Displacement as a Function of Time

To describe an oscillating system quantitatively, physics tracks its DISPLACEMENT -- how far, and in which direction, the oscillating quantity has moved away from its own equilibrium (mean, or rest) position -- as a function of time $t$. Exactly what "displacement" means depends on the physical system being described: for a block attached to a spring, the displacement $x(t)$ is simply its linear distance, along the line of motion, from the point where the spring exerts no force at all; for a swinging pendulum, the displacement is more naturally expressed as an angle $\theta(t)$ measured from the vertical (the position at which the pendulum hangs at rest); for a vibrating string or membrane, the displacement is the perpendicular distance of a chosen point from its straight, undisturbed position. Whatever the physical quantity actually oscillating, once its equilibrium position has been fixed as the reference point (where displacement is defined to be zero), the ENTIRE subsequent motion is completely and exactly captured by writing displacement as an explicit function of time, $x(t)$.\n\nThe single most important such function in the whole of oscillatory motion is the sinusoidal function\n\n$x(t) = A\sin(\omega t + \phi) \qquad \text{or equivalently} \qquad x(t) = A\cos(\omega t + \phi)$\n\nThe two forms differ only in where the stopwatch is started (a shift of exactly $\pi/2$ in the phase constant $\phi$ converts one into the other), and either form may be used, whichever proves more convenient for a given set of initial conditions. This particular function is singled out for special attention not merely because it happens to be mathematically simple and easy to work with, but for a much deeper physical reason, wor …