Group velocity and phase velocity

[Naman Jain Kota]

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

 

[blue_leaf77]

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 $k$.

 

[Naman Jain Kota]

How did they put the linearization equation?
Was there approximate and then it could really be applied for wavefunction approaching delta functions

 

[blue_leaf77]

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").

 

[Naman Jain Kota]

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