We had a new sink fitted, a couple of years ago. I've just noticed something curious about it. If I fill it with water, dip my finger in at just the right place, and waggle the finger up and down, then of course the water goes up and down as well. But it also goes up and down almost as strongly at a second position, where there is no finger.
The secret is the shape of the bowl. It's not a rectangle or a circle. It's an ellipse.
Gardeners know that you can make an elliptical flowerbed by sticking two stakes in the ground, placing a loop of string over them, and stretching it taut with a third stake which you scrape along the ground. The first two stakes are the foci of the ellipse. The gardener's trick tells us that if you add the distances from any point on an ellipse to the two foci, you always get the same number.
This property of the ellipse has an interesting implication if you play pool on an elliptical table. If a ball starts at one focus and bounces off the cushion, then it will pass through the other focus. To see why, look at the point where the ball hits the cushion. The path taken by the ball consists of two straight lines joining that point to each focus. These two lines meet the cushion at the same angle, which is how a pool ball bounces.
Why are the angles the same? Because the gardener's string is stretched taut, so it follows the shortest path from focus to cushion and back to the other focus. If those angles were different, then a nearby path would be shorter. So the angles have to be the same.
This explains my sink. When I stick my finger in at one focus, it creates waves that travel through the water. The disturbances that make up the waves radiate out until they hit the rim, where they bounce just like a pool ball, coming together at the other focus. Because the gardener's string has a fixed length, these disturbances all get there at the same time. So they reinforce each other, and the water at the second focus goes up and down by a large amount.
This focusing effect has important uses. Some involve a parabola, which is an ellipse with one focus moved away to infinity. Instead of being a closed oval, it's U-shaped. Straight lines emanating from infinity are parallel, and they all bounce off the parabola to hit the second focus, the one that isn't at infinity. So a TV satellite dish shaped like a paraboloid-the surface formed by revolving a parabola around its axis-makes a parallel beam of incoming radio waves converge at the focus. That concentrates the incoming wave energy in one spot, creating a strong signal. The dish picks this up and sends it to your TV.
Radio telescopes, used by astronomers to study the distant cosmos, often use the same trick. Electronics can even make an array of detectors behave like a paraboloid without actually being one. The most sensitive radio telescope yet constructed is LOFAR (LOw Frequency ARray). A few weeks ago it imaged two jets associated with the massive black hole at the centre of our Galaxy. It will eventually synthesise signals from 5,000 detectors in six European countries, simulating a parabolic dish 1000 miles across.
What about genuine ellipses? They're used in lasers. Make a tube with an elliptical cross-section and pump in light from a glass rod at one focus. After being reflected, it piles up at the other focus, where there's no glass rod in the way. This technique can be used to ‘pump' the laser, triggering its ability to amplify light.
There's even a quantum version, where the signal is the wave-function of an atom-a wave of probabilities that bounces just like a water wave or a light wave. Put a real atom at one focus of an elliptical mirror that can reflect quantum waves, and you'll find a copy of the atom's wave-function at the other focus-but no atom. In 2000 researchers at IBM did exactly that using an atom of cobalt. This ‘quantum mirage' lets experimenters probe atoms and molecules, and manipulate quantum states.
When I waggled my finger in the sink, I created a similar mirage with water. It's a vivid illustration of how the same mathematics can show up in many different applications. And it's extraordinary what scientific secrets lurk in your bathroom sink.
