A dispersion relation is a rule that connects a wave’s angular frequency \(\omega\) to its wavenumber \(k\). It is usually written as
\(\omega=\omega(k)\). This relationship contains deep information about how a medium responds to oscillations at different spatial scales.
When \(\omega\) depends linearly on \(k\) (for example \(\omega=vk\)), all wavelengths travel at the same speed and the medium is said to be
nondispersive. When the dependence is nonlinear (for example \(\omega=\sqrt{gk}\)), different wavelengths travel at different speeds,
which produces dispersion.
Two important velocities are defined from \(\omega(k)\). The phase velocity is the speed of an individual crest (a point of constant phase):
\[
v_p=\frac{\omega}{k}.
\]
The group velocity describes the speed of a wave packet envelope (and often the speed of energy transport):
\[
v_g=\frac{d\omega}{dk}.
\]
In a nondispersive medium where \(\omega=vk\), these are equal: \(v_p=v_g=v\). In a dispersive medium they differ, and that difference is what causes
a packet to change shape over time.
Deep-water gravity waves provide a classic example of dispersion. For waves on sufficiently deep water (depth much larger than wavelength),
the dispersion relation is
\[
\omega^2=gk,\qquad \text{so}\qquad \omega=\sqrt{gk},
\]
where \(g\) is gravitational acceleration. From this, the phase speed becomes
\[
v_p=\frac{\omega}{k}=\sqrt{\frac{g}{k}}.
\]
Using \(k=2\pi/\lambda\), you can rewrite this as
\[
v_p=\sqrt{\frac{g\lambda}{2\pi}}.
\]
The group velocity is found by differentiating:
\[
v_g=\frac{d\omega}{dk}=\frac{1}{2}\sqrt{\frac{g}{k}}=\frac{1}{2}v_p.
\]
This result explains a familiar ocean observation: the “wave groups” (the envelope) move more slowly than the individual crests, so crests seem to
appear at the back of a group, travel forward through it, and disappear at the front.
Wave packets and spreading. Real signals are not infinite sinusoids; they are localized packets made by superposing many wavenumbers near a
central value \(k_0\). If the dispersion relation is nonlinear, then the different \(k\) components travel with slightly different phase speeds.
Over time, this causes the packet to spread. A quick qualitative rule is: if \(v_g<v_p\), the envelope drifts behind the carrier crests.
But spreading is controlled more precisely by the curvature of the dispersion relation (second derivative \(d^2\omega/dk^2\)): stronger curvature
means faster broadening.
Nondispersive examples. An ideal string under constant tension supports waves with \(\omega=vk\), where \(v=\sqrt{T/\mu}\) is constant.
Light in vacuum also obeys \(\omega=ck\), so \(v_p=v_g=c\). (In materials, light can be dispersive; then \(v_g\) depends on frequency and can be
different from \(v_p\).)
This calculator previews these ideas by computing \(\omega\), \(v_p\), and \(v_g\) for a chosen model and showing plots of \(\omega(k)\) or
\(v_p(k)\) and \(v_g(k)\). The slow animation illustrates a wave packet whose carrier and envelope move at different speeds when dispersion is present.
University-level extensions include finite-depth water waves, capillary waves, waveguides, plasma dispersion, and even relativistic dispersion relations
used in quantum and high-energy physics.
