Editing Group velocity (section)

==In three dimensions==
{{See also|Plane wave}}
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:<ref>[https://books.google.com/books?id=cC0Kye7nHEEC&pg=PA239 Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis, p239]</ref>
*One dimension: <math>v_{\rm p} = \omega/k, \quad v_{\rm g} = \frac{\partial \omega}{\partial k}, \,</math>
*Three dimensions: <math>\mathbf{v}_{\rm p} = \frac{\omega}{k} \hat{\mathbf{k}}, \quad \mathbf{v}_{\rm g} = \vec{\nabla}_{\mathbf{k}} \, \omega \,</math>
where <math display="block">\vec{\nabla}_{\mathbf{k}} \, \omega</math> means the [[gradient]] of the [[angular frequency]] {{mvar|ω}} as a function of the wave vector <math>\mathbf{k}</math>, and <math>\hat{\mathbf{k}}</math> 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.