Phase Velocity — Physica

Equivalent forms

v_p = \lambda f

v_p = \frac{\omega}{k}

v_p = \frac{\lambda}{T}

The crest speed is just the product of how often (f) and how far apart (lambda) — two everyday quantities multiply to a velocity.

Symbol	Name	SI unit	Dimension	Range
$f$	Frequency Oscillations per second	Hz	T^-1	1 – 1000
$lambda$	Wavelength Distance between successive crests	m	L	0.01 – 10
$v_p$	Phase velocityoutput Speed of a point of constant phase	m/s	L T^-1	0 – 5000

Unit systems

SI:: f in Hz, lambda in m, v_p in m/s
CGS:: lambda in cm, v_p in cm/s
Imperial:: v_p in ft/s

Where it holds

Defines the speed of a single Fourier component. For a wave packet carrying information, the group velocity is the relevant signal speed; in anomalous dispersion v_p can exceed c without violating relativity.

Discovery

Lord Rayleigh · 1877

Rayleigh, in his 'Theory of Sound', sharply distinguished the velocity of a single wave's phase from the velocity of a group of waves (a wave packet), resolving long-standing confusion about which speed carries energy and signal.

Try this

How fast does a single crest of a wave actually move?

Watch one crest of a pure sine wave and time how fast it slides forward. That speed — the phase velocity — is set entirely by the wave's frequency and wavelength.

Research status: stable

Common misconceptions

Phase velocity is not the speed of energy or information; a single infinite sine wave carries neither. v_p > c is allowed and observed in plasmas.

Derivation

A traveling wave is

y = A \cos (k x - \omega t)

.

A point of constant phase satisfies

k x - \omega t =

const, so

dx/dt = \omega /k = v_p

.

Since

\omega = 2\,\mathrm{pi} f

and

k = 2\,\mathrm{pi}/\lambda

,

v_p = \omega /k = f \lambda

.

Limiting cases

Non-dispersive medium⟶ v_p constant for all f — packets keep their shape

Dispersive medium⟶ v_p depends on f; phase and group velocities differ

Group Velocity→Wave Speed Equation→Wave Equation→Optical Dispersion→