10.2 Phase and group velocity

A dispersion relation carries two distinct velocities: the speed of an individual crest and the speed of a packet’s envelope. They coincide only when $\omega$ is linear in $k$ ; whenever it is not, the medium is dispersive, and a wave packet spreads and reshapes as it travels. This lesson separates the two velocities and derives the envelope speed.

Two velocities

Phase velocity $v_p = \omega/k$ — the speed of a single Fourier mode’s crest.
Group velocity $v_g = d\omega/dk$ — the speed of the envelope of a packet built from a narrow band of modes near a carrier $k_0$ .

where

$v_p$: phase velocity — speed of a wave crest m/s
$v_g$: group velocity — speed of the packet envelope m/s
$k_0$: carrier wavenumber of the packet rad/m

The two are equal precisely when $\omega = ck$ with $c$ constant, for then $\omega/k = d\omega/dk = c$ . Any curvature in $\omega(k)$ makes them differ, and a packet — being a superposition of modes with slightly different phase velocities — cannot hold a rigid shape.

The envelope moves at dω/dk

▶ Group velocity from a stationary-phase argument Derivation

Build a packet from a narrow band of plane waves,

\psi(x,t) \;=\; \int A(k)\,e^{i(kx - \omega(k)t)}\,dk,

with $A(k)$ sharply peaked at $k_0$ . Expand $\omega(k)$ to first order about $k_0$ : $\omega(k)\approx\omega(k_0) + (k-k_0)\,\omega'(k_0)$ . Substituting and factoring out the carrier,

\psi(x,t) \;\approx\; e^{i(k_0 x - \omega(k_0)t)}\int A(k)\,e^{i(k-k_0)(x - \omega'(k_0)t)}\,dk.

The integral depends on position and time only through the combination $x - \omega'(k_0)t$ : it is the envelope, translating rigidly at speed $\omega'(k_0)$ . The prefactor is the carrier, moving separately at $\omega(k_0)/k_0$ .

So $v_g = \omega'(k_0)$ carries the envelope — and with it the energy and any signal the packet encodes — while $v_p$ carries the crests through it.

Dispersion ω(k):

Carrier wavenumber k₀ = 1.00

The ω(k) curve is the dispersion relation. The red dashed line is the chord from the origin to k₀ — its slope is the phase velocity v_p. The green dashed line is the tangent at k₀ — its slope is the group velocity v_g. In the packet panel, the green vertical line tracks the envelope centroid; the red ticks track an individual wave crest. For linear dispersion they move together; for nonlinear, they separate, and the envelope also broadens in time.

Three cases make the distinction vivid. For linear $\omega(k)$ the packet is rigid, $v_p = v_g$ . For quadratic $\omega\propto k^2$ the envelope outruns the crests, $v_g = 2v_p$ . For square-root $\omega\propto\sqrt{k}$ the envelope lags, $v_g = v_p/2$ ; crests then appear at the rear of the packet, sweep forward through it, and vanish at the front.

⏳ The history — Rayleigh, group velocity, and the wake of a ship

Rayleigh’s Theory of Sound (1877) was the first systematic English-language treatise on acoustics, and it remains in print and readable. Rayleigh set the concept of group velocity on a firm footing while studying water waves, explaining the curious observation that crests on a propagating disturbance are born at the back of a packet, march forward through it, and die at the front — behaviour that demands the phase velocity differ from the group velocity.

The wake of a ship is the cleanest illustration. The V-shaped Kelvin wedge carries crests moving at the phase velocity of deep-water waves inside an envelope moving at the group velocity. For deep-water gravity waves $v_p = 2v_g$ , so the crests travel twice as fast as the wake pattern itself, continually overtaking its trailing edge and vanishing at its leading one.

The same envelope-versus-carrier analysis reappears in the WKB method of the last lesson of this chapter, and in every dispersive medium from optical fibres to quantum wave packets.

Read the original: The Theory of Sound, Vol. I (Lord Rayleigh (J.W. Strutt), 1877)