Group Velocity

Prerequisites: Fourier Transform. Full path (6 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

A single wave of one wavelength that goes on forever carries no message: every crest looks like every other crest. To send a signal, or to describe a localized particle, we need a wave packet, a lump of wave with a beginning and an end. A packet is built by adding many waves with slightly different wavelengths. Where they are in step they reinforce each other, elsewhere they cancel.

In many media, waves of different wavelength travel at different speeds. This is called dispersion. Then the crests inside the packet and the packet as a whole move at different speeds. The speed of the crests is the phase velocity. The speed of the envelope, the lump itself, is the group velocity.

You can see this when you throw a stone into a deep pond. Watch a single ripple crest: it appears at the back of the group of ripples, moves forward through the group and dies out at the front. For gravity waves on deep water (wavelengths above a few centimetres, where surface tension does not matter) the group moves at half the speed of the individual crests.

The group velocity is usually the more important of the two. In ordinary media the energy of the wave and any signal it carries travel with the envelope, at the group velocity. For a quantum particle described by a wave packet, the group velocity is the velocity of the particle in the classical sense.

There are media, for example near an absorption line, where the formula for the group velocity gives a value larger than the speed of light or even a negative value. There the packet is strongly reshaped as it propagates and the peak of the output is built from the front part of the input. No signal travels faster than light.

• When group velocity is an optical illusion | Gravity and Levity

Concrete

Consider waves in one dimension with a dispersion relation $\omega = \omega(k)$, which says what angular frequency $\omega$ a plane wave $e^{i(kx-\omega t)}$ of wavenumber $k$ has. A wave packet is a superposition

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

where the amplitude $A(k)$ is sharply peaked around some $k_0$. Only wavenumbers close to $k_0$ contribute, so we expand

$$\omega(k) \approx \omega_0 + v_g (k-k_0) + \tfrac{1}{2}\,\omega''(k_0)\,(k-k_0)^2 , \qquad \omega_0 = \omega(k_0), \quad v_g = \left.\frac{d\omega}{dk}\right|_{k_0} .$$

Dropping the quadratic term for the moment and inserting the expansion gives

$$\psi(x,t) \approx e^{i(k_0 x - \omega_0 t)} \int dk\, A(k)\, e^{i(k-k_0)(x - v_g t)} = e^{i(k_0 x - \omega_0 t)}\, F(x - v_g t) .$$

The first factor is a carrier wave whose crests move with the phase velocity

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

The second factor $F$ is the envelope. It keeps its shape and moves rigidly with the group velocity

$$v_g = \frac{d\omega}{dk} .$$

The quadratic term $\omega''(k_0)$, the group velocity dispersion, makes different parts of the packet move at slightly different group velocities, so the envelope spreads over time.

Examples

• Light in vacuum, $\omega = ck$: $v_p = v_g = c$. There is no dispersion and the packet keeps its shape.
• Deep water gravity waves, $\omega = \sqrt{gk}$: $v_g = \frac{1}{2}\sqrt{g/k} = \frac{1}{2} v_p$, the pond example from above.
• A free non-relativistic quantum particle, $\omega = \hbar k^2/2m$ (from $E = \hbar\omega$, $p = \hbar k$): $v_g = \hbar k/m = p/m$, the classical velocity, while $v_p = p/2m$ is half of it.
• A free relativistic particle, $\hbar\omega = \sqrt{\hbar^2 c^2 k^2 + m^2 c^4}$: $v_g = c^2 k/\omega = c^2 p/E$, which is again the particle velocity $v < c$, while $v_p = c^2/v > c$. The product is $v_p v_g = c^2$.
• Light in a medium with refractive index $n(\omega)$, $k = n(\omega)\,\omega/c$: differentiating gives $dk/d\omega = (n + \omega\, dn/d\omega)/c$, so

$$v_g = \frac{c}{n + \omega \frac{dn}{d\omega}} .$$

In the normal case $dn/d\omega > 0$ and $v_g < c/n$.

In three dimensions with $\omega = \omega(\vec k)$ the same argument gives $\vec v_g = \nabla_{\vec k}\, \omega$, which need not point along $\vec k$, as in anisotropic crystals.

Abstract

For a linear, translation invariant wave equation in $d$ space dimensions, each branch of the dispersion relation is a function $\omega(\vec k)$ on wavevector space, and the group velocity is its gradient $\vec v_g(\vec k) = \nabla_{\vec k}\,\omega$. The precise statement behind the heuristic expansion is the method of stationary phase. For

$$\psi(\vec x,t) = \int \frac{d^d k}{(2\pi)^d}\, A(\vec k)\, e^{i(\vec k\cdot \vec x - \omega(\vec k) t)}$$

with smooth $A$, take $t \to \infty$ at fixed $\vec x/t$. The phase is stationary at the wavevectors $\vec k_s$ with $\nabla_{\vec k}\omega(\vec k_s) = \vec x/t$. If the Hessian $H_{ij} = \partial^2\omega/\partial k_i \partial k_j$ is non-degenerate there,

$$\psi(\vec x,t) \approx \sum_{\vec k_s} \frac{A(\vec k_s)}{(2\pi t)^{d/2}\, |\det H(\vec k_s)|^{1/2}}\, e^{i(\vec k_s\cdot\vec x - \omega(\vec k_s)t) - i\frac{\pi}{4}\operatorname{sgn} H(\vec k_s)} ,$$

where $\operatorname{sgn} H$ is the signature of $H$. So at late times the component with wavevector $\vec k$ is found near $\vec x = \vec v_g(\vec k)\, t$, and the amplitude decays as $t^{-d/2}$ because the packet spreads. Where $\det H = 0$ (caustics) the decay is slower and the formula needs higher order terms.

For slowly varying media, the geometric optics (WKB) limit writes the wave as $a\, e^{iS}$ with local wavevector $\vec k = \nabla S$ and frequency $\omega = -\partial_t S$. The local dispersion relation $\omega = \omega(\vec k, \vec x)$ then acts as a Hamiltonian on the phase space $(\vec x, \vec k)$, and wave packets follow the rays

$$\frac{d\vec x}{dt} = \frac{\partial \omega}{\partial \vec k}, \qquad \frac{d\vec k}{dt} = -\frac{\partial \omega}{\partial \vec x} .$$

The first equation says that rays move at the group velocity. With $\omega = E/\hbar$ and $\vec k = \vec p/\hbar$ these are Hamilton's equations of the corresponding particle, which is how classical mechanics emerges from wave mechanics in this limit. The same equation with $\omega$ replaced by a band energy $\varepsilon_n(\vec k)/\hbar$ gives the velocity of Bloch electrons in a crystal.

For conservative linear waves, the energy averaged over a period is transported at the group velocity in a uniform medium. In a slowly varying medium the conserved quantity is the wave action density $\mathcal A = \mathcal E/\omega$ (energy density over intrinsic frequency), which obeys $\partial_t \mathcal A + \nabla\cdot(\vec v_g \mathcal A) = 0$.

Path to this page

1. Calculus
2. Imaginary Numbers
3. Trigonometric Functions
4. Exponential Function
5. Euler's Formula
6. Fourier Transform