Group velocity and dispersion

Every wave packet is secretly two waves at once. Zoom in and you see fast ripples racing along at the phase velocity; zoom out and you see the overall "blob" — the envelope carrying the energy — drifting at the group velocity. For an ideal string or light in vacuum these two speeds happen to coincide, and a pulse glides along without changing shape. But most real media are not so obliging: a medium is called dispersive when the phase velocity itself depends on wavenumber, and then the envelope speed and the ripple speed pull apart. The consequence is not subtle — a sharp pulse sent into a dispersive medium spreads out as it travels, because its different wavenumber components drift out of step with one another. This article turns that qualitative story into two clean formulas, $v_p=\omega/k$ and $v_g=d\omega/dk$ , shows how a given dispersion relation $\omega(k)$ decides whether $v_g=v_p$ or not, and works both cases numerically — a non-dispersive string and a dispersive water-wave relation — so you can read off dispersive behaviour from $\omega(k)$ on sight.

The core insight

A wave packet's envelope moves at the group velocity $v_g=d\omega/dk$ , which equals the phase velocity $v_p=\omega/k$ only when $\omega(k)$ is linear (non-dispersive); when $\omega(k)$ is nonlinear, $v_g\neq v_p$ and the packet spreads as it propagates.

The core idea

Two different speeds live inside one wave packet.

Phase velocity is the speed of a single crest of the underlying oscillation, for a plane wave $y=A\cos(kx-\omega t)$ :

v_p = \frac{\omega}{k}.

where: $\omega$ is angular frequency (rad/s) and $k$ is wavenumber (rad/m).

Group velocity is the speed of the envelope — the shape that actually carries the packet's energy and information — for a narrow band of wavenumbers centred on $k_0$ :

v_g = \left.\frac{d\omega}{dk}\right|_{k_0}.

where: the derivative is evaluated at the packet's central wavenumber $k_0$ . The two-wave beat picture makes this concrete. Superposing two waves of nearly equal $k$ produces an envelope factor $\cos\!\big(\tfrac{\Delta k}{2}x-\tfrac{\Delta\omega}{2}t\big)$ , and following a point of constant envelope phase gives $x=\dfrac{\Delta\omega}{\Delta k}t+\text{const}$ — the envelope itself travels at $\Delta\omega/\Delta k$ . Narrowing the band ( $\Delta k\to 0$ ) turns that ratio into the derivative $d\omega/dk$ , which is exactly $v_g$ . Since a packet's energy sits where the envelope is large, $v_g$ , not $v_p$ , is the physically meaningful "speed of the wave" for any localized signal.

The dispersion relation $\omega(k)$ decides everything. It is fixed by the medium (tension and density for a string, elastic constants for sound, Maxwell's equations for light in vacuum), and its shape tells you at once whether the two velocities agree.

Non-dispersive: $\omega=vk$ with $v$ constant gives $v_p=v$ and $v_g=v$ — every wavenumber travels at the same speed, so a packet keeps its shape.
Dispersive: $\omega(k)$ is nonlinear, so $v_p=\omega/k$ varies with $k$ and in general $v_g\neq v_p$ ; a packet built from a band of $k$ 's spreads out because its components drift apart in phase.

A useful bridge between the two velocities, when $v_p(k)$ is known instead of $\omega(k)$ , comes from writing $\omega=v_p k$ and differentiating:

v_g = v_p + k\frac{dv_p}{dk}.

where: the second term vanishes exactly when $v_p$ is $k$ -independent, recovering $v_g=v_p$ ; if $v_p$ falls with $k$ ("normal dispersion"), then $v_g<v_p$ .

Figure

Two wave packets after travelling the same distance one keeping its narrow shape on a non dispersive string and one spread wide on a dispersive medium

Derivation AGroup velocity from a superposed two-wave beat

The goal is to show, using only the addition of two cosine waves and no calculus beyond a single limiting step, that the envelope of a beating pair of waves travels at $\Delta\omega/\Delta k$ , and that this ratio becomes the derivative $d\omega/dk$ once the band of wavenumbers is narrowed to a point — establishing $v_g=d\omega/dk$ as the speed of energy transport rather than an assumed definition.

Step 1. Superpose two waves of equal amplitude with slightly different wavenumber and frequency, $k_1=k_0-\tfrac{\Delta k}{2}$ , $k_2=k_0+\tfrac{\Delta k}{2}$ , and similarly $\omega_1,\omega_2$ about $\omega_0$ : $y=A\cos(k_1x-\omega_1t)+A\cos(k_2x-\omega_2t)$ .

Step 2. Use the cosine sum identity to factor this into carrier times envelope: $y=2A\cos\!\big(\tfrac{\Delta k}{2}x-\tfrac{\Delta\omega}{2}t\big)\cos(k_0x-\omega_0t)$ . The second factor is the fast carrier oscillating at $k_0,\omega_0$ ; the first factor is the slow envelope, oscillating at the much smaller $\Delta k,\Delta\omega$ .

Step 3. Track a point of constant envelope phase, i.e. hold $\tfrac{\Delta k}{2}x-\tfrac{\Delta\omega}{2}t=\text{const}$ . Solving for $x$ gives $x=\dfrac{\Delta\omega}{\Delta k}t+\text{const}$ , so the envelope pattern moves at speed $\Delta\omega/\Delta k$ — this is the beat pattern's velocity, distinct from the carrier's phase velocity $\omega_0/k_0$ .

