Related types of frequency

• Angular frequency, usually denoted by the Greek letter ω (omega), is defined as the rate of change of angular displacement, θ, (during rotation), or the rate of change of the phase of a sinusoidal waveform (notably in oscillations and waves), or as the rate of change of the argument to the sine function:

${\displaystyle y(t)=\sin <!--\; \; -->\left(\mathrm{\theta <!--\; \theta \; -->}(t)\right)=\sin <!--\; \; -->(\mathrm{\omega <!--\; \omega \; -->}t)=\sin <!--\; \; -->(2\mathrm{\pi <!--\; \pi \; -->}ft)}$

${\displaystyle \frac{\mathrm{d}\mathrm{\theta <!--\; \theta \; -->}}{\mathrm{d}t}=\mathrm{\omega <!--\; \omega \; -->}=2\mathrm{\pi <!--\; \pi \; -->}f}$

Angular frequency is commonly measured in radians per second (rad/s) but, for discrete-time signals, can also be expressed as radians per sampling interval, which is a dimensionless quantity. Angular frequency (in radians) is larger than regular frequency (in Hz) by a factor of 2π.

• Spatial frequency is analogous to temporal frequency, but the time axis is replaced by one or more spatial displacement axes. E.g.:

${\displaystyle y(t)=\sin <!--\; \; -->\left(\mathrm{\theta <!--\; \theta \; -->}(t,x)\right)=\sin <!--\; \; -->(\mathrm{\omega <!--\; \omega \; -->}t+kx)}$

${\displaystyle \frac{\mathrm{d}\mathrm{\theta <!--\; \theta \; -->}}{\mathrm{d}x}=k}$

Wavenumber, k, is the spatial frequency analogue of angular temporal frequency and is measured in radians per meter. In the case of more than one spatial dimension, wavenumber is a vector quantity.