# Where does the simple harmonic motion equation come from and what does it mean?

We know that the displacement, x, is described by the equation x = Acos(ωt + Φ), where A is the amplitude of oscillation, ω is the angular frequency and Φ is the phase shift. The velocity, v, is the time derivative of displacement (v = dx/dt), so differentiating both sides with respect to time t gives v = -ωAsin(ωt + Φ). The acceleration, a, is the time derivative of velocity (a = dv/dt), so differentiating both sides with respect to time again gives a = -ω2Acos(ωt + Φ) = -ω2x. This is the defining equation of simple harmonic motion: it states that the acceleration is proportional (since ω2 is a constant) and in the opposite direction (due to the negative sign) to the displacement.
This can be more easily visualised by sketching the curves for displacement, velocity and acceleration. Assuming the phase shift Φ = 0, the displacement x = Acos(ωt) and is described by a cosine curve. The velocity is described by an upside-down sine curve, and the acceleration is described by an upside-down cosine curve. So the acceleration curve is the same as the displacement curve, but reflected in the x-axis.