Dispersion
When wave speed depends on frequency, a pulse made of many frequencies spreads and reshapes as it travels, and phase and group velocities part ways.
Dispersion
Dispersion is what happens when different frequencies travel at different speeds. In the ideal Wave Equation every frequency moves at the same c, so a pulse keeps its shape forever. Real media rarely cooperate: deep-water waves, light in glass, and electrons in a crystal all carry high and low frequencies at different speeds. A pulse is a superposition of many frequencies, so if they march at different rates the pulse spreads out and distorts as it goes. The rainbow from a prism is dispersion: glass bends blue more than red because it is slower for blue.
The dispersion relation
All the physics lives in one function — the dispersion relation \omega(k), tying angular frequency to wavenumber. A non-dispersive medium has the straight line \omega = ck; anything curved means dispersion. From it come two distinct velocities:
The phase velocity is how fast a single crest moves; the group velocity is how fast the packet of energy and information moves. When \omega=ck they are equal and nothing disperses. When \omega(k) curves, they differ — sometimes dramatically, with crests sliding through the envelope faster than the envelope itself advances.
Group velocity carries the message
A crucial subtlety: the group velocity is what carries energy and signal, not the phase velocity. It is entirely possible — in fact common — for individual crests to race ahead while the bundle they belong to lags behind, the crests appearing at the back of the packet, sweeping forward, and vanishing at the front. The object that makes this concrete is the Wave Packet: a localized envelope (moving at v_{\text{group}}) filled with oscillations (moving at v_{\text{phase}}).