Separate carrier speed from envelope speed

Add equal-amplitude cosine waves with wavenumbers just above and below a central value. The trigonometric sum becomes a rapidly oscillating carrier multiplied by a slowly varying cosine envelope. A carrier crest keeps its fast phase constant and moves with central angular frequency divided by central wavenumber. The envelope instead keeps the difference phase constant and moves with the frequency difference divided by the wavenumber difference. These are distinct ratios even though both are built from the same two components.

As the two wavenumbers approach one another, their frequency-difference ratio tends to the derivative of the dispersion relation. This derivative is group velocity. The argument assumes a differentiable real-frequency branch and components close enough that a local linear expansion is meaningful. In nondispersive linear acoustics, angular frequency is sound speed times wavenumber, so phase and group speeds coincide. A nonconstant phase speed is a warning to examine the derivative rather than reuse the phase-speed formula.

Keeping only the constant and linear terms makes a narrow-band envelope translate without changing shape. The quadratic term gives different offsets from the central wavenumber different extra phases. Over time those phases can broaden, chirp, or otherwise deform the envelope. A useful small-distortion condition compares the accumulated quadratic phase across the bandwidth with one radian. It is an approximation check, not an exact universal threshold at which every packet suddenly stops being recognizable.

For example, a dispersion law proportional to wavenumber squared has group speed twice its phase speed. Components at the high-wavenumber side also have larger group speed, so a sufficiently broad pulse spreads. If the spectrum is not narrow, or if absorption and gain strongly reshape it, a single envelope velocity may be a poor descriptor. The peak that survives attenuation can move because different parts are suppressed, not because the same identifiable piece of energy raced along that path.

A causal front is the earliest response to a newly introduced disturbance, not the maximum of a pre-existing smooth pulse. Determining it requires the full causal propagation problem and the relevant high-frequency behavior, not just one derivative near a carrier. A local quadratic approximation may even have mathematical tails outside the region where it is physically valid. An unusual group velocity in an absorbing or dispersive model does not by itself prove equally unusual information transport or violate a causal theory.

The waves lab offers a useful control experiment: its two components have the same wavenumber and angular frequency. Predict that changing their constant relative phase will alter the resultant amplitude without creating a slowly moving beat envelope. Test this at several phase settings. The absence of a wavenumber-difference control is deliberate here: the display demonstrates why a phase offset alone is not dispersion. Calculate group velocity from the stated dispersion relation on paper rather than treating this equal-frequency lab as a packet solver.