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Derivation Phase Velocity of a Wave

Phase velocity of a wave crest
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 \(k\).

Illustration : Phase velocity of a point A of the wave.

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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