What Is Resonance? Why a Push at the Right Moment Does So Much

Anything that can be displaced and pulled back, a pendulum, a guitar string, a tuning fork, a bridge deck, a column of air in a bottle, will swing back and forth at a rate set by its own properties if it is disturbed and left alone. A short pendulum swings faster than a long one; a taut string vibrates faster than a slack one; a small bell rings higher than a large one. That rate is the object's natural frequency, and most objects have several, one for each pattern in which they can vibrate.

Pluck the string or tap the glass and it oscillates at its natural frequency and gradually dies away as friction and air resistance drain the energy. The rate of that dying away, the damping, turns out to matter as much as the frequency itself.

Now apply a repeating push from outside. If the push cycles far from the natural frequency, the object is forced to move at the push's rate, sluggishly and out of step, and its motion stays small. As the driving frequency approaches the natural one, each push begins to arrive just as the object is moving in the same direction, so the pushes reinforce the motion instead of fighting it. The energy from each cycle is added to the energy already stored, and the amplitude grows.

How large it grows depends on damping. With no friction the amplitude would increase without limit; in reality it climbs until the energy lost each cycle equals the energy added, and settles there. A lightly damped system such as a bell or a crystal has a sharp, tall resonance peak, responding strongly to a narrow band of frequencies. A heavily damped one, a car's suspension or a door closer, barely resonates at all, which is the whole point of the damping.

A radio receives thousands of stations at once and picks out one by resonance. A circuit containing a coil and a capacitor has an electrical natural frequency, set by turning the tuning knob, and it responds strongly to the one station broadcasting at that frequency while the rest pass through almost unnoticed. Quartz watches keep time because a sliver of quartz resonates at exactly 32,768 times a second when driven electrically, and the circuit counts the cycles.

Musical instruments are resonance made audible. A violin string on its own is nearly silent; it is the body of the violin, driven by the string through the bridge, resonating at the frequencies the string supplies, that moves enough air to be heard. Blow across a bottle and the air column inside resonates at the frequency set by its length. The human voice works the same way, with the throat and mouth reshaping to resonate at different frequencies and so form different vowels. Magnetic resonance imaging drives the nuclei of hydrogen atoms in the body at their natural frequency in a strong magnetic field and listens to the reply.

The same effect is dangerous when it is not wanted. An opera singer can shatter a wine glass by holding the note that matches the glass's natural frequency, loudly enough and long enough for the amplitude to exceed what the glass can bend. Soldiers have been ordered to break step on bridges since the nineteenth century after marching in rhythm set several bridges swaying; in 1831 the Broughton suspension bridge near Manchester collapsed under troops for that reason. The Millennium Bridge in London had to close two days after opening in 2000 when crowds unconsciously fell into step with its sideways sway, feeding it, and dampers had to be fitted.

The Tacoma Narrows Bridge, which twisted itself to pieces in a moderate wind in 1940 and was filmed doing it, is the famous case, though engineers now describe it as a related instability called aeroelastic flutter rather than simple resonance. Buildings in earthquake zones are designed so that their natural frequencies avoid the frequencies of ground shaking, and some tall towers carry a huge pendulum near the top, a tuned mass damper, that swings against the building's motion and drains it.

Resonance is easy to demonstrate because it only needs a rhythm and something that can swing:

• •Push a swing at its own rhythm, then try pushing twice as often and watch the height fall
• •Hold a taut string and hum near it until it buzzes at one particular pitch
• •Slide a wet finger round the rim of a glass to drive it at its ringing frequency
• •Hang two pendulums of the same length from one string and set one going; the other starts on its own

Atoms and molecules have natural frequencies too, and they are the most precise ones known. An electron in an atom can only occupy certain energy levels, and it jumps between two of them when it absorbs radiation of exactly the matching frequency. An atomic clock exploits this. It tunes a microwave signal until caesium atoms respond most strongly, meaning the signal has hit their resonance, and then counts its cycles. Since 1967 the second has been defined as 9,192,631,770 of those cycles, because every caesium atom resonates at the same frequency, which no pendulum or quartz crystal can match.

Resonance is also invoked in a common misunderstanding about microwave ovens. They are often said to work by hitting the resonant frequency of water molecules, but that is not quite right. The frequency used, about 2.45 gigahertz, lies well below water's strongest molecular resonances. Instead the rapidly flipping electric field drags polar water molecules back and forth, and their jostling against neighbours turns the energy into heat. The frequency was chosen partly because it was set aside for such uses and partly because it penetrates a few centimetres into food, whereas a sharp water resonance would heat only the outer surface.

Resonance works on the scale of planets too, where the repeated push is gravity. Jupiter's moons Io, Europa and Ganymede are locked in a pattern in which Io goes round four times for every two orbits of Europa and every one of Ganymede, so the three keep meeting at the same points and tugging on each other in the same rhythm. Those regular tugs keep Io's orbit slightly stretched, so that Jupiter's gravity squeezes and relaxes the moon as it goes round. The flexing heats its interior, and Io is the most volcanically active body in the solar system as a result.

The same effect can clear space as well. In the asteroid belt there are gaps, first noticed by Daniel Kirkwood in the nineteenth century, at distances from the Sun where an asteroid would complete exactly three orbits for every one of Jupiter's, or five for every two. An asteroid there feels Jupiter's pull at the same point in its orbit again and again, its path is pumped out of shape, and it is eventually thrown out of the region. Saturn's rings show a similar pattern: the dark Cassini Division lies close to where ring particles would circle twice for every orbit of the moon Mimas.

Resonance is the large response of a system to a driving force that repeats at the system's own natural frequency, when each push adds to the motion the last one left. Damping limits how big it grows. It is what lets a radio select a station and a violin be heard, and it is what engineers design out of bridges and towers.