Where v = fλ comes from

7 minAS Level7.1.47.1.5

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By the end of this lesson

By the end you can derive $v = f\lambda$ from the definitions of speed, frequency and wavelength, and use it to find any one of $v$ , $f$ or $\lambda$ given the other two.

One wavelength per period

Watch a single crest. To move from one crest position to the next, the wave must travel exactly one wavelength, $\lambda$ .

The time it takes to do that is exactly one period, $T$ , because the pattern repeats every period.

So in one period, the wave covers a distance $\lambda$ . That is all the derivation needs.

The derivation in three lines

Start from the definition of speed:

$v = \frac{\text{distance travelled}}{\text{time taken}}$

A crest travels one wavelength $\lambda$ in one period $T$ , so

$v = \frac{\lambda}{T}$

Frequency is the reciprocal of period, $f = 1/T$ , so replacing $1/T$ by $f$ :

$v = f\lambda$

That is the wave equation. It is not a new law — it is the definition of speed applied to one cycle of the wave.

Definition

The wave equation

For a progressive wave,

$v = f\lambda$

where $v$ is the wave speed in $\text{m s}^{-1}$ , $f$ is the frequency in Hz, and $\lambda$ is the wavelength in metres.

It can be rearranged:

$f = \frac{v}{\lambda} \qquad \lambda = \frac{v}{f}$

The equation works for any progressive wave: water, sound, light, seismic.

Worked example: Worked example: speed from frequency and wavelength

A water wave has frequency $50\ \text{Hz}$ and wavelength $0.20\ \text{m}$ .

Calculate its speed.

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