Period, frequency and displacement

The period T is the least time after which the motion repeats; its SI unit is the second. Very fast or very slow motions use handier units: a quartz crystal vibrates with periods of microseconds, Mercury's orbital period is 88 earth days, and Halley's comet returns every 76 years.

Frequency ν = 1/T is the number of repetitions per unit time. Its unit s⁻¹ is called the hertz (Hz), after Heinrich Rudolph Hertz: 1 Hz = one oscillation per second. A frequency need not be a whole number.

Worked example: a heart that beats 75 times a minute has ν = 75/60 s = 1.25 Hz and T = 1/1.25 Hz = 0.8 s.

In this chapter displacement means the change with time of whatever quantity is oscillating: the position of a block on a spring measured from equilibrium, the angle of a pendulum from the vertical, the voltage across a capacitor in an AC circuit, the pressure change in a sound wave, or the electric and magnetic fields in a light wave.

A displacement variable can be positive or negative, and its zero is chosen for convenience, usually at the equilibrium position.

f(t) = A cos ωt takes the same value whenever ωt grows by a whole multiple of 2π, so it is periodic with T = 2π/ω. A sin ωt has the same period.

A sin ωt + B cos ωt is periodic with the same period and equals D sin(ωt + φ), where D = √(A² + B²) and φ = tan⁻¹(B/A).

Fourier's theorem: any periodic function can be written as a sum of sine and cosine functions of different periods with suitable coefficients.

A sum of periodic terms repeats after the least time that is a whole multiple of every term's period, so sin ωt + cos 2ωt + sin 4ωt has period 2π/ω. Functions such as e^(−ωt) or log(ωt) never take a value again and are not periodic.