Interpreting the Wave-Number in the Formula for Group Velocity

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The group velocity of a wave packet is defined by the formula v_g = dω/dk, where the wave number k influences the velocity. The discussion highlights the complexity of interpreting this relationship, particularly when ω and k are not linearly related, leading to varying slopes in the group velocity. It is noted that while different points in the wave packet travel at different velocities, the wave number is typically associated with the entire packet rather than individual segments. Suggestions for determining the wave number include using the average wavelength or analyzing the frequency spectrum of the wave packet. The conversation emphasizes the challenges posed by dispersion, which causes the wave packet to lose its shape and complicates the selection of an appropriate wave number.
ShayanJ
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You people know that group velocity of a wave packet is calculated with the formula v_g=\frac{d \omega}{d k}.But this gives an expression which,in general,is a function of k.My problem is,I can't think of an interpretation for it.What is that wave-number appearing in the expression for group velocity?
Thanks
 
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I'm not sure what is confusing about this. As long as ω and k are not related linearly, the slope dω/dk will change with ω or k.
 
I know that different points of the wave packet move with different velocities,but I want to know how can I find the wave number associated to each point of the wave packet.
 
The wave number is associated with the entire packet, not with different parts of the packet.

A couple of ways to get the wave number are:

1. Use the wavelength, or perhaps the average wavelength, for the oscillations within the wave packet.

2. Look at the frequency spectrum of the wave packet, and use the peak or average frequency in that spectrum to determine the wave number.
 
I can't accept what you say!
In the presence of dispersion,any wave packet,in general,will lose shape which means different parts of it move with different velocities.
Also computing group velocity with a wave number which can be chosen between some alternatives seems a little arbitrary!
 

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