Dispersion relation

The dispersion relation $\omega(\mathcal K)$ relates angular frequency with wavenumber. In a linear dispersion relation, $\omega = v_p \mathcal K$. One common nonlinear dispersion relation is $\omega = v_p \mathcal K \sqrt{1 + \alpha \mathcal K^2}$. A nonlinear dispersion relation is called dispersive.

Phase velocity

We will find the velocity of a traveling wave. Over one period $T = \frac{2\pi}{\omega}$ the wave travels one wavelength $\lambda = \frac{2\pi}{\mathcal K}$, giving us the phase velocity

This is true generally whether the dispersion is linear or not. A linear dispersion relation results in a constant phase velocity, while a nonlinear relation results in a phase velocity that is a function of frequency.

Group velocity

Consider the superposition of two traveling waves $A \sin(\mathcal K_1 x - \omega_1 t)$ and $A \sin(\mathcal K_2 x - \omega_2 t)$. When the difference in wavenumbers is very small we get a beat pattern, as shown below.

Group velocity is the velocity of this beat envelope. The velocities of the two component waves are different, and indeed this is visible in the animation. In this case the phase velocities are very close to the group velocity, but in the case where we have many superpositioned waves at different frequencies it is more convenient to consider the group velocity.

As we have shown above, the velocity of the wave is its angular frequency over its wavenumber. For the beat envelope, this is $v = \frac{\omega_1-\omega_2}{\mathcal K_1 - \mathcal K_2}$. Since we have assumed the difference in the wavenumbers is very small, we get