What is a wave? Frequency, wavelength and sound, explained

A wave is a disturbance that travels through a material or through space, carrying energy from one place to another without carrying the material with it. Sound is a wave of pressure in air, a ripple is a wave on water, and light is a wave of electric and magnetic fields. Every wave is described by the same few quantities, amplitude, wavelength, frequency and speed, linked by one formula: speed = frequency × wavelength.

In short

• A wave moves energy, not stuff. The water in an ocean wave mostly goes round in circles while the wave crosses the sea.
• Frequency is how many waves pass each second, wavelength is the distance between crests, and their product is the wave's speed: v = f × λ.
• Sound is a wave of squeezed and stretched air. It needs a material to travel through, and it travels faster in water and steel than in air.

What is a wave, and what actually moves?

The defining fact about a wave is that the pattern travels while the material stays roughly where it was. A stadium crowd doing "the wave" shows it: each person only stands and sits, yet a bulge races around the stands. Flick one end of a rope tied to a wall and a hump runs along it, though every bit of rope only moves up and back down.

Ocean waves are the classic case people get wrong. A wave rolling toward the beach looks like water rushing landward, but, as NOAA explains, waves transmit energy, not water, across the ocean: the water moves mostly in circles as each wave passes. A floating bottle bobs up, forward, down and back, and ends up almost where it started. Only when a wave reaches shallow water and breaks does the water itself surge up the beach.

Waves come in two shapes of motion. In a transverse wave, the material moves at right angles to the direction the wave travels, like the rope. In a longitudinal wave, it moves back and forth along the direction of travel, squeezing together and spreading apart. Sound in air is longitudinal. Light is a transverse wave, but of electric and magnetic fields, and needs no material at all.

What are amplitude, wavelength and frequency?

Four numbers describe a simple wave.

• Amplitude is how far the disturbance swings from rest: the height of a water wave's crest above the calm surface, or how much a sound wave squeezes the air. Bigger amplitude means more energy; the energy a wave carries grows with the square of its amplitude.
• Wavelength, written λ (the Greek letter lambda), is the distance from one crest to the next, in meters.
• Frequency, f, is how many crests pass a point each second, in hertz (Hz). One hertz is one wave per second. The time between crests, the period, is 1 ÷ f.
• Speed, v, is how fast the crests move, in meters per second.

These are tied together by v = f × λ: a wave's speed equals its frequency times its wavelength. The logic is simple: if f crests pass each second and they are λ meters apart, the pattern moves f × λ meters each second. Drawn as a graph, the simplest wave is a sine curve, the smooth repeating function from trigonometry, and any more complicated wave can be built by adding sine waves together.

A worked example: middle C in air and water, and an FM station

Middle C on a piano has a frequency of about 261.63 Hz. Sound travels through air at 20 °C at about 343 m/s. Rearranging v = f × λ gives λ = v ÷ f.

1. In air: λ = 343 ÷ 261.63 = 1.31 meters. Each squeeze of air is about 1.3 m behind the last.
2. In fresh water at 20 °C, where sound travels at about 1,481 m/s: λ = 1,481 ÷ 261.63 = 5.66 m. The frequency is set by the piano string, so it doesn't change when the sound enters the water; the wavelength stretches instead.
3. An FM radio station broadcasting at 100 MHz, 100 million hertz, sends a wave that travels at the speed of light, about 3.0 × 10⁸ m/s: λ = 300,000,000 ÷ 100,000,000 = 3.0 m. A radio wave of 3 m and a sound wave of 1.3 m are entirely different things, one of fields and one of air, but the formula is the same.

The same arithmetic gives the 12.5 cm waves of 2.4 GHz Wi-Fi, and explains why a simple FM antenna, a quarter of a wavelength long, is about 75 cm.

How does sound travel?

Sound is a longitudinal pressure wave. A vibrating object, a guitar string or a loudspeaker cone, pushes on the air next to it, squeezing it into a region of higher pressure, then pulls back, leaving a region of lower pressure. Each layer of air pushes the next, and the pattern of squeezes and stretches travels outward. The air molecules themselves only jostle back and forth.

What you hear depends on the wave. Frequency is pitch: a young adult with healthy hearing hears roughly 20 Hz to 20,000 Hz, and the upper limit falls with age. Amplitude is loudness. The shape of the wave, the mix of frequencies in it, is why a piano and a violin playing the same note sound different.

Because sound is a disturbance of a material, it can't cross a vacuum: in space, no one hears an explosion. And it travels faster in water than in air, and faster again in steel, about 5,900 m/s. That surprises people, since water is far denser than air and you might expect the heavier material to respond more sluggishly. Speed depends on two things working against each other: stiffness, how strongly the material springs back when squeezed, which speeds sound up, and density, which slows it down. Water is about 800 times denser than air but more than 10,000 times harder to compress, so stiffness wins.

What happens when two waves meet?

When two waves overlap, their disturbances simply add, and then each carries on unchanged, a rule called superposition. Where two crests meet, the result is a bigger crest: constructive interference. Where a crest meets a trough, they cancel: destructive interference. Noise-canceling headphones use this, measuring incoming sound with a microphone and playing the same wave inverted, so the two cancel in your ear.

Waves also bend around obstacles and spread through gaps, which is called diffraction, strongest when the gap is about the size of the wavelength. Sound waves a meter long bend easily around a doorway, which is why you can hear people in the next room without seeing them; light's wavelengths are less than a thousandth of a millimeter, so it doesn't. Interference and diffraction are the fingerprints of a wave, and it was by finding them in light that physicists proved light is a wave.

What is the Doppler effect?

The Doppler effect is the change in frequency you hear when a source of waves moves toward or away from you. An ambulance siren sounds higher as it approaches and drops in pitch the moment it passes. As the source moves toward you, it is partly catching up with the waves it has just sent, so the crests bunch up: shorter wavelength, higher frequency. Moving away, they spread out.

For a sound source moving straight toward a still listener, the heard frequency is f × v ÷ (v − s), where v is the speed of sound and s the speed of the source; moving away, it is f × v ÷ (v + s). Take a siren at 700 Hz on an ambulance doing 25 m/s, about 90 km/h, in air at 343 m/s. Approaching: 700 × 343 ÷ 318 = 755 Hz. Leaving: 700 × 343 ÷ 368 = 652 Hz. The drop of about 100 Hz is the familiar "neee-oww".

The same effect works for light. Galaxies moving away from us have their light stretched toward longer, redder wavelengths, a redshift, and that measurement is how astronomers found that the universe is expanding (for distant galaxies the stretch comes from space itself expanding, but nearby it behaves like a Doppler shift), a key piece of evidence for the Big Bang. Police radar guns and weather radar measure speed with the Doppler shift of radio waves.

Waves are one of the main ways energy gets from place to place: sunlight, sound and earthquakes all deliver it as waves. Light and radio are the most important family, covered across the electromagnetic spectrum.

Check yourself

• Without looking back: if a wave carries energy across the ocean, why doesn't the water travel with it?
• When middle C passes from air into water, which changes, its frequency or its wavelength, and why?
• Explain why an ambulance siren drops in pitch as it passes you.

Sources

• Why does the ocean have waves? — National Ocean Service, NOAA
• 16.1 Traveling Waves, in University Physics Volume 1 — OpenStax, Rice University
• 16.5 Interference of Waves, in University Physics Volume 1 — OpenStax, Rice University
• 17.7 The Doppler Effect, in University Physics Volume 1 — OpenStax, Rice University
• Speed of sound — Wikipedia
• C (musical note) — Wikipedia