A harmonic wave is a disturbance that varies in space and time according to a sinusoidal function—a periodic trigonometric function of the form
(1) y(x, t) = A cos(kx – ωt + φ)
that describes a smooth, regular oscillation between a maximum and a minimum value. Phase velocity and group velocity represent two distinct metrics for measuring the propagation of such a wave, depending on whether the focus is on an individual oscillation or on the overall envelope structure of the signal.
When considering an ideal monochromatic wave, the phase velocity (vf = ω / k) quantifies the rate at which a point of constant phase moves through space—that is, the displacement speed of a single crest or trough of the sinusoid. However, physical waves capable of transmitting information are never purely monochromatic; they consist of a superposition of spectral components across a range of frequencies (1), forming what is known as a wave packet. The group velocity (vg = dω / dk) precisely measures the propagation speed of the overall envelope of this wave packet.
The core distinction between these two quantities lies in their underlying physical significance. While the phase velocity describes strictly the motion of the oscillatory state and can, in specialized media such as certain plasmas or metamaterials, exceed the speed of light in vacuum c, the group velocity represents the effective speed at which physical energy and signal information travel.
In non-dispersive media—where all spectral components propagate at the same speed—the two values are identical (vf = vg). Conversely, in dispersive media, the frequency dependence of the medium’s properties causes a dispersion relation where the two velocities diverge, typically resulting in a group velocity that is lower than the phase velocity (vg < vf).
In the image the red dot propagates with phase velocity while the green dots propagate with group velocity, source Wikipedia.
