Consider a linear evolution model of waves that admits as fundamental solutions the oscillators e^{i(kx-\omega t)}, where k and \omega are bound to satisfy a certain constraint. For instance, the free linear
Schrödinger equation in one dimension: i\hbar\partial_t\Psi+\frac{\hbar^2}{2m}\partial_x^2 \Psi=0 satisfies the above description with \omega=\frac{\hbar}{2m}k^2. The group velocity v_g is defined by the equation: v_\text{g} \ \equiv\ \frac{\partial \omega}{\partial k}\,. Here, is the wave's
angular frequency (usually expressed in
radians per second), and is the angular
wavenumber (usually expressed in radians per meter). The
phase velocity is: . The
function , which gives as a function of , is known as the
dispersion relation. • If is
directly proportional to , then the group velocity is exactly equal to the phase velocity. A wave of any shape will travel undistorted at this velocity. • If
ω is a linear function of
k, but not directly proportional , then the group velocity and phase velocity are different. The envelope of a
wave packet (see figure on right) will travel at the group velocity, while the individual peaks and troughs within the envelope will move at the phase velocity. • If is not a linear function of , the envelope of a wave packet will become distorted as it travels. Since a wave packet contains a range of different frequencies (and hence different values of ), the group velocity will be different for different values of . Therefore, the envelope does not move at a single velocity, but its wavenumber components () move at different velocities, distorting the envelope. If the wavepacket has a narrow range of frequencies, and is approximately linear over that narrow range, the pulse distortion will be small, in relation to the small nonlinearity. See further discussion
below. For example, for
deep water gravity waves, \omega = \sqrt{gk}, and hence . This underlies the
Kelvin wake pattern for the bow wave of all ships and swimming objects. Regardless of how fast they are moving, as long as their velocity is constant, on each side the wake forms an angle of 19.47° = arcsin(1/3) with the line of travel.
Derivation One derivation of the formula for group velocity is as follows. Consider a
wave packet as a function of position and time . Let be its
Fourier transform at time , \alpha(x, 0) = \int_{-\infty}^\infty dk \, A(k) e^{ikx}. By the
superposition principle, the wavepacket at any time is \alpha(x, t) = \int_{-\infty}^\infty dk \, A(k) e^{i(kx - \omega t)}, where is implicitly a function of . Assume that the wave packet is almost
monochromatic, so that is sharply peaked around a central
wavenumber . Then,
linearization gives \omega(k) \approx \omega_0 + \left(k - k_0\right)\omega'_0 where \omega_0 = \omega(k_0) and \omega'_0 = \left.\frac{\partial \omega(k)}{\partial k}\right|_{k=k_0} (see next section for discussion of this step). Then, after some algebra, \alpha(x,t) = e^{i\left(k_0 x - \omega_0 t\right)}\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}. There are two factors in this expression. The first factor, e^{i\left(k_0 x - \omega_0 t\right)}, describes a perfect monochromatic wave with wavevector , with peaks and troughs moving at the
phase velocity \omega_0/k_0 within the envelope of the wavepacket. The other factor, \int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}, gives the envelope of the wavepacket. This envelope function depends on position and time
only through the combination (x - \omega'_0 t). Therefore, the envelope of the wavepacket travels at velocity \omega'_0 = \left.\frac{d\omega}{dk}\right|_{k=k_0}~, which explains the group velocity formula.
Other expressions For light, the refractive index , vacuum wavelength , and wavelength in the medium , are related by \lambda_0 = \frac{2\pi c}{\omega}, \;\; \lambda = \frac{2\pi}{k} = \frac{2\pi v_\text{p}}{\omega}, \;\; n = \frac{c}{v_\text{p}} = \frac{\lambda_0}{\lambda}, with the
phase velocity. The group velocity, therefore, can be calculated by any of the following formulas, \begin{align} v_\text{g} &= \frac{c}{n + \omega \frac{\partial n}{\partial \omega}} = \frac{c}{n - \lambda_0 \frac{\partial n}{\partial \lambda_0}}\\ &= v_\text{p} \left(1 + \frac{\lambda}{n} \frac{\partial n}{\partial \lambda}\right) = v_\text{p} - \lambda \frac{\partial v_\text{p}}{\partial \lambda} = v_\text{p} + k \frac{\partial v_\text{p}}{\partial k}. \end{align} ==Dispersion==