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:Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis, p239 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.
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:Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis p239 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.