In Three Dimensions
See also: Plane waveFor waves traveling through three dimensions, such as light waves, sound waves, and matter waves, the formulas for phase and group velocity are generalized in a straightforward way:
- One dimension:
- Three dimensions:
where means the gradient of the angular frequency as a function of the wave vector, and is the unit vector in direction k.
If the waves are propagating through an anisotropic (i.e., not rotationally symmetric) medium, for example a crystal, then the phase velocity vector and group velocity vector may point in different directions.
Read more about this topic: Group Velocity
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