovery", 1959 (included in "The Essential Tension" 『科学革命における本質的緊張』)
- National Astronomical Observatory of Japan (国立天文台), ed., "Chronological Scientific Tables" (理科年表), Maruzen Publishing (丸善出版) (specific heat of water)
*This article is a popular-science explanation for general readers. The figures given are rough guides to help you understand the mechanism. Some historical episodes rest on the recollections of those involved, and details are disputed. When you visit a waterfall, keep away from slippery rocks and the pool at its foot, and follow local signs and no-entry notices.