Dispersion

Dispersion is the phenomenon where the phase velocity $v_{p}$ of a wave is dependent on the frequency $\omega$ of the wave, $v_{p}\equiv v_{p}(\omega)$. This phenomenon depends on the nature of the medium the wave is traveling in. A medium that exhibits dispersion is said to be dispersive. In other words, dispersion occurs when the individual components of a wave's Fourier series (or Fourier transform) propagate at different speeds. As such, monochromatic waves cannot show dispersion, as they're made up of only a single frequency.

In dispersive media, it is useful to analyze the relation between the wavevector $\mathbf{k}$ and the angular frequency $\omega$, which is called the dispersion relation $\omega(k)$¹. In nondispersive media, the dispersion relation is always

where $v_{p}$ is the phase speed. In dispersive media, the relation must be nonlinear

You can prove this is always nonlinear with some algebra. Just move the speed to the left to get $\omega/v_{p}(\omega)=k$. Assume for instance that $v_{p}(\omega)\propto \omega^{p}$ for some $p$. Then $\omega/\omega^{p}\propto k$. This is manifestly nonlinear unless $p=0$, in which case $v_{p}$ wouldn't depend on $\omega$ in the first place.

As an example, in regular glass the phase speed of an electromagnetic wave is $v_{p}=c/n$, where $c$ is the speed of light and $n$ is the refractive index. It is well-known that $n$ is frequency dependent in glass (think of the classic prism that turns a beam of light into a rainbow), so the dispersion relation is

which is nonlinear². Now, it should be noted that relations like this one, they are linear if you set a specific value of $\omega$, that is, if you're working with monochromatic waves. However, this is more of a consequence of dispersion not applying to monochromatic waves than something about the material specifically. Other dispersion relations are entirely nonlinear regardless of whether you are working with monochromatic waves or not, such as some wave guides or plasma which behave like $\omega \propto \sqrt{ k }$.

Dispersion causes waves to change their profile over time while traveling through dispersive media, usually by spreading out ("dispersing"). This is most obvious when analyzing wave packets, localized waveforms with a "shape". The outer profile of a wave packet, the envelope, moves at a different speed than each individual sinusoidal component of the wave's Fourier series, which is called the group velocity

In nondispersive media, this is manifestly equal to the phase speed. In fact, you can alternatively define dispersion as occurring when phase speed and group speed do not coincide.