Derivation Phase Velocity of a Wave
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:
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:
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:
Substitute the wavelength 2
and period 3
into formula 1
to express the phase velocity with angular wavenumber \(k\) and angular frequency \(\omega\):
Here the factor \(2\pi\) is cancelled and you get the formula: