Wavelength and Angular Wave Number

The wavelength $\lambda$ of a progressive wave is the distance, measured along the direction of propagation, between two nearest points of the medium that are in exactly the same phase of oscillation at a given instant -- most easily pictured as the distance from one crest to the very next crest (for a transverse wave) or from one compression to the very next compression (for a longitudinal wave). The angular wave number $k$ is defined as $k = \frac{2\pi}{\lambda}$ with SI unit radian per metre ($\text{rad m}^{-1}$); it measures the rate at which the phase of the wave $(kx-\omega t)$ changes with position $x$ at a fixed instant, exactly analogous to the way angular frequency $\omega$ (below) measures the rate at which phase changes with time at …