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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?

pmu2022
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A good place to read up this argument is in Sir James Lighthill's book "Waves in Fluids" I took the course from him that turned into this book, and I recall him discussing whether this claim was generally true, or only a rule of thumb. I think the discussion made it into the book. Certainly it has many non-trivial applications of the group-velocity.

mike stone
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The answer was given by Biot:

Biot, M. A. (1957). General Theorems on the Equivalence of Group Velocity and Energy Transport. Physical Review, 105(4), 1129–1137. doi:10.1103/PhysRev.105.1129

Sergey Kuznetsov
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