Phase velocity

The phase velocity of a wave is one of two definitions of its velocity, the other being group velocity. It is the velocity at which the phase of a single-frequency component of the wave propagates in space. Imagine fixing a point on the profile of the wave so that its amplitude never changes: the velocity at which that point propagates is the phase velocity.

The phase velocity is only well-defined on waves with a single frequency component, that is monochromatic waves. Consider a monochromatic plane wave of angular frequency $\omega$ and wavenumber $k$ (equivalently, wavevector $\mathbf{k}$ of norm $\lvert \mathbf{k} \rvert\equiv k$). Its phase velocity is

where $\lambda$ and $T$ are the wavelength and period. If the wave is composed of multiple layered frequencies, each of them will have its own phase velocity. The speed of the wave as a whole is then described by group velocity. This is important when dealing with dispersive media, where the two do not match.