What are waves and resonance?

Flick one end of a long rope and a bump runs away from your hand. Keep shaking and you send a whole train of bumps. That moving pattern is a wave. Watch the red bead: it only goes up and down. The rope stays where it is. What travels is the shape, and the energy it carries.

Three numbers describe it. The frequency, f, is how many times a second your hand shakes, in hertz (Hz). The wavelength, λ (lambda), is the distance from one crest to the next. The speed, v, is how fast a crest moves.

Each shake sends out one wavelength, and you make f of them every second. So in one second the wave moves f wavelengths: v = f × λ. That one line works for ropes, sound, light and ripples on a pond.

Here is the surprise. You can't make the wave faster by shaking faster. The speed belongs to the rope: pull it tighter, or use a lighter rope, and waves run faster, v = √(tension ÷ mass per metre). Shake faster and the crests just crowd closer together. The wavelength is the thing that gives: λ = v ÷ f.

A rope wave is transverse: the rope moves across the way the wave goes. Switch to the slinky and the coils move back and forth along the wave, bunching and spreading. That is a longitudinal wave, and it is exactly how sound moves through air.

A loudspeaker cone jerks forward and squeezes the air in front of it. That squeezed layer pushes the next one, and so on. Then the cone pulls back and leaves a thinner patch behind. The result is a pressure wave: bunches of air (compressions) and gaps (rarefactions) running away at the speed of sound, about 343 m/s in air at 20 °C. Each bit of air only jiggles back and forth. It is a longitudinal wave, like the slinky.

Your ear hears the frequency as pitch. The A above middle C is 440 Hz, so its wavelength is 343 ÷ 440 = 78 cm. Go up an octave and the frequency doubles. People hear roughly 20 Hz to 20,000 Hz: wavelengths from 17 m down to under 2 cm. Every piano key is 1.0595 times the one below it (see PianoClear).

Now pin a string at both ends. Waves run along it, bounce off the ends and overlap. Only certain patterns survive: those that fit a whole number of half-wavelengths into the string. They are standing waves. Some points never move (nodes); others swing the most (antinodes).

That gives the harmonics: fₙ = n × v ÷ 2L. A guitar's low E string, 648 mm long, rings at 82 Hz, 165 Hz, 247 Hz… all at once, and the mix is its tone. Press a fret and you shorten L, so the pitch rises; the 12th fret halves the string and plays an octave (see GuitarClear).

A flute does the same with a column of air. Open at both ends, it follows the same rule with v = 343 m/s. Blow harder and it jumps to the second harmonic, an octave up (see FluteClear).

In 1865 James Clerk Maxwell worked out that a wobbling electric field makes a wobbling magnetic field, which makes an electric field again, and the pair can travel on their own through empty space. His maths said they go at about 300,000 km/s: the speed of light. So light is an electromagnetic wave. In 1887 Heinrich Hertz made invisible ones with sparks, and radio was born.

All of them obey the same line as the rope: c = f × λ, with the same speed c. Only the frequency changes, and with it the wavelength and the name: radio, microwaves, infrared, visible light, ultraviolet, X-rays.

A microwave oven works at 2.45 GHz, so λ = 3 × 10⁸ ÷ 2.45 × 10⁹ = 12.2 cm. The waves bounce off the metal walls and form a standing wave, with hot spots about half a wavelength, 6.1 cm, apart. That's why the turntable turns (see MicrowaveClear). Stop it, melt a bar of chocolate, measure the gap between melted spots and you can work out the speed of light.

Why can you see through the door but the microwaves can't get out? The metal mesh has holes about 1–2 mm wide. A hole lets a wave through only if it is wider than about half a wavelength. Microwaves are 12 cm long, so they bounce back. Light is 0.0005 mm long and sails through.

Your TV makes every colour from tiny red, green and blue sub-pixels, about 630, 530 and 460 nanometres (see TVClear). The camera sorts light by the same wavelengths with red, green and blue filters, and its lens bends light because light slows down in glass (see CameraClear).

Pull a swing back and let go. It swings to and fro at its own pace, about once every 3 seconds for a 2.4 m swing. That pace is its natural frequency. Every object that can wobble has one: a guitar string, a bridge, a wine glass, a building.

Now push it. Push at a random rhythm and your pushes fight each other. Push in step with the swing, once each time it comes back, and every push adds a little more. The swing climbs higher and higher. That is resonance: driving something at its natural frequency so small pushes build a big motion.

What stops it growing forever? Damping: friction and air drag take a little energy each swing. At resonance the swing settles where the energy you add equals the energy lost. Less damping means a taller, sharper peak.

A washing machine meets resonance every spin. The tub hangs on springs, so it has a natural frequency of a few hertz. A lump of wet clothes shakes it as the drum turns. As the drum spins up through about 200 rpm, it passes that frequency and the machine shudders, then calms down at full speed. A badly unbalanced load can make it walk across the floor (see WasherClear).

A wine glass rings at one clear note when you tap it. Sing or play that exact note loudly enough and the rim flexes more and more until it cracks. Musical instruments use resonance on purpose: a tabla's black spot, the syahi, tunes its drumhead so its overtones fall into a musical series, which C. V. Raman explained in 1920 (see TablaClear). Your inner ear uses it too: each part of the cochlea resonates with a different pitch (see EarClear). Even MRI is resonance: radio waves at exactly 64 MHz tip the hydrogen nuclei in your body in a 1.5 tesla magnet (see MRIClear).

When two waves meet they simply add. Crest on crest makes a bigger crest; crest on trough makes nothing at all. This is interference. Drop two pebbles in a pond and you see calm lines fanning out between the ripples, where the waves cancel. In 1801 Thomas Young used exactly this to show that light is a wave: two slits make bright and dark bands. Noise-cancelling headphones do it on purpose, playing an upside-down copy of the noise.

Two notes that are nearly the same pitch interfere in time. They drift in and out of step, so you hear a wah-wah-wah: beats, as many per second as the difference in frequency. Tune a guitar, a harmonium or a tanpura by listening for the beats to slow down and stop.

Myth-buster. In 1940 the Tacoma Narrows Bridge twisted itself apart in a 64 km/h wind, and many books say "resonance". But the wind was steady: there was no rhythm to match. The swirls of air shed by the deck came about 0.9 times a second, not the bridge's 0.2. What really happened is flutter: above a critical speed, the twisting deck bent the airflow so that the wind pushed with each twist. It is negative damping, and each twist fed the next.

Last surprise. Sound needs a medium, something to squeeze. Pump the air out of a jar and a ringing bell goes silent, as Robert Boyle and Robert Hooke showed in 1660. Light doesn't: its electric and magnetic fields carry themselves. That is why sunlight reaches us across 150 million km of empty space, while space itself is silent.