When I come out of the tearoom with my full mug and take a couple of brisk steps down the corridor, I start the sloshing. If I want to get back to my office without spilling the tea, I have to prevent this sloshing from growing. This is the crux of the problem. As I walk, I can’t help rocking the mug slightly. If the pace of that rocking matches the natural frequency of the sloshing, the sloshing will grow. When you push a child on a swing, you push in a regular rhythm that matches the rate of the swing, and so the swing gets bigger. The same happens with the tea. This is called resonance. The closer the external push is to the natural frequency of the sloshing, the more likely it is that tea will be spilled. The problem for all thirsty humans is that it just so happens that most people walk at a pace that is very close to the natural sloshing frequency of the typical mug. The faster you walk, the closer to it you get. It’s almost as if the system were designed to slow me down, but it’s just an inconvenient coincidence.
So it turns out there isn’t really a satisfactory solution. If I use the tiny mug, it sloshes too fast for my walking pace to make the sloshing worse and the tea won’t spill. But I want more than a thimbleful of tea. If I use the larger mug, my brisk walking is very close to its natural frequency, and disaster is just three steps down the corridor. The only solution is to slow down, so that the rocking from the walking is much slower than the sloshing frequency.? I feel better for having tried, but the lesson here is that I can’t beat the time-dependency of the physics.
Anything that swings—oscillates—will have a natural frequency. It’s fixed by the situation, and the relationship between how hard the pull to equilibrium is and how fast things are going when they get there. The child on the swing is just one example, along with a pendulum, a metronome, a rocking chair, and a tuning fork. When you’re carrying a shopping bag and it seems to be swinging at a rate which doesn’t match your steps, that’s because it’s just swinging at its natural frequency. Big bells have deep notes because their size means that they take a long time to squish and stretch and squish again, so they ring with a low frequency. We get a huge amount of information about the size of objects by listening to them, and it’s because we can hear how long they take to vibrate.
These special timescales are really important for us, because we can use them to control the world. If we don’t want the oscillation to grow, we have to make sure that the system isn’t pushed at its natural frequency. That’s the game with the tea. But if we want an oscillation to continue without much effort, we choose to nudge it along at its natural frequency. And it’s not just people who use this. Dogs do, too.
Inca is poised and ready, focused on the tennis ball like a sprinter waiting for the starting gun. As I lift the plastic arm holding the ball, she tenses, and then the ball sails over her head and she’s off, a slim bundle of enthusiasm and seemingly endless energy. Her owner, Campbell, and I chat while Inca rushes happily across the scrubby grass. She doesn’t bring the thrown ball back to us, because she’s already got a second tennis ball in her mouth (apparently this is a “spaniel thing”), but when she reaches it she stands guard until we catch her up and lob the first ball farther ahead. After half an hour of non-stop chasing, she finally sits down, tail cheerily swishing the grass, and looks up at us, panting.
I kneel down and stroke her back. All that running around has made her hot. She isn’t sweaty because dogs don’t sweat, but she still has to get rid of all that excess heat. The panting looks like hard work, presumably using lots of energy and generating even more heat. It seems like a bit of a paradox. Inca is untroubled by my ponderings but quite happy to be stroked, and a strand of saliva drips from her wide-open mouth. After I’ve been out running, my breathing rate comes down back to normal quite gradually, but when Inca stops panting it happens very suddenly. Big brown eyes look up at me, and I wonder how much longer she needs to recover before it’s time for more tennis balls.
By far the most efficient way to lose heat is to evaporate water. That’s why we sweat. Turning liquid water into a gas takes a huge amount of energy, and conveniently the gas then floats away, taking that energy with it. Since dogs don’t sweat, they don’t produce water on their skin that can evaporate, but they have plenty of water in their nasal passages. Panting is all about pushing as much air as possible over the wet insides of their noses, to get rid of heat quickly. As if to demonstrate the point, Inca starts panting again. I reckon she’s taking about three breaths each second, which seems like a lot of hard work. But the really clever bit is that it isn’t. Her lungs act as an oscillator. This is the most efficient rate for her to breathe at because it’s the natural frequency of her lungs. As she breathes in, she’s stretching the elastic walls of the lungs, and after a while, the elastic pushes back strongly enough to turn the cycle around. Just as the lungs get back to their unstretched size, she puts in a tiny bit of energy to send them off on the cycle again. The downside is that when she’s breathing this fast, she’s not really replacing the air deep in her lungs, so she isn’t actually taking on board much extra oxygen while all this is going on. That’s why she doesn’t breathe like this all the time. But just at the moment the need to lose heat trumps her need for oxygen, and by pushing her lungs at exactly the right frequency, she’s getting as much air through her nose for as little effort as possible. So the panting is generating a tiny amount of heat compared to what she’s losing. She’s breathing in through her nose, but she’s got her mouth wide open because the dribbling is also cooling her. When the saliva evaporates, that helps lose a bit of heat energy too. The panting stops again, and Inca eyes the abandoned tennis ball. One inquiring look at Campbell is enough (he’s well trained) and the game begins again.
The natural frequency of something depends on its shape and what it’s made of, but the biggest factor is its size. This is why smaller dogs pant faster. They’ve got tiny lungs, which naturally inflate and deflate many more times each second. Panting is a very efficient way of losing heat if you’re small. But it gets less efficient as you get bigger, and that may be why larger animals sweat instead (especially hairless ones like us).
Every object has a natural frequency, and often more than one if there are different possible patterns of vibration. As the objects get bigger, those frequencies generally get lower. It can take quite a push to make a really massive object move, but even a building can vibrate, very very slowly. A building can in fact behave a bit like a metronome, a sort of upside-down pendulum—the base is fixed while the top moves. Higher up, the wind is faster than at ground level, and this is enough to give tall narrow buildings the sort of shove that will start them swaying at their natural frequency. If you’ve ever been in a tall building on a very windy day, you’ve probably felt this. A single cycle can take a few seconds. It’s disconcerting for humans inside, so the architects of these buildings spend a lot of time working out how to reduce the swaying. They can’t remove it completely, but they can change the frequency and flexibility to make it less noticeable. If you feel it happening, don’t worry—the building will have been designed to bend, and it isn’t going to fall.