Wave packets Group Velocity For the the group velocity of a wave packet, the group velocity is the partial derivative of omega with respect to the wavenumber, what does this mean? I thought that for some given wave packet both the angular frequency and wave number should be fixed, so what differentiation $d$ is there?
 A: Supposedly if you have a wave packet you don't have a "single frequency" or "single wavelength". In order for you to have a wave-packet you have a superposition of waves of different frequencies and wave-lengths.
Imagine you superpose two sinusoidal progressive waves which have different frequencies and wavelengths in a dispersive medium (there will be wave dispersion). The waves have the same amplitude (just to make it easier).
Let's define the waves this way:
$$ y_{1}=Acos(k_{1}x-\omega_{1}t) $$
$$ y_{2}=Acos(k_{2}x-\omega_{2}t) $$
As they are in a dispersive medium their velocity might be different:
$$\frac{\omega_{1}}{k_{1}}= v_{1}$$ $$ \frac{\omega_{2}}{k_{2}}=v_{2}$$
If we write $$ \psi(x,t)=y_{1}+y_{2}$$
we can get:
$$ \psi(x,t)=Acos(k_{1}x-\omega_{1}t)+Acos(k_{2}x-\omega_{2}t)$$ $$=2Acos(\frac{k_{1}-k_{2}}{2}x+\frac{\omega_{2}-\omega_{1}}{2}t)cos(\frac{k_{1}+k_{2}}{2}x-\frac{\omega_{2}+\omega_{1}}{2}t) $$
That last one came from this trigonometric identity: $$cos(a)+cos(b)=2cos(\frac{a+b}{2})cos(\frac{a-b}{2})$$
If we write $$ \omega_{0}= \frac{\omega_{2}+\omega_{1}}{2};\Delta \omega=\frac{\omega_{2}-\omega_{1}}{2};k_{0}= \frac{k_{2}+k_{1}}{2};\Delta k=\frac{k_{2}-k_{1}}{2}$$
The expression before will be simplified:
$$ \psi(x,t)= 2Acos(\Delta k\cdot x+\Delta \omega \cdot t)cos(k_{0}x-\omega_{0}t)$$
You will know the phase velocity when you compute $v_p=\omega_0/k_0$ and you will get the group velocity when you compute $v_{g}=\Delta \omega/\Delta k$.
The idea that the group velocity is the partial derivative of omega in respect to the wavenumber comes from the idea that the initial frequencies $\omega_{1}$ and $\omega_{2}$ are very close values (and so is the wave number). Then you will have beats:
$v_{g}=lim(\Delta \omega /\Delta k)=lim(\frac{\omega_{2}(k_{2})-\omega_{1}(k_{1})}{k_{2}-k_{1}}) $ where $\omega_{1}≈\omega_{2}$, leading to:
$$v_{g}=\frac{∂w}{∂k}$$
And there you have it. To conclude, in a wave-packet you don't have a single frequency or wavenumber, but several. This brings the necessity to compute the phase velocity (which is the velocity at which the waves inside the wave-packet envelope travel) and the group velocity (the velocity of the envelope that contains the waves). I'm sorry for any bad english.
Hope I helped. Happy new year.