Step 4. Let the band narrow, $\Delta k\to 0$ , with $k_1,k_2$ both approaching a genuine wave packet's central wavenumber $k_0$ . The ratio $\Delta\omega/\Delta k$ then becomes, by definition of the derivative, $d\omega/dk$ evaluated at $k_0$ . Since the envelope is where the packet's energy is concentrated, this derivative is the group velocity $v_g=\left.\dfrac{d\omega}{dk}\right|_{k_0}$ — the actual transport speed of a realistic, localized signal.

Worked example

Setup. Reconsider the two dispersion relations from the Pass-1 article and extend them with one further check. (a) A string obeys $\omega(k)=v_0k$ with $v_0=20\ \text{m/s}$ ; find $v_p,v_g$ at $k=5\ \text{rad/m}$ . (b) Deep-water surface waves obey $\omega(k)=\sqrt{gk}$ with $g=9.8\ \text{m/s}^2$ ; find $v_p,v_g$ at $k_0=0.5\ \text{rad/m}$ , and confirm $v_g=v_p+k\,dv_p/dk$ .

Step 1. Non-dispersive string: $v_p=\omega/k=v_0k/k=v_0=20\ \text{m/s}$ , and $v_g=d\omega/dk=d(v_0k)/dk=v_0=20\ \text{m/s}$ . Both speeds equal $20\ \text{m/s}$ , confirming $\omega=v_0k$ is linear and the string is non-dispersive; a packet here keeps its shape indefinitely.

Step 2. Deep-water phase velocity: $\omega(k_0)=\sqrt{(9.8)(0.5)}=\sqrt{4.9}=2.214\ \text{rad/s}$ , so $v_p=\omega/k_0=2.214/0.5=4.427\ \text{m/s}$ .

Step 3. Deep-water group velocity: $d\omega/dk=\tfrac{1}{2}\sqrt{g/k}$ , so at $k_0=0.5$ , $v_g=\tfrac{1}{2}\sqrt{9.8/0.5}=\tfrac{1}{2}\sqrt{19.6}=\tfrac{1}{2}(4.427)=2.214\ \text{m/s}$ . Note $v_g=\tfrac12 v_p$ — the signature result for deep-water gravity waves.

Step 4. Cross-check with the bridge formula. Since $v_p(k)=\sqrt{g/k}$ , $dv_p/dk=-\tfrac12\sqrt{g}\,k^{-3/2}=-\dfrac{v_p}{2k}$ , so $v_p+k\,dv_p/dk=v_p-\tfrac{v_p}{2}=\tfrac{v_p}{2}=2.214\ \text{m/s}$ , matching Step 3 exactly.

The water-wave packet spreads as it travels, since $v_g\neq v_p$ there, while the string packet does not, since $v_g=v_p=20\ \text{m/s}$ .

Competitive stretch

For a relation of the form $\omega=Ak^2$ (found for some bending-wave and quantum-particle-like problems), $v_p=Ak$ and $v_g=d\omega/dk=2Ak=2v_p$ — here the group velocity exceeds the phase velocity, the opposite ordering from the water-wave case. The sign of $dv_p/dk$ , not just whether it is zero, sets whether $v_g$ sits above or below $v_p$ .

Common traps

Assuming $v_g=v_p$ always; it only holds for a linear $\omega=vk$ — always inspect the functional form of $\omega(k)$ first.
Computing $v_g$ by dividing $\omega(k_0)$ by $k_0$ ; that formula gives $v_p$ , not $v_g$ . Group velocity needs differentiation of $\omega$ with respect to $k$ , evaluated afterward at $k_0$ .
Forgetting that $v_g$ , not $v_p$ , is the speed of energy and information transport for a real, localized pulse — "the speed of the wave" in an experiment means $v_g$ .
Treating pulse-spreading in a dispersive medium as an error or noise; it is the expected, correct consequence of $\omega(k)$ being nonlinear, not something to be corrected away.

Ask the tutor: if differentiating $\omega(k)$ to get $v_g$ , or seeing physically why $v_g\neq v_p$ makes a packet spread, still feels shaky, ask the AI tutor to work a third dispersion relation and interpret the group velocity.

Why is group velocity the one that matters for signals, if phase velocity is the one in the basic wave formula?

Because a real signal is never a single infinite plane wave — it is a localized pulse or packet built from a band of wavenumbers, and its energy and information sit in the envelope, not in any one internal crest. The envelope moves at $v_g=d\omega/dk$ , so that is the speed at which you could actually clock a pulse arriving, or a message being sent. $v_p$ describes only how the internal ripples slide under that envelope; it can even exceed $c$ in some media without transporting anything faster than light, precisely because it isn't the transport speed.

If v_p and v_g both come from the same omega of k, why do they ever disagree?

$v_p=\omega/k$ is a ratio evaluated at one point, while $v_g=d\omega/dk$ is a slope at that point — and a ratio and a slope only match when the function is a straight line through the origin, i.e. $\omega=vk$ . As soon as $\omega(k)$ curves (any nonlinear dependence, like $\sqrt{gk}$ or $Ak^2$ ), the local slope of the curve differs from the ratio $\omega/k$ to that point, so $v_g\neq v_p$ . The dispersion relation's shape is the whole story.

Is a dispersive medium doing something wrong physically, since the packet loses its shape?

No — dispersion is a normal, well-defined medium property, not a malfunction. It arises whenever different wavenumbers genuinely experience different restoring forces or propagation physics in that medium, so they are entitled to travel at different phase speeds. Once you accept $\omega(k)$ is nonlinear for a given medium, pulse-spreading is a correct and predictable consequence, calculable exactly from $v_g(k)$ across the packet's band — engineers even use it deliberately, for example to compress or stretch optical pulses in fibre-optic systems.

Group velocity and dispersion

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads