Group Velocity - in Three Dimensions

In Three Dimensions

For 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.