# Group Velocity of Non-Dispersive Wave Packet

waley

## Homework Statement

I know that for a dispersive wave packet, the group velocity equals the phase velocity, which is given by v=w/k. But how do I calculate the group velocity of a non-dispersive wave packet? I'm supposed to be giving an example with any functional form.

## Homework Equations

group velocity is defined by (partial derivative of w(k))/(partial derivative of k)

## The Attempt at a Solution

Since w is not dependent on k, the partial derivative should equal to zero? But I'm not sure how to go from there.

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blue_leaf77
Homework Helper
Could you possibly mean a photon wavepacket, i.e. a light pulse? I think you got your definition reversed - nondispersive wavepacket has its group and phase velocities equal but the dispersive one has them different. A non-dispersive lightpulse is that propagating in a non-dispersive medium, a medium whose refractive index does not vary with frequency (or wavelength). In this case if you differentiate both sides of the equation ##k = \omega n(\omega)/c ## you will get the group velocity in terms of the refractive index (as a function of ## \omega##).

waley
Could you possibly mean a photon wavepacket, i.e. a light pulse? I think you got your definition reversed - nondispersive wavepacket has its group and phase velocities equal but the dispersive one has them different. A non-dispersive lightpulse is that propagating in a non-dispersive medium, a medium whose refractive index does not vary with frequency (or wavelength). In this case if you differentiate both sides of the equation ##k = \omega n(\omega)/c ## you will get the group velocity in terms of the refractive index (as a function of ## \omega##).
Ah, sorry I realized after that I indeed got my definitions reversed. And I also forgot to add that I'm asking in terms of sound waves - would your equation still hold?
How exactly would I differentiate both sides - Is the derivative of k supposed to be the group velocity, and my "functional form" would be something of n in terms of w that I differentiate with respect to w?

waley
Actually, would it be sufficient enough if I say that for the non-dispersive relation, since the group velocity equals the phase velocity, I just solve the phase velocity with the equation v=w/k? So if I have a functional form of, let's say, cos(x), the phase and group velocity would just be cos(x)/k?