What is the (Angular) Wavenumber?
Wavenumber \(\nu\) (pronounced "Nu") is defined as:
Here \( \lambda \) is the wavelength. So the wavenumber is simply the reciprocal of the wavelength. The SI unit of the wavenumber is: \( [\nu] ~=~ \frac{1}{\mathrm m} \).
You can read from the definition 1
of the wavenumber: The larger the wavelength \( \lambda \) of the function is, the smaller the wavenumber!
The wavenumber \( \nu \), is a spatial frequency and is similar to the temporal frequency \( f \), which we can read from, for example, the unit: While the temporal frequency has the SI unit \( \frac{1}{\mathrm s} \), the wavenumber has the unit \( \frac{1}{\mathrm m} \). For example, let's take a sine function:

Frequency states how many times the period \(T\) fits into one second.

Wavenumber states how many times the wavelength \(\lambda\) fits into one meter.
With the help of the relation \( v_{\text p} ~=~ \lambda \, f \) you can express the definition 1
of the wavenumber with the help of the frequency of the wave \( f \) and its propagation velocity (more exactly: phase velocity) \( v_{\text p} \):
As you can see from the equation 2
, the frequency \(f\) and the wavenumber \(\nu\) are related by the phase velocity \(v_{\text p}\) of the wave.

The faster the wave propagates, the smaller the wave number and the larger the frequency.

The slower the wave moves, the larger the wavenumber and the smaller the frequency.

The larger the frequency \( f \) of the wave, the larger the wavenumber \(\nu\).

The smaller the propagation speed \( v_{\text p} \) of the wave, the larger the wavenumber \(\nu\).
Angular wave number
Die Kreiswellenzahl \(k\) unterscheidet sich von der Definition 1
der Wellenzahl \(\nu\) durch den Faktor \(2\pi\):
Note that in the literature, especially in solid state physics, the angular wavenumber is simply called the wavenumber. Nevertheless, you should keep in mind that what is actually meant is the angular wavenumber with the factor \(2\pi\).
If you compare definition 1
of wavenumber and definition 3
of angular wavenumber, you will see that they are related as follows: