# The speed of information propagation (classical, not quantum)

It seems to be a common statement in textbooks that:

"For a linear wave equation with the dispersion relation $\omega_k$, the propagation speed of information is given by the group velocity."

which is usually demonstrated by the propagation of a wave package centered around a specific $k$ (e.g., wikipedia). But this seems far away from a proof that there is no way to transmit information faster than that.

Generically, assuming there is an upper-bound for the group velocity, is there a rigorous way to prove that it limits the speed of information propagation? Or how should one attack this problem?

Update: thanks to @freecharly, group velocity can be larger than the signal speed according to Milonni's book. I would change the question then: for a generic dispersion relation, how does one decide the maximum speed for information propagation?

• Such statement would be nonsense because it is easy to find dispersive examples where the group velocity is actually negative and neither things nor information can travel backward in time, or whatever negative group velocity might mean. – hyportnex May 10 '18 at 22:21
• One can then change the question into: for generic dispersion relation, how does one decide the maximum speed for information propagation? – Yen-Ta Huang May 10 '18 at 22:29
• in RF communications one can view the leading pulse edge as the beginning of the information and normally that propagates with the group velocity but for that one must assume that the medium is not very dispersive. For this interpretation one must have that the group velocity be positive and the signal itself be narrow band so that the envelope is reasonably stable and does not change over the time of observation. – hyportnex May 10 '18 at 22:41
• Envelop type argument is more like a heuristic than a proof to me: that couldn't (logically) exclude the possibility that some creative guys create a non-envelop type wave to transmit information in some way, that is faster than the envelope propagation. Plus, even without stable envelop, I don't see why one couldn't try to send information in other ways, which could have an upper bound for the speed of information propagation. – Yen-Ta Huang May 10 '18 at 22:54

• @Yen-TaHuang - What do you mean by a "generic linear wave equation"? A system described by a simple wave equation $$\Delta \phi=\frac {1}{v^2} \frac {\partial^2 \phi}{\partial t^2}$$ where v is the speed of propagation? Here you have a linear dispersion equation and group and phase velocities are equal. – freecharly May 10 '18 at 21:54
• Any linear PDE with finite order space and time derivative, with translational symmetry. For example, I would accept $\partial_t^2 \psi=(\partial_x^2+\lambda \partial_x^4 + m^2) \psi$ – Yen-Ta Huang May 10 '18 at 22:03