Phase velocity
Speed of a wave's constant-phase surface.
Phase velocity is the speed at which a wavefront—a surface of constant phase—travels through a medium. For any single-frequency component of a wave, a specific phase, such as a crest or trough, appears to move at this velocity. In the case of a simple sinusoidal wave, the phase velocity can be expressed as the ratio of the wavelength to the wave’s time period, or equivalently as the angular frequency divided by the wavenumber. This definition applies not only to isolated waves but also to more complex signals, such as beats, where the phase velocity of the carrier wave determines the phase velocity of the entire wave set.
The behavior of phase velocity is closely tied to the phenomenon of dispersion, where the frequency of a wave is a function of its wavenumber. In electromagnetics and optics, the refractive index of a medium is defined as the ratio of the speed of light in vacuum to the phase velocity. When the refractive index varies with frequency, the medium is dispersive; when it does not, the medium is non-dispersive, and the group velocity equals the phase velocity. For surface gravity waves on deep water, the phase velocity is twice the group velocity. In such wave groups, new waves appear to emerge at the back of the group, grow in amplitude toward the center, and vanish at the front. The water particle velocities in these waves are much smaller than the phase velocity. Notably, the phase velocity of light waves is not a physically meaningful quantity for information transfer, as it does not relate to the speed of signals. In some wave packets, the phase velocity and group velocity can even travel in opposite directions, with the envelope moving rightward while the crests and troughs move leftward.
- definition
- Speed of a wavefront, surface of constant phase
- formula_sinusoidal
- v_p = λ / T
- formula_angular
- v_p = ω / k
- relation_to_refractive_index
- n = c / v_p = ck / ω
- non_dispersive_condition
- ∂n/∂ω = 0
- group_velocity_formula
- v_g = c / (n + ω ∂n/∂ω)
Lore & Background
The phase velocity of a wave is defined as the speed at which any surface of constant phase, such as a wavefront, propagates. For a single-frequency component of a wave, this is the velocity at which a particular phase—for instance, a crest—appears to travel. In the case of a simple sinusoidal wave, the phase velocity can be expressed in terms of the wavelength and the time period, or equivalently through the angular frequency and the wavenumber. This concept extends beyond isolated waves to beats or signals composed of multiple waves, where the phase velocity of the carrier wave determines the phase velocity of the overall wave set.
In dispersive media, such as those encountered in electromagnetics and optics, the frequency is a function of the wavenumber, meaning the phase velocity and group velocity depend on both the medium and the frequency. The ratio of the speed of light to the phase velocity defines the refractive index. The group velocity equals the phase velocity only when the refractive index is independent of frequency, a condition known as a non-dispersive medium. In dispersive media, various properties change with frequency, and the relationship between frequency and wavenumber is called the dispersion relation.
A notable example occurs with surface gravity waves on deep water. In this case, the phase velocity is twice the group velocity. Water particle velocities are generally much smaller than the phase velocity. Visualizations of such waves show that new wave crests appear to emerge at the back of a wave group, grow in amplitude toward the center, and vanish at the front. Additionally, wave packets can exhibit a phase velocity greater than the group velocity, or the group velocity and phase velocity may travel in opposite directions, with the envelope moving one way while the peaks and troughs move the other. The phase velocity of light waves is not a physically meaningful quantity and is unrelated to information transfer.
Reader's Guide
Phase velocity is a fundamental concept in wave physics, describing the speed of a wave's phase fronts. For a simple sinusoidal wave, it is given by v_p = λ / T or v_p = ω / k. The definition extends to beats or signals composed of multiple waves, where the phase velocity of the carrier determines the phase velocity of the wave set. In electromagnetics and optics, the frequency is a function of wavenumber, so phase velocity and group velocity depend on the medium and frequency. The refractive index is defined as n = c / v_p. Group velocity equals phase velocity only when the refractive index is independent of frequency (non-dispersive medium). The relation ω(k) is known as the dispersion relation of the medium.
Did You Know?
- Phase velocity is the speed of any wavefront, a surface of constant phase.
- For a simple sinusoidal wave, phase velocity equals wavelength divided by time period.
- The phase velocity of light waves is not a physically meaningful quantity and is not related to information transfer.
- Group velocity equals phase velocity only when the refractive index is independent of frequency.
More in Optics And Photonics 1-24
Spotted an error? Know more?
This is a living reference — every entry is fact-audited, and reader corrections feed straight into our audit queue. Suggest an edit · See this site's audit record