Periodic Motion: Period, Frequency and Periodic Functions

A motion is called PERIODIC if it repeats itself, identically, after equal intervals of time. The smallest such interval of time after which the motion FIRST repeats -- so that the system returns to exactly the same state of displacement and velocity it was in before -- is called the PERIOD of the motion, conventionally written $T$ and measured in seconds (s). Closely related to the period is the FREQUENCY, the number of complete repetitions (oscillations, revolutions, or cycles) occurring in one second:\n\n$f = \frac{1}{T}$\n\nmeasured in hertz (Hz), where $1\ \text{Hz}$ means one complete cycle per second. A closely related quantity used throughout the mathematics of oscillatory motion is the ANGULAR FREQUENCY,\n\n$\omega = 2\pi f = \frac{2\pi}{T}$\n\nmeasured in radians per second (rad/s). Angular frequency is preferred in most formulas because it lets a repeating pattern be written directly as a sine or cosine of $\omega t$, with a full cycle corresponding to the angle increasing by exactly $2\pi$ radians.\n\nMathematically, a function $f(t)$ is called PERIODIC with period $T$ if\n\n$f(t+T) = f(t) \quad \text{for every value of } t$\n\nthat is: whatever the function's value at some instant $t$, it returns to exactly that same value -- and keeps changing in exactly the same way from there onward -- one full period $T$ later, and this keeps happening forever, for every later cycle too.\n\nThe familiar sine and cosine functions are the simplest and most important periodic functions: both $\sin(\omega t)$ and $\cos(\omega t)$ repeat with period $T = 2\pi/\omega$. But they are far from the ONLY periodic functions that occur in nature or in the laboratory. A triangular wave (rising and falling in straight-line ramps), a square wave (jumping abruptly between two fixed values), and the far more irregular-looking position-versus-time graph traced out by a real vibrating guitar string are all periodic functions too, even though none of them looks anything like a simple sine curve. It is a remarkable mathematical fact -- first proved by Joseph Fourier -- that ANY periodic function, however complicated or irregular its shape within one p …