# Wavenumber

Wavenumber, k, sometimes called the propagation number, is a quantification of spatial scale. It can be thought of as a spatial analog to the temporal frequency, and is often called spatial frequency. It is often defined as the number of wavelengths per unit distance, or in terms of wavelength, λ:

${\displaystyle k={\frac {1}{\lambda }}\ }$

This is analogous to frequency f, which is the reciprocal of period T; that is, f = 1/T. In a sense, period can be thought of as a temporal 'wavelength—the length of an oscillation in time.

If you've explored the applications of frequency in geophysics, you'll have noticed that we sometimes don't use ordinary frequency f, in Hertz. Because geophysics deals with oscillating waveforms, ones that vary around a central value (think of a wiggle trace of seismic data), we often use the circular or angular frequency. In this way, we can also express the close relationship between frequency and phase, which is an angle. So in many geophysical applications, we want the angular wavenumber. It is expressed in radians per metre:

${\displaystyle k={\frac {2\pi }{\lambda }}\ }$

The relationship between angular wavenumber and angular frequency is analogous to that between wavelength and ordinary frequency—they are related by the velocity V:

${\displaystyle k={\frac {\omega }{V}}\ }$

It's unfortunate that there are two definitions of wavenumber. Some people reserve the term spatial frequency for the ordinary wavenumber, or use ν (that's a Greek nu, not a vee — another potential source of confusion!), or even σ for it. But just as many call it the wavenumber and use k, so the only sure way through the jargon is to specify what you mean by the terms you use. As usual!

Here are two spectra: the FFTs of an image of some waves, and a binary image of some particles: