Group velocity and phase velocity

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The discussion centers on the derivation of group velocity defined as dw/dk and its approximation under specific circumstances. It is noted that this approximation applies when the wave function's amplitude A(k) is sharply peaked, allowing for a first-order linearization of the phase. If A(k) is not sharply peaked, higher-order terms must be considered, which can alter the pulse shape, leading to phenomena like "chirping." The conversation also questions the validity of group velocity as dw/dk, suggesting that combining sine wave functions can create envelopes that challenge this definition. Overall, the group velocity's derivation and its limitations in certain contexts are key points of contention.
Naman Jain Kota
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Could you please explain the derivation of
group velocity = dw/dk

I read ut here https://en.m.wikipedia.org/wiki/Group_velocity

Is it approximation, if so under what circumstances
 
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Isn't the derivation already available in that link? The group velocity is defined to be the velocity at which the envelope of a wave travels. It's an approximation in the sense that it was derived involving approximating the phase in first order of ##k##.
 
How did they put the linearization equation?
Was there approximate and then it could really be applied for wavefunction approaching delta functions
 
Yes, the linearization is an approximation which only applies to cases where ##A(k)## is sharply peaked. If this is not the case, higher orders of the phase must be included in the calculation and they usually lead to the modification of the shape of the pulse (so-called "chirping").
 
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If i add two sine wave functions (real parts) envelopes are formed. So group velocity must not be dw/dk. Asking coz my proff said that which i feel is wrong
 

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