Wave packet

A wave packet is a finite-in-extent disturbance built by adding together plane waves whose wavenumbers cluster around a central value k₀. The packet has two characteristic speeds: the phase velocity of its carrier (v_p = ω₀/k₀) and the group velocity of its envelope (v_g = dω/dk at k₀). The narrower the spread in k, the longer the packet is in space — the Fourier uncertainty that sits at the heart of quantum mechanics.

Under a non-linear dispersion relation, the packet broadens as it propagates because different Fourier components travel at slightly different group velocities. The broadening is governed by the curvature β = d²ω/dk², and for a Gaussian packet σ(t) = σ₀·√(1 + (βt/σ₀²)²).