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What is the (Angular) Wavenumber?

Important Formula

Formula: Relationship Between Wave Number and Angular Wave Number

$$k ~=~ 2\pi \, \nu$$ $$k ~=~ 2\pi \, \nu$$ $$\nu ~=~ \frac{k}{2\pi}$$

Rearrange formula

What do the formula symbols mean?

Angular wavenumber

$$ k $$ Unit $$ \frac{\mathrm{rad}}{\mathrm{m}} $$

Angular wave number indicates the covered phase angle of a wave per meter.

Wavenumber

$$ \nu $$ Unit $$ \frac{1}{\mathrm{m}} $$

Wavenumber indicates how many times the wavelength of a periodic wave fits into one meter.

Wavenumber $\nu$ (pronounced "Nu") is defined as:

$$ \begin{align} \nu ~=~ \frac{1}{\lambda} \end{align} $$

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} $.

Wavelength as the distance between two wave crests.

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.

Frequency of a sinusoidal wave here is $f = 3.5 / \mathrm{s}$ (Hz).
Wavenumber states how many times the wavelength $\lambda$ fits into one meter.

Wavenumber of a sinusoidal wave here is $\nu = 2.5 / \mathrm{m}$.

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} $:

$$ \begin{align} \nu ~=~ \frac{f}{v_{\text p}} \end{align} $$

Phase velocity indicates how fast a wave crest propagates from A to B.

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 $2\pi$:

$$ \begin{align} k ~=~ \frac{2\pi}{\lambda} \end{align} $$

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:

$$ \begin{align} k ~=~ 2\pi \, \nu \end{align} $$