The group velocity of a wave packet tells you how fast the packet envelope moves. If a wave is described by a dispersion relation
\(\omega=\omega(k)\), then the group velocity is defined by
\[
v_g=\frac{d\omega}{dk}.
\]
This is different from the phase velocity,
\[
v_p=\frac{\omega}{k},
\]
which measures how fast an individual crest or phase point travels. In many practical situations, especially for pulses and signals, the group velocity is
the more physically important quantity because it is closely connected to energy transport and signal propagation.
To understand why, imagine building a localized wave packet by adding many sinusoidal waves with nearby wavenumbers centered around \(k_0\). The individual
waves interfere to form a carrier oscillation inside a slowly varying envelope. If the medium is nondispersive, the relation \(\omega(k)\) is linear,
so all wavenumbers travel with the same speed. In that case,
\[
v_g=v_p,
\]
and the packet keeps its shape as it moves. A classic example is an idealized string with \(\omega=vk\), where \(v\) is constant.
In a dispersive medium, however, different wavenumbers travel differently. Then the carrier and the envelope no longer move together, and the packet
can broaden over time. A standard example is the deep-water gravity-wave relation
\[
\omega=\sqrt{gk},
\]
where \(g\) is gravitational acceleration. Differentiating gives
\[
v_g=\frac{d\omega}{dk}=\frac{1}{2}\sqrt{\frac{g}{k}}.
\]
Since
\[
v_p=\frac{\omega}{k}=\sqrt{\frac{g}{k}},
\]
we get the famous result
\[
v_g=\frac{1}{2}v_p.
\]
This explains why, for ocean swell in deep water, the visible crests seem to run through a wave group while the group itself moves more slowly.
This tool also includes a power-law model,
\[
\omega=a k^n.
\]
Differentiating gives
\[
v_g = a n k^{n-1},
\qquad
v_p = a k^{n-1}.
\]
Therefore,
\[
v_g = n\,v_p.
\]
This is a very useful algebraic model because it lets you see immediately how the exponent \(n\) affects dispersion. If \(n=1\), the system is
nondispersive. If \(n\neq 1\), then \(v_g\) and \(v_p\) differ.
A related quantity is the curvature of the dispersion relation,
\[
\frac{d^2\omega}{dk^2}.
\]
While \(v_g\) tells you how fast the envelope moves, the second derivative indicates whether different nearby Fourier components spread apart and cause the
packet to widen. The more nonlinear the relation \(\omega(k)\), the more pronounced the spreading can become.
In engineering and physics, group velocity matters in acoustics, optics, water waves, plasmas, waveguides, quantum mechanics, and signal transmission.
In optics, for example, the group velocity can differ substantially from the phase velocity in a dispersive material, which is essential in fiber optics and
pulse shaping. In waveguides and transmission systems, group velocity often determines how information moves through the medium, while phase velocity alone
does not fully capture signal behavior.
This calculator focuses on the most important introductory cases. It computes \(v_g\) directly from the chosen relation, compares it with \(v_p\), plots the
relevant functions versus \(k\), and animates a wave packet so that you can visually see the difference between carrier motion and envelope motion.
