# Prove that group velocity is the velocity of energy transport in wave

Generally, the group velocity $$v_g = \dfrac{\partial \omega}{\partial k}$$ of a wave is the velocity of energy transport. In "Introduction to Solid State Physics", Kittel following is stated:

The transmission velocity of a wave packet is the group velocity, given as the gradient of the frequency with respect to K. This is the velocity of the energy propagation in the medium.

I haven't found any rigorous treat of this relation. How can I prove that for a chain of linear springs (1D harmonic approximation of Phonons) this relation holts? It is easy in this case to describe the energy of the system, which is the sum of the kinetic energy of the particles on every site plus the potential energy of the springs. How to I get from there to the transmission velocity of the energy?

• The group velocity can be shown to be the speed at which a wavepacket moves when it propagates freely: physics.oregonstate.edu/~roundyd/COURSES/ph427/… This is an approximation that breaks down in some circumstances such as in tunnelling experiments: winful.engin.umich.edu/wp-content/uploads/sites/376/2018/01/… Sep 4 '19 at 8:45
• You need to look in the fourier domain of a pulse, which is made up of a (possibly infinite) superposition of modes of the system. Group velocity is usually more used in the context of light pulses in the presence of dispersion. Standard derivation is easily found en.wikipedia.org/wiki/Group_velocity#Derivation Sep 4 '19 at 11:43
• Does this answer your question? Understanding group velocity In particular, seem the derivation in my answer Oct 13 at 7:41
• I don't think the marked duplicate actually answers the question "rigorously", or at least not obviously. In particular, @RogerVadim's answer shows that a Gaussian wavepacket propagates at $v_g$, which is suggestive but doesn't immediately prove that the speed of energy propagation is $v_g$ for any other type of pulse. Oct 13 at 18:16
• @MichaelSeifert this is not true for an arbitrary shape of impulse, because in dispersive media this shape may change. Oct 13 at 19:06