A wave packet is a localized superposition of many nearby wavenumbers. Instead of filling all space like a single pure sinusoid, a packet has a concentrated envelope and therefore can model signals, pulses, and localized disturbances. In many physical systems, the packet does not simply translate rigidly. If the medium is dispersive, different Fourier components travel differently, so the packet spreads with time.
The key quantities are the phase velocity
\[
v_p=\frac{\omega}{k},
\]
and the group velocity
\[
v_g=\frac{d\omega}{dk}.
\]
The phase velocity describes how individual crests move, while the group velocity describes how the envelope or energy-bearing packet center moves. In a non-dispersive medium, these are equal and the packet keeps its shape. In a dispersive medium, they differ, so the carrier wave can slip inside the envelope.
To model packet spreading, one often expands the dispersion relation near a central wavenumber \(k_0\):
\[
\omega(k)\approx \omega_0 + v_g (k-k_0) + \frac{\beta_2}{2}(k-k_0)^2.
\]
The coefficient \(\beta_2\) controls the curvature of the dispersion relation and therefore the strength of the spreading. A Gaussian packet is especially convenient because its form remains Gaussian during evolution, though its width changes with time.
If the initial envelope is
\[
\exp\!\left[-\frac{(x-x_0)^2}{2\sigma_0^2}\right],
\]
then an approximate spreading law is
\[
\sigma(t)=\sigma_0\sqrt{1+\left(\frac{\beta_2 t}{2\sigma_0^2}\right)^2}.
\]
The envelope center moves to
\[
x_c(t)=x_0+v_g t,
\]
while the carrier oscillation has phase motion related to
\[
x_p(t)=x_0+v_p t.
\]
For the sample case of a water-wave-style packet with initial width \(10\text{ m}\), central wavenumber \(k_0=1\text{ rad/m}\), and a dispersive medium, the width after \(t=10\text{ s}\) becomes larger than the initial width. This means the packet broadens while it travels. At the same time, if \(v_p>v_g\), the internal oscillation peaks move faster than the packet envelope.
This distinction matters in many real applications. In signal transmission, dispersion can distort pulses. In water waves, one sees wave groups where individual crests appear and disappear inside the envelope. In optics, similar mathematics describes pulse broadening in fibers. In quantum mechanics, a free-particle wave packet also spreads in time due to dispersion in the Schrödinger relation.
The simulator here uses a Gaussian-envelope approximation rather than a full numerical Fourier transform, but it captures the essential physics clearly: the packet center travels with \(v_g\), the carrier phase moves with \(v_p\), and the width grows when the dispersion coefficient is nonzero.
This tool therefore provides a clean introduction to wave packet evolution, the difference between phase and group motion, and the visual effect of dispersive spreading.
