Simple harmonic motion: amplitude, phase and angular frequency

A particle moving back and forth between x = −A and x = +A is in SHM when its displacement is x(t) = A cos(ωt + φ) with A, ω and φ constant, that is, when displacement is a sinusoidal function of time. SHM is not just any periodic motion.

Amplitude A is the magnitude of the greatest displacement from the mean position; it can always be taken as positive.

Phase is the time-dependent argument (ωt + φ). Given the amplitude, the phase fixes the particle's state of motion, its position and velocity, at time t.

The phase constant (phase angle) φ is the value of the phase at t = 0. If A is known, φ follows from the displacement at t = 0.

Angular frequency ω is linked to the period by ω = 2π/T = 2πν; its SI unit is the radian per second. It follows from x(t) = x(t + T), because cosine first repeats when its argument grows by 2π.

Two SHMs can share ω and φ but differ in amplitude, share A and ω but differ in phase constant, or share A and φ but differ in ω; halving the period doubles the frequency.

The particle moves fastest as it passes x = 0 and stops momentarily at the extremes x = ±A. The period stays the same whatever instant is chosen as t = 0.

sin ωt − cos ωt = √2 sin(ωt − π/4) is SHM with period 2π/ω, amplitude √2 and phase angle −π/4 (or 7π/4).

sin² ωt = ½ − ½ cos 2ωt is periodic with period π/ω; it is a harmonic motion about the point ½, not about zero.

An angle written without a unit is in radians: sin(15) is the sine of 15 radians, and a value in degrees must carry the degree sign.