Periodic motion

A motion that repeats itself at regular intervals is periodic. The hands of a clock, the earth spinning once a day and going round the sun once a year, and your heartbeat are all periodic.

• The period T is the least time after which the motion repeats.
• The frequency ν = 1/T is the number of repeats per second, in hertz (1 Hz = 1 s⁻¹).

An oscillation (or vibration) is a special periodic motion: to and fro about a mean position, where the net force on the body is zero. The body goes out to an extreme position on one side, comes back through the mean, goes out the same distance on the other side, and returns: that is one oscillation. A pendulum, a mass on a spring, a car engine's piston and a plucked guitar string all oscillate.

Every oscillation is periodic, but not every periodic motion is an oscillation: the earth going round the sun repeats, but it does not swing back and forth.

In maths, sin ωt and cos ωt repeat when ωt grows by 2π, so their period is T = 2π/ω. Functions made of a single sine or cosine are called harmonic; every harmonic function is periodic, but many periodic graphs (a saw-tooth, a square wave) are not harmonic.

• A sum of a sine and a cosine with the same ω is still one harmonic function: 3 sin ωt + 4 cos ωt = 5 sin(ωt + φ), with tan φ = 4/3. Its period is still 2π/ω.
• A sum with different ω's is periodic if one period fits a whole number of times into the others: sin ωt + sin 3ωt repeats after 2π/ω, the longest of its periods.
• Fourier showed that any periodic function can be built from sines and cosines.

A function that only grows or only shrinks can never come back to an earlier value, so it is not periodic: e^(–ωt) only falls, and log(ωt) only rises.