# Derivation Phase Velocity of a Wave

**Level 2**requires school mathematics. Suitable for pupils.

The **phase velocity** \( v_{\text p} \) of a wave is the velocity with which a point of the wave moves. Here we want to express the phase velocity using the **angular frequency** \(\omega\) and the **angular wavenumber** \(k\).

So consider any point on the wave, for example the top of a wave crest (point A in illustration 1). We want to find out how fast this point moves from A to B.

The velocity is distance between A and B per time. This distance corresponds by definition to the **wavelength** \(\lambda\). And the time after which the point \(A\) arrives at \(B\) is by definition the **period** \(T\). Thus the phase velocity is:

**Phase velocity as wavelength per period**

The angular wavenumber \(k\) is the angle travelled per length. Within one wavelength \(\lambda\) the angle \(2\pi\) is travelled: \( k = \frac{2\pi}{\lambda} \). Rearranged for the wavelength yields:

**Wavelength is 2Pi divided by wavenumber**

The **angle frequency** \(\omega \) is the angle traveled per time. Within a period \(T\) the angle \(2\pi\) is covered: \(\omega = \frac{2\pi}{T} \). Rearranged for the period we get:

**Period is equal to 2Pi divided by angular frequency**

Substitute the wavelength 2

and period 3

into formula 1

to express the phase velocity with angular wavenumber \(k\) and angular frequency \(\omega\):

**Phase velocity expressed with wavelength and wavenumber**

Here the factor \(2\pi\) is cancelled and you get the formula:

**Phase velocity expressed with frequency and angular wavenumber**