My name is Alexander FufaeV and here I write about:

What is the Phase Velocity of a Wave?

Important Formula

Formula: Wave in Vacuum

$$\class{blue}{v_{\text p}} ~=~ \class{red}{\lambda} \, f$$ $$\class{blue}{v_{\text p}} ~=~ \class{red}{\lambda} \, f$$ $$f ~=~ \frac{ \class{blue}{v_{\text p}} }{ \class{red}{\lambda} }$$ $$\class{red}{\lambda} ~=~ \frac{ \class{blue}{v_{\text p}} }{ f }$$

Rearrange formula

What do the formula symbols mean?

Phase velocity

$$ \class{blue}{v_{\text p}} $$ Unit $$ \frac{\mathrm m}{\mathrm s} $$

Phase velocity is the speed of propagation of the same phase of a (monochromatic) wave.

In the case of electromagnetic waves, the phase velocity is the speed of light: $ c = \lambda \, f $, with the value $ c = 299 \, 792 \, 458 \, \frac{\mathrm m}{\mathrm s} $.

Frequency

$$ f $$ Unit $$ \mathrm{Hz} = \frac{ 1 }{ \mathrm{s} } $$

Frequency of the wave. Indicates how often the wave (e.g. light wave) oscillates per second.

Wavelength

$$ \lambda $$ Unit $$ \mathrm{m} $$

Wavelength is the smallest distance between two points of the same phase. For example, the distance between two wave crests.

The phase velocity $v_{\text p}$ describes how fast the phase $ (k\,x ~-~ \omega\,t ~-~ \varphi) $ of the wave (with wavelength $ \lambda $) propagates.

Phase velocity of a wave crest

To determine the phase velocity, you can look at a wave crest, for example. As soon as it has traveled the distance $ \lambda $, exactly the period duration $ T $ has passed. (Because a period duration is exactly the time at which the phase has passed through an angle of $ 2\pi $, which corresponds to a wavelength). Velocity is defined as distance per time, that is, in the case of phase velocity:

$$ \begin{align} \class{blue}{ v_{\text p} } ~=~ \frac{ \class{red}{ \lambda } }{ T } \end{align} $$

With $ T ~=~ \frac{1}{f} $ for the phase velocity you can rewrite the above equation:

$$ \begin{align} \class{blue}{v_{\text p}} ~=~ \class{red}{\lambda} \, f \end{align} $$