# Derivation Phase Velocity of a Wave

Level 2 (suitable for students)
Level 2 requires school mathematics. Suitable for pupils.
Updated by Alexander Fufaev on

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 wave number $$k$$.

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
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The 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
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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
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Substitute the wavelength 2 and period 3 into formula 1 to express the phase velocity with wavenumber $$k$$ and angular frequency $$\omega$$:

Phase velocity expressed with wavelength and wavenumber
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Here the factor $$2\pi$$ is cancelled and you get the formula:

Phase velocity expressed with frequency and wave number
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