The **phase velocity** of a wave is the rate at which the wave propagates in some medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

- $v_{\mathrm {p} }={\frac {\lambda }{T}}.$

Equivalently, in terms of the wave's angular frequency ω, which specifies angular change per unit of time, and wavenumber (or angular wave number) k, which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν_{p},

- $v_{\mathrm {p} }={\frac {\omega }{k}}.$

To understand where this equation comes from, consider a basic cosine wave, *A* cos (*kx*−*ωt*). After time t, the source has produced *ωt/2π = ft* oscillations. After the same time, the initial wave front has propagated away from the source through space to the distance x to fit the same number of oscillations, *kx* = *ωt*.

Thus the propagation velocity *v* is *v* = *x*/*t* = *ω*/*k*. The wave would have to propagate faster when higher frequency oscillations are distributed less densely in space unless the wave length is compensatorily shortened.^{[2]} Formally, *Φ* = *kx*−*ωt* is the phase, where

- ${\frac {\partial x}{\partial t}}=-{\frac {\partial \Phi /\partial t}{\partial \Phi /\partial x}}.$

Since *ω* = −d*Φ*/d*t* and *k* = +d*Φ*/d*x*, the wave velocity is *v* = d*x*/d*t* = *ω*/*k*.

## Relation to group velocity, refractive index and transmission speed

A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line).

Since a pure sine wave cannot convey any information, some change in amplitude or frequency, known as modulation, is required. By combining two sines with slightly different frequencies and wavelengths,

- $\begin{array}{}\end{array}$
Equivalently, in terms of the wave's angular frequency ω, which specifies angular change per unit of time, and wavenumber (or angular wave number) k, which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν_{p},

- $v_{\mathrm {p} }={\frac {\omega }{k}}.$

To understand where this equation comes from, consider a basic cosine wave, *A* cos (*kx*−*ωt*). After time t, the source has produced *ωt/2π = ft* oscillations. After the same time, the initial wave front has propagated away from the source through space to the distance x to fit the same number of oscillations, *kx* = *ωt*.

Thus the propagation velocity *v* is *v* = *x*/*t* = *ω*/*k*. The wave would have to propagate faster when higher frequency oscillations are distributed less densely in space unless the wave length is compensatorily shortened.^{[2]} Formally, *Φ* = *kx*−*ωt* is the phase, where

- ${\frac {{\text{d}}\omega }{{\text{d}}k}}={\frac {c}{n}}-{\frac {ck}{n^{2}}}\cdot {\frac {{\text{d}}n}{{\text{d}}k}}~.$