Waves and Sound: Frequency, Wavelength, Doppler Effect

Energy travels, water stays

Watch a float on the sea. A wave comes through, the float rises and falls, then sits almost where it started. The wave itself has moved on. That is what a wave is: a disturbance that carries energy from one place to another without carrying the material along. The water moves in small circles while the shape travels. The same holds for a wave on a rope, a wave in a stadium and the ripples from a stone. For a typical ocean swell the time between two crests is about 8 s, and the crests are about 100 m apart.

Three numbers describe a wave

The amplitude A is how far the wave rises above the middle level. The wavelength λ (lambda) is the distance from one crest to the next, in metres. The frequency f is how many crests pass per second, in hertz (Hz), and the period is T = 1 / f. A wave moves one wavelength in one period, which gives v = f·λ. The swell with T = 8 s has f = 1 / 8 = 0.125 Hz, and with λ = 100 m its speed is v = 0.125 × 100 = 12.5 m/s, about 45 km/h. Change the frequency in the sandbox and watch the wavelength adjust.

Sound needs something to shake

Sound is a pressure wave. A loudspeaker, a guitar string or your vocal cords push the air; the air molecules bunch together, push their neighbours, then spread apart again. The bunching and spreading travel outward at the speed of sound, while each molecule only jiggles back and forth. That is why sound cannot cross a vacuum. In the animation the dots stand for air: look for the dense bands moving along while each single dot barely leaves its place.

How fast is sound?

In air at 20 °C, sound travels at about 343 m/s, roughly 1235 km/h. The speed grows by about 0.6 m/s for each degree, so at 0 °C it is about 331 m/s. That gives a thunderstorm rule: count 3 s between a flash and its thunder, and the lightning struck about 343 × 3 ≈ 1030 m away, close to a kilometre. A note of 440 Hz, the A used to tune orchestras, has a wavelength λ = v / f = 343 / 440 ≈ 0.78 m.

Pitch and loudness

Frequency sets the pitch. The A above middle C is 440 Hz, and one octave higher is 880 Hz, exactly double. Amplitude sets the loudness: a taller wave pushes the air harder. Healthy young ears hear from about 20 Hz to 20 000 Hz. At 20 Hz the wavelength in air is 343 / 20 ≈ 17 m, and at 20 000 Hz it is 343 / 20 000 ≈ 1.7 cm. Above that range is ultrasound, below it is infrasound. Try the sandbox: move the frequency and amplitude sliders, then press Listen to hear a short tone. Keep your speaker volume low.

The Doppler effect

An ambulance siren sounds higher as it comes toward you and lower as it leaves. The siren does not change; the waves do. Ahead of the moving source the crests are squeezed together, so you meet them more often, and behind it they are stretched. Christian Doppler described the effect in 1842. For a source moving at speed u toward a still listener, the heard frequency is f′ = f·v / (v − u), and moving away it is f′ = f·v / (v + u). A 1000 Hz siren at 20 m/s (72 km/h) is heard at 1000 × 343 / 323 ≈ 1062 Hz approaching and 1000 × 343 / 363 ≈ 945 Hz leaving.

Waves and sound