Dispersion relation and their origins & meaning

[Urmi Roy]

Hi Everyone,

I'm trying to understand dispersion relations in general.

I know that for a simple wave like a light wave there is a 'constant phase' so the dx/dt is equal to the ratio of the angular frequency (omega) by the wave vector (k).

However what does a 'constant phase' mean? How can I visualize the propagation of a wave with this characteristic, compared to other waves?

On the other hand, then considering a crystal, the dispersion relation is defined as the 'delta_omega/delta_k'
So I can't really visualize what this relation means.

I'd appreciate any help in helping me intuitively understand these concepts.

 

[atyy]

In the case of light, the linear dispersion relation means that light of every frequency travels at the same speed. So if you have a traveling bump, the bump will move with constant speed without changing shape.

If the dispersion relation is not linear, then a bump will generally change shape, eg. become more dispersed or spread out over time.

There are some good examples using water waves in http://www.people.fas.harvard.edu/~djmorin/waves/dispersion.pdf (p20).

 

[jfizzix]

If you were to take the average value of $\frac{d\omega}{dk}$ (averaged over all values of $k$), what you would have is the velocity of the mean position of the wavepacket. This velocity is what's known as the group velocity, and is distinct from the phase velocity, which is only defined for single plane waves. You can have an average phase velocity for a beam of light, though this too, is distinct from group velocity.

Note: $\frac{d\omega}{dk}$ is often considered just the group velocity, where $k$ is given to be the value of the peak of the spatial frequency spectrum.

 

[PhilDSP]

Hi Urmi Roy,

Surfaces of constant phase for the waveform move at the phase velocity $\frac{\omega}{k}$. Waves with dispersive relations often have a phase velocity that moves in a different direction than the group velocity.

https://books.google.se/books?id=LD...ant phase moves at the phase velocity&f=false

http://www.physics.utoronto.ca/~phy485/ModOpt/dloads/DispPrimer.pdf
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