phase and group velocity
Watch waves on water and you notice something strange: the individual crests seem to move at one speed, but the whole packet of ripples — the bunch, the envelope — travels at a different speed, often slower. These are two genuinely different velocities. The phase velocity is the speed of a single crest; the group velocity is the speed of the packet, the bundle of waves of nearly equal wavelength that carries the signal and the energy. In a dispersive medium they disagree, and the difference is not a trick of the eye.
Both are read straight off the dispersion relation omega(k). The phase velocity is v_p = omega / k — divide frequency by wavenumber, and you get the speed at which a point of constant phase (a crest) moves. The group velocity is v_g = d(omega)/dk — the derivative of the curve — and it is the speed at which a wave packet, built by superposing waves with wavenumbers clustered near some k, actually travels. You can derive v_g by adding two nearby waves: the carrier wiggles at the phase speed while the slow beat envelope it produces moves at the derivative speed. When omega = c k exactly (non-dispersive), v_p = omega/k = c and v_g = d(omega)/dk = c agree, so crests and packet march together. When omega(k) curves, they part ways.
Group velocity is the one that carries information and energy, which is why physics insists on the distinction: a phase velocity can even exceed the speed of light in some media without violating relativity, because no signal rides on a single infinite crest, whereas the group velocity, which carries the actual message, stays subluminal. For PDEs this is the practical meaning of dispersion: solve a wave-like equation by transform and the inverse integral concentrates, for large time, near the point moving at the group velocity — which is exactly what the method of stationary phase makes precise. So group velocity is also the answer to 'where does the energy go for large t?'
For deep-water waves omega = sqrt(g k), so v_p = omega/k = sqrt(g/k) and v_g = d(omega)/dk = (1/2) sqrt(g/k) — exactly half the phase speed. Watch an ocean swell and you can see individual crests advancing through the group from the back and dying out at the front, because each crest outruns the packet it belongs to.
Phase velocity = omega/k moves the crests; group velocity = d(omega)/dk moves the packet and the energy.
A common confusion is that one of these is 'the real' speed of the wave. Neither is universally; the phase velocity moves crests, the group velocity moves signals and energy — and only in a non-dispersive medium do they coincide.