Why does v = fλ?

The universal wave equation v = fλ is one of the fundamental equations in Physics. It relates the speed of a wave, v, the frequency, f, and the wavelength λ.

Waves are periodic disturbances that propagate energy through a medium. They are caused by an oscillating source that creates the original disturbance. The frequency of the resulting wave is equal to the frequency of the source. The period $T$ of the oscillation is the inverse of the frequency $f$. One period is the time that it takes for the source to complete a full cycle and return to its original position. The particles in the medium propagate the wave through individual oscillations with this same period.

The wavelength $\lambda$ of a wave is defined as the distance between two successive points on a wave that are oscillating in phase. An example of points on a transverse wave that are in phase with each other are crests - at a crest the particles are at the top of their oscillatory motion. Similarly, all points on troughs are in phase with each other. So the distance between adjacent crests or troughs is one wavelength. For a longitudinal wave, a wavelength is the distance between the centers of two adjacent compressions or rarefactions.

Now we can think about how far a wave travels in the time that it takes the source disturbance to repeat itself, or in other words, complete one cycle. The time for one cycle is the period $T$. Waves will travel away from the source at a speed $v$. This speed depends on the nature of the medium and is independent of the frequency of the source. If we consider a wave that travels away from the oscillating source, we can use the speed equation $v = \dfrac{s}{t}$ to determine that the distance $s$ that the wave travels in one period $T$ is given by

$\hspace{3em} s = vT$

This distance is the separation between two repeating points on the waveform, and so is equal to the wavelength $\lambda$

$\hspace{3em} λ = vT$

Solving for the speed gives

$\hspace{3em} v = \dfrac{\lambda}{T}$

And we know that the period is the inverse of the frequency

$\hspace{3em}T = \dfrac{1}{f}$

Substituting in, we arrive at the wave equation

$\hspace{3em} v = f \lambda$