Extras

Phase & Group Velocity

A crest and the packet it belongs to need not travel at the same speed — and in a dispersive medium they don’t.

Builds on Chapter 3 · Spectra, Modulation & Phase — phase.

Two sinusoids of slightly different frequency add up to a fast carrier under a slow envelope. The crests travel at the phase velocity vp = ω/k; the envelope travels at the group velocity vg = dω/dk. In a dispersive medium the two differ — and crests slide through the envelope.

Preset:
Group velocity v_g = 1.00 · v_p
Play The carrier always moves right at the phase velocity; the envelope moves at the group velocity as set.
What This Shows (Plain-Language Guide)

Add two sinusoids whose wavenumbers are k ± Δk and whose frequencies are ω ± Δω, and the sum factors into a product: a fast carrier cos(k·x − ω·t) multiplied by a slow envelope cos(Δk·x − Δω·t). The result is a train of wave packets.

  • The carrier crests — track the marked dot — move at the phase velocity v_p = ω / k.
  • The envelope — track the triangle on its peak — moves at the group velocity v_g = dω / dk = Δω / Δk.

When the medium is non-dispersive (frequency exactly proportional to wavenumber), v_g = v_p and the two markers move together — the packet is rigid. When the medium is dispersive, they differ, and individual crests can be seen born at one edge of the packet, travel through it, and vanish at the other edge.

The Textbook Case

Deep-water waves have v_g = ½ v_p: crests move twice as fast as the group, so they appear at the back of a wave group, race forward through it, and disappear at the front. Energy travels at the group velocity, not the crest velocity.

Why It Belongs Here

Phase velocity is a fourth kind of “phase” — after position-within-a-cycle, relative phase between two signals (the Lissajous / correlation demo), and spectral phase across frequency. Group delay (−dφ/dω) is the same idea for filters: different frequencies can be delayed by different amounts, which is dispersion in a signal-processing guise. A filter’s frequency-dependent delay is applied by convolution with its impulse response — the flip-and-slide picture of that operation is in the Convolution Explorer.