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How to find the speed of sound in a monoatomic crystal

My question is about phase and group velocities in a monoatomic crystal. I want to find the speed of sound, so I am using the equation for vibrational modes: ...
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Relationship between phase velocity and group velocity with De Broglie postulates

If I have to show that the group velocity of a free particle is twice the phase velocity, is the following argument correct (avoiding to use the wave function and the momentum operator): For a ...