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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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  • The group velocity can be shown to be the speed at which a wavepacket moves when it propagates freely:

    http://physics.oregonstate.edu/~roundyd/COURSES/ph427/wave-packets.html

    This is an approximation that breaks down in some circumstances such as in tunnelling experiments:

    http://winful.engin.umich.edu/wp-content/uploads/sites/376/2018/01/physics_reports_review_article__2006_.pdf

     – alanf Sep 04 '19 at 08:45
  • 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 https://en.wikipedia.org/wiki/Group_velocity#Derivation – Feel My Black Hole Sep 04 '19 at 11:43
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    Does this answer your question? Understanding group velocity In particular, seem the derivation in my answer – Roger V. Oct 13 '21 at 07:41
  • 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. – Michael Seifert Oct 13 '21 at 18:16
  • @MichaelSeifert this is not true for an arbitrary shape of impulse, because in dispersive media this shape may change. – Roger V. Oct 13 '21 at 19:06

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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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  • This is a link-only answer, which is not acceptable in our answering policy. Please add context around the link so your fellow users will have some idea what it is and why it’s there. Always quote the most relevant part of an important link, in case the target site is unreachable or goes permanently offline. – SuperCiocia Oct 13 '21 at 18:03