Period, Frequency and Angular Frequency

The period $T$ of a progressive wave is the time taken by any one particle of the medium to complete one full cycle of its oscillation, and the frequency $f=1/T$ is the number of complete oscillations that particle makes per second, measured in hertz (Hz). The angular frequency is $\omega = \frac{2\pi}{T} = 2\pi f$ with SI unit radian per second ($\text{rad s}^{-1}$); it measures the rate at which the phase $(kx-\omega t)$ changes with time $t$ at a fixed position, just as $k$ measures its rate of change with position at fixed time. Because the wave advances exactly one wavelength $\lambda$ in exactly one period $T$ (that is how far the whole pattern shifts before it repeats), the wave's speed of propagation $v$ is $v = \frac{\lambda}{T} = f\lambda = \frac{\omega}{k}$ This relation, $v=f\lambda$, is purely kinematic -- a direct consequence of how wavelength, period and speed are defined -- and it holds for every …